Numerical Procedures for Analyzing Dynamical Processes

17AD-A248 175i iI-fli I i~Ih NUMERICAL PROCEDURES FOR ANALYZING DYNAMICAL PROCESSES fTIC FINAL REPORT Project Period: October 1, 1990 - February 29, 1992 Celso Grebogi Edward Ott James A. Yorke University of Maryland College Park, MD 20742-3511 92-07966 .q2 q 20 04 7 111111111111I11I11 Prepared for the Office of Naval Research (DARPA) under grant number N00014-88-K-0657 DISC LAIMR NOTICE \s \ THIS DOCUMENT IS BEST QUALITY AVAILABLE. THE COPY FURNISHED TO DTIC CONTAINED A SIGNIFICANT NUMBER OF PAGES WHICH DO REPRODUCE LEGIBLY. NOT OFFICE OF NAVAL RESEARCH PUBLICATIONS / PATENTS / PRESENTATIONS / HONORS FOR 1 OCTOBER 1990 THROUGH 29 FEBRUARY 1992 CONTRACT: N00014-88-K-0657 R&T NO.: b41u001- -- 04 TITLE OF CONTRACT: Numerical Procedures for Analyzing Dynamical Processes NAME(S) OF PRINCIPAL INVESTIGATOR(S): Celso Grebugi Edward Ott James A. Yorke NAME OF ORGANIZATION: University of Maryland ADDRESS OF ORGANIZATION: Laboratory for Plasma Research College Park, Maryland 20742-3511 Reproduction in whole, or in part, is permitted for any purpose of the United States Government. This document has been approved for public release and sale; its distribution is unlimited. 2 INTRODUCT-ON The following report summarizes our activities under the Office of Naval Research (DARPA) Contract No. N0014-88-K-0657. We have organized this report under the following five categories: I. Deliverables: computer tape and disk with instructions, and summary of accomplishments related to the proposed projects. II. List of publications for the period of this report. III. Appended respective reprints and preprints. N I ," .. . .. A 3 qUo I. Deliverables: Computer Tape and Disk with Instructions arid 3ummary of Accomplishments Related to the Proposed Projects PROJECT I Deliverable: \XVQ are ddivering a tape with a software package for UNIX workstations with documentat'on for analyzing low dimensional dynamical behavior from time series. In particur.z the Lyapunov exponent code will, together with the di- niension code. permit the user to distinguish between periodic, chaotic, and random processes, "Random processes" here means behavior whose dimension is too high to compute. The code computes the information dimension of the time series. We are also including in the same tape a noise-redaction code with documentation. Summary of Project 1: Nonlinear Noise Filteringof Experimental Datafrom Chaotic Processes Many attempts have been made to apply ideas from dynamical sytems to the analysis of experimental data including estimates of attractor dimension and measurement of Lyapunov exponents. Ai, essential problem is that noise often complicates the analysis. For example, noise obscures the fractal structure of the attractor, so that estimates of the attractor dimension cmn be difficult to obtain. Various methods have been proposed to estimate the noise levels in the data, and these are useful for determining the smallest scales at which dimension measurements are feasible. However, up until now no systematic method has been developed for noise reduction. We have developed a method which we believe is a potential breakthrough in the analysis of experimental data. Typically, attractors are reconstructed from a scalar time series of experimental data using time delays. Conventional signal filtering techniques are not useful in this case, because they exanine only portions of the signal which are close in time. We examine points on an attractor which are close in 4 phase space; the corresponding parts of the original signal in general are far apart intime. Our method is a linearization technique which uses the dynamics of the reconstructed attractor to estinate and correct errors in the trajectories. The method relies on the assumption that in a small neighborhood about a point on the trajectory, the dynamics on the attractor is nearly linear. in other words, given a point xi on the attractor, its image is xi+1 = f(xi) for some nonlinear, unlivown function f. We assur- that it is possible to find a matrix A and a vector b such that x,+, - Axi + b. The mehod has two steps: first, to compute the matriecs A and vectors b for each point on the trajectory, and second, to find a new trajectory near the original one which best satisfies the linear approximation. We believe that a reliable procedure like the one outlined above will be invaluable for the analysis of experimental data. 5 PROJECT 2 Deliverable: We are delivering a disk containing a Dynamics code (for IBM com- I patible PCs and for UNIX/X window workstations) with a Manual for computing and evaluating dynamical processes. The source code contains 20,000 lines of code. I In particular, the program will compute stable and unstable manifolds as described below. Summary of Project 2: A fast Reliable Method for the Numerical Computation of Stable Manifolds of Chaotic P? ocesses 3 Saddle points often play a crucial role in the dynamics of a particular map f. A schematic illustration of a saddle p,:int in two dimensions is given in the following 3 figure: __- --_ S I |I Because p is an unstable fixed point, any point p' eventually moves away from I p as f is iterated, even though f(p) = p. For example, in Fig. 1, initial conditions slightly to the right of the curve labeled S move toward p for a few iterates, then are repelled to the right thereafter, eventually approaching the curve U. Initial conditions slightly to the left of S wili move close to p, then off to the left. The cirve S is the stable manifold of p: it is the set of initial conditions which are attracted to p. The cure U is the unstable manifold. If f is invertible then U is the stable manifold of p for the inverse map f-1. More generally, U is the set of points whose preimages tend to p. * 6 I I In many cases, the stable and unstable manifolds wind around in complicated ways. Because the manifolds arc intertwined so closely, initial conditions can approach and be repelled from the saddle point repeatedly, leading to complex behavior. Stable manifolds of fixed points often form part of the boundary between two basins of attraction. In this case, the structure of the stable manifold determines how sensitive the system is to small errors in measuring an initial condition. In addition, it is often important to know whether the stable and unstable manifolds cross at a point other than the saddle point p. Such homoclinic intersections are often of interest, especially in cases where the map depends on a parameter. Hence, I 5 a knowledge of the structure of the stable and unstable manifolds is essential to understanding the dynamics. We have developed efficient, reliable numerical methods to calculbie them. ! I I I I I I II I7 II II. LIST OF PUBLICATIONS FOR * 1 THE PERIOD OF THIS REPORT 1. "Noise Reduction: Finding the Simplest Dynamical System Consistent with the Data," E. J. Kostelich and J. A. Yorke, Physica 41D, 183 (1990). 1 2. "Shadowing of Physical Trajectories in Chaotic Dynamics: Containment and Refinement," C. Grebogi, S. M. Hammel, J. A. Yorke, and T. Sauer, Phys. I Rev. Lett. 65, 1527 (1990). 3. Antimonotonicity: Concurrent Creation and Annihilation of Periodic Orbits," Bulletin AMS 23, 469 (1990). 4. "Chaotic Scattering in Several Dimensions," Q. Chen, M. Ding, and E. Ott, Phys. Lett. 145A, 93 (1990). 5. "Cross-sections of Chaotic Attractor-,," Q. Chen and E. Ott, Phys. Lett. 147A, 450 (1990). 6. "Rigorous Verification of Trajectories for the Computer Simulation of Dynam- I ical Systems," T. Sauer and J. A. Yorke, Nonlinearity 4, 961 (1991). 7. "Analysis of a Procedure for Finding Numerical Trajectories Close to Chaotic Saddle Hyperbolic Sets," H. E. Nusse and J. A. Yorke, Ergod. Th. & Dynam. Sys. 11, 189 (1991). 3 8. "Embedology," T. Sauer, J. A. Yorke, and M. Casdagli, J. Stat. Phys. 65, 579 (1991). 9. "A Numerical Procedure for Finding Accessible Trajectories of Basin Boundaries," H. E. Nusse and J. A. Yorke, Nonlinearity 4, 1183 (1991). 3 10. "Calculating Topological Entropies of Chaotic Dynamical Systems," Q. Chen, E. Ott, and L,Hurd, Phys. Lett. 156A, 48 (1991). 3 11. "On the Tendency Toward Ergodicity with Increasing Number of Degrees of Freedom in Hamiltonian," L. Hutd, C. Grebogi, and E. Ott, submitted for * publication. 8 12. "Metamorphoses: Sudden Jumps in Basin Boundaries," K. T. Alligood, L. Tedeschini-Lalli, and J. A. Yorke, submitted for publication. 13. "Accessible Saddles on Fractal Basin Boundaric ,," K. '. Alligood and J. A. Yorke, submitted for publication. 14. "The Analysis of Experimental Data Using Time-Delay Embedding Methods," E. J. Kostelich and J. A. Yorke, submitted for publication. 15. "When Cantor Sets Intersect Thickly," B. R. Hunt, I. Kan, and J. A. Yorke, submitted for publication. 16. "Border-Collision Bifurcations Including 'Period Two to Period Three' for Piecewise Smooth Systems," H. E. Nusse and J. A. Yorke, submitted for publication. I9 I I I I I I I 3 III. APPENDED RESt'ECTIVE REPRINTS AND PREPRINTS U U I I I I I I I I 10 Phvsica D A1(1990) 183-196 North-Holiand NOISE REDUCTION: FINDING THE SIMPLEST DYNAMICAL SYSTivl CONSISTENT WITH THE DATA Eric J. KOSTELICHab and James A. YORKE Institute for Phvsical Science and Technology, Uniersitv of Manland. College Park, MD 20742. LSA hCenter /or Nonlinear Dvnanics. Department of Phi-sics. Untverstiv ol Texas. A usttn. TX 78712. USA Department of Mathematics. Universitt of Maryland. Colleqe Park. MD 20742. USA Received 3 March 1989 Rev:scd manuscnpt received I October 1989 Accepted 18 October 1989 Communicated by R Westervelt I A novel method is desLnbed for noist. rcduttion II Laotii, experimental data %.hoednami.s arc low dimensional In ad lition. we show how the approaLh allow,. cpenmentalists to use n n of tle same tethniqucs that hae been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations. I. Introduction Numerical computation and computer graphics have been essential tools for investigating the behavior of nonlinear maps and differential equations. The pioneering work of Lorenz 1251 was made possible by numencal integration on a computer, allowing him to take nearby pairs of initial conditions ankd compare the trajectories. H16non 1191 discovered ,he complex dynamics of his celebrated quadratic map with the aid of a programmable calculator. A variety of classical and modern techniques has been exploited to find pertodic orbits, their stable and unstable manifolds [141, basins of attraction [26], fractal dimension [27], and Lyapunov exponents [10, 31. 37]. In some cases, numerical methods can establish rigorou ly the existence of initial conditions whose trajectories have essentially the same intricate structure that one sees on a computer screen [18]. 'Curient address Department of Mathematics, Anzona State University. Tmpe, AZ 85287, USA 0167-2789/90/$03 50 (North-Holland) Elsevier Science Publishers B V Until recently, experimentalists have not been able to apply most of these methods to the analysis of experimental data, since they do not in genetal have explicit equations to model the behavior of thei: apparatus. In ,ases where it is possible to lind accurate models of the ph)sical system, quantitative predictions about the behavtor of actual experiments are possible 117]. However, all that is available in a typicl experiment is the time-dependent output (e.g. voltage) from one or more probes, which is a function of the dynamics. One fundamental problem in the analysis of experimental data concerns the correspondence between the dynamics that goerns the behavior of the apparatus and the discretely sampled time series that Limprises the data. Another question is how to unimize the effect of noise. In this paper, we show how the time dely embedding method, row commonly used to reconstruct an attractor from experimental data, yields a novel procedure for reducing noise in data whose dynamcs can be characterized as low dimensional. Moreover, we I IS4 EL. Kworkh OcdJA. Yar.ug / "'damm i show how the approach can be extended to allow experimentalists access to many of the analytical measurementsL In the embedding method. one generates a set of mdimensional points ow tools mentioned above, Section 2 reviews the time delay embedding method and some of its applications. Section 3 introduces some of the problems associated with traditional filters and outlines our noise reduction method. coordinates are vralucs in the time series separated by a conswant delay [I . For example. wlen m =3.I the reconstructed attractor is the set of points I x,= ;.. s,.,s..j) where ir is the time delay. Takens [341 has shown that under suitable hvpoheses. hsprocedure yields a set whose properties are equivalent to those of the original attractor provided that the embedding dimension 2. The time delayv es merln As stated in section 1. one problem in analyzing experimental data is how to relate the measurements with the dynamics. Before the early 1980s. power spectra were the principal method for analyz:ng such data. For instance. Fenstermacher et al. 1131 relied heavily on power spectra to detect transitions from periodic to weakly turbulent flow between concentric rotating cylinders. However. Fourier analysis alone is inadequate for describing the dynamics. Other methods also have been used to analvze time series output from dynamical systems. Lorenz 1251 used next amplitude maps to describe some features of the dynamics; that is. he plotted -,-,. against :,, where z, is the nth relative maximum of the third coordinate of the numerically calculated solution. Such maps are often useful. not only for investigating features of the Lorenz at- called crnamic in that information about the dynamics is stored in the compu;er for analysis. With each data vector x,, one stores the "next'" method has come into common use as a way of cal system underlying that data this approach, one supposes thaerientaldata.In they i bhaio goned b as linear approximation provides an estimate of the Jacobian of the map at x, 1111. Eckmann et al. [101 use linear maps computed in this way to integrate solution traveling along an attractor" t (which is not observable directly). However, one assumes there is a smooth function that maps points on the attractor to real numbers (the experimental ='Existing numerical methods requirc the attractor to be low dimensional I I In principle, the embedding method allows one to study the dvnamics i detail The earliest applications may be called staic in that the analsis focuses on the geome:ric properties of the set of points on the reconstructed attractor, For exampie. phase portraits and Poincari sections are used in ref. 151 to help determine the transition between quasiperiodic and chaotic flow in a CouetteTaylor experiment. Another important application is the estimation of attractor dimension from experimental data. for which tte:e i%a largx literature [271. In addition, various .nfornmtion theoretic notions can be used to find good choices of embedding dimension and time delay 1151. More recent applications of the embedding method are quite different in nature and can be vector, for example. x, s for some 8 > 0. This makes it possible to compute a linear approxima- that the dynam ica l behavio r is govern ed by a I i is large enough. tractor 1321, but also for instance in experiments on intermittency in oscillating chemical reactions 1301. In the past decade, the time delay embedding reconstructingdata. an attractor from a time series of experimental In l tion of the dynamics in a neighborhood of x,. assuming that there is a low-dimensional dynami- a s t o a i t o a q ai n '2 . In particular. a n i d t ep s t v a set of variational equations and find the positive w macrial wasirst presentd by D Rucik at a Nobel smosiin 19D Wolf ct al. [371 have proposed a diffcrcnt me'hod in which nearb' pairs of points arc followed to cstimatc the largest Lyapunov cxponent I I I Ei. Kti*h :3 diJA. Yare/.Vese mAwim In fact. the time delay embedding method pro- new time series with some of the high-frequency %idesa powerful set of tools for anahag the components removed. This isthe basic idea be- dynamics the breadth of which may not have hind Wee and other bandpass filters 1291 been realized by Eckmann and Rudle. In the remainder of this paper. we dicus two novel applications that are possiW specifically: However. as noted previously, power spectral analyss insufficient to characte.re the dynamics when the data are chaotic. Since the power at spectrum of a low-dimensional chaotic signal resembles that of a noisy one, the suppression of certain frequencies can alter the dynamics of the (1)Noise edtm.Sim one can ap the dynamics at each point it becomes ossible identify and correct inaccuracies in trajectories arising from random errors in the oinal series. Numerical evidence rigna th ie reduction procedure described below improves the yof other analyses such as Lyapnov accuracpofone an lyse s. ucats , exponents and dimension calculaaons. (2) Simplicial approximaions. Linear approximations can be computed at each point on a grid in a neighborb~od of the attractor 'o form a in aneihbor~odof te atracor o fom a simplicial approximation of the dynamical system. This can be used to locate unstable periodic orbits We consider noise reduction in section 3. 3. Noise reduction The ability to extract information from timevarying signals is limited by the presence of noise, Recent experiments to study the transition to turbulence in systems far from equilibrium, like those by Fenstermacher et al. [131, Behringer and Ahlers [21, and Libchaber et al. [241, succeeded largely because of instrumentation that enabled them to quantify and reduce the noise. However, it is often expensive and time consuming to redesign experimental apparatus to improve the signal to noise ratio. An important question, therefore, is how the experimental data can be filtered or otherwise preprocessed before it is analyzed further. One common approach is to use Fourier analysis: one might model the noise as a collection of highfrequency components and subtract them from a power spectrum (or Fourier transform) of the input data. The transform can be inverted to yield a I It5 e have shown that a simple low-pass filter effectively introduces an extra Lyapunov exponent that depends on the cutoff frequency. If the cutoff frequency is sufliciently low. then the filter can increase the fractal dimension of the reconstructed attractor. This resut also has been confirmed by Mitschke et al. suit also an coni c cyrit . 1281 with data from an electronic circuit. We now consider a different approach and show Wow cosidera differ et can so exploited to reduce the noise, at least in cases where the time series can be viewed as a dynamical system with a low-dimensional attractor. Our objective is to use the dynamics to detect and correct errors in trajectories that result from noise. This is done in two steps once an embedding dimension m and a time delay T have been fixed. In the first step. we consider the motion of an ensemble of points in a small neighborhood of each point on the attractor in order to compute a linear approximation of the dynamics there. In the second step, we use these approximations to consider how well an individual trajectory obeys them. That is, we ask how the observed trajectory can be perturbed slightly to yield a new trajectory that satisfies the linear maps better. The trajectory adjustment is done in such a way that a new time series is output whose dynamics are more consistent with those on the phase space attractor, This approach is fundamentally different from traditional noise reduction methods. Because we consider the motion of points on a phase space attractor, we are using information in the original signal that is not localized in a time or frequency domain. Points that are close in phase space correspond to data that in general are widely and F. 136 EJ. Kadc a .,d.A. 'YVoii w irr I oomu irregularly spaced in tme .due to de se ti ve-dependeuc on initial conditions on chaotic at- :: tractors. In contrast. Kalman 14 and similar filters examine data that are closely spaced in time: bandpass filters operate in the frequency domain.I Fig. 1. Schematic diagram for the first stage of the noise reduction method. A collection of points in an c-ball about the reference point xf is used to find a linear approximation of 4. Edmann-Ruele linearkiabo the dynamcs there. The discrete sampling of the original signal means that the points on the reconstructed attractor can be treated as iterates of a nonlinear map f whose exact form is unknown. We assume that f is nearly linear in a small neighborhood of each attractor point x and write We mention three difficulties in computing the local linear approximations in the subsections below. f(x) = Ax + b =- L(.) for some m x m matrix A and m-vector b. (The matrix A is the Jacobian of f at x.) This approximation, which we call the Eckmann-Ruelle linearization at x. can be computed with least-squares methods similar to those described in refs. [11, 101. Given a reference point Xrf, let (x, ),"- be a collection of the n points which are closest to xre1. With each point x, we store =4 the next point (i.e., the image of x,), denoted y, , The kth row a k of A and the kth component bk of b are given by the least-squares solution of the equation yA = bk + a k x, (1) where Yk is the k th component of y and the dot denotes the dot product. Fig. I illustrates the idea'. "The points x, are points on the attractor which are not consecutive in time. The subscnpt i merely enumerates all the points on the attractor contained witun a small distance (of Xrer. In this notation. x, and y, are consecutive in time. *SFarmer and Sidorowich [12] observe that the Eckmann- Ruelle bineanzation can be used for prediction Given a reference point x,, find the Eckmann-Ruelle hneanzation A,x + b,, compute x,, I = A,x, + b,, and repeat the process to get the predicted trajectory I I 4.1. 111 conditioned least squares There is a particular problem when one tries to compute solutions to eq. (1) with a finite data set of limited accuracy that has not been addressed in previous papers [10, 311. Suppose for example that all the points in a neighborhood of x,,f lie nearly along a single line. i.e., the attractor appears one dimensional within the available resolution. Although it is possible to measure the expansion along the unstable manifold at xrf there are not enough points in other directions to measure the contraction. Hence it is not possible to compute a 2 x 2 Jacobian matrix accurately. Any attempt to do so will result in an estimate of the Jacobian whose elements have large relative errors. This kind of least-squares problem is il conditioned. The ill conditioning can be avoided by changing coordinates so that the first vector in the new basis points in the unstable direction 6 . A one-dimensional approximation of the dynamics is computed using the new coordinates; that is, we approximate the dynamics only along the unstable manifold. We recover the matrix A by changing coordinates back to the original basis. For example, if we are working in the plane and the unstable direction is the line y = x, then we rotate the coordinate axes by 45', The dynamics are approximated by a one-dimensional linear map 6 This is done by computing the right singular vectors (9)of the n x m matrix whose jth row is x, The procedure is called principal component analysis in the statistical hterature. I i I E.. Kostehch and J.A. Yorke/ Noise reducion computed along the line Y = x. Then we rotate back to the original coordinates. (The resulting matrix A has rank 1 in this example.) This approach substantially enhances the robustness of the numerical procedure. 4.2. Finding nearest neighbors A second problem is finding an efficient way to locate all of the points closest to a given reference point. The dynamical embedding method imposes stringent requirements on any nearest-neighbor strigen onanyalgorithm reuireent algorithm. The storage overhead for the corre- I ~ 187 in the same box number. The search is extended if necessary to adjacent boxes. Only a crude partition is needed for this algorithm to work efficiently (typically we choose B = 40), and the grid4 is extended only to the first three coordinate axes. *"hen the embedding dimension is larger than three, a preliminary list of nearest neighbors is obtained using only the first three coordinates of each attractor point. The final list is extracte by computi each point in the preliunary list. Although there are circumstances where this can perform poorly (e.g.. when most of th can p o r e oo r ted ina ostndf sponding datather structures must be ttrctorpoits, small, because there are te ns ofthouand of ar th oten usands of of attracto r points. the boxes), attractorthepoints are concentrated of distribution of points ona ihandful typical The algorithm must be fast, since there is one nearest-neighbor problem for each linear map to be computed, We solve this problem by partitioning the phase space into a grid of boxes that is parallel to the coordinate axes. Each coordinate axis is divided into B intervals. (Fig. 2 illustrates the grid in two dimensions.) Each point on the attractor is asdimesios.) achpoit ontheattacto isassigned a box number according to its cotrdinates. For example, a point on the plane whose first coordinate falls in the jth interval (counting from 0) along the x axis and whose second coordinate falls in the k th interval along the y axis is assigned to box signd number kB +j. The list of box t boxnumer B along +j Th lis otbox numbers is sorted, carrying a pointer to the attractors is sufficiently uniform that the running time is very fast. Memory use is also efficient: a set of N attracto. points requires 3N storage locations. In contrast, the tree-search algorithm advocated in ref. [12] requires several times more storage (although the lookup time is probably slightly less). Because N = 10 in typical applications, we believe that the box-grid approach (or some variant) is the most practical. A survey of ome varaniheos al . A svey of other nearest-neighbor algorithms is given in ref. [3]. original data point. Given a reference point xrf, its box number is found using the above formula. A binary search in the list of box numbers then locates the address of xmf and all the other points ordinary least squares to compute the linear maps. In the usual statitical problem of fitting a straight line, one has observations (x,, I where x, is is measured. One assumes known exactly and y', that y, = ao + a~x, + c,, w'ere the (, are independent errors drawn from the same normal distribution. (Analogous assumptions hold in the BI- B _B +21BB B-_B+2 _B+ _ 1- _1 J+ SBIever, 2B+-1 0 1 __. B. -1 B- Fig. 2. Box numbenng scheme in two dimensions. The attractoris normalized to fit in the unit square. The bottom row of boxes rests against the x axis and the leftmost row of boxes against the Y axis. t r c o s i suf i e ly nf r m ha t e r n i g There potential difficulty Tria is a etial dicuty inthethe use se of multivariate case.) In the present situation, howboth x, and v,are measured with error. It can be shown that the ordinary least-squares method produces biased estimates of the parameters a0 and a, in this case [16, 23]. In practice this does not seem to be a serious problem, but statistical procedures to handle this situation (the socalled "errors in variables" methods) may provide 188 EJ. Kostehcl and J.A Yorke/ Noise reduction -The trajectory adjustment can be iterated. That is, once a new trajectory .1, has been found, one can replace each x, in eq. (2) by .,and compute a Fig. 3. Schematic diagram of the trajectory adjustment procedure. The trajectory defined by the Fquence (x, ) is perturbed to a new trajectory given by { i, ) which ismore consistent with the dynamics. In this example we show what the perturbed trajectory might look like J the dynanucs were approximately honzontal translation to the nght. an alternative approach to noise reduction. We consider this question in the appendix. 5. Trajectory adjustment by rnnimizing new sequence { .}. We place an upper limit on the distance a point can move. Points which seem to require especially large adjustments can be flagged and output unchanged. (This may be necessary if the input time series contains large "glitches" or if nonhnearities are significant over small distances in certain regions of the attractor.) When the input is a time series, we modify the above procedure slightly since we require a time series as output. The trajectory adjustment is done The Eckmann-Ruelle linearization procediire described above is computed and the resulting maps are stored for a sequence of reference points along a given trajectory (for the results quoted here, the sequence usually contains 24 points). We now consider how to perturb this trajectory so that it is more consistent with the dynamics. The objc:tive is to choose a new sequence of points 1, to minimize the sum of squares so that changes to the coordinates of x, (corresponding to particular time series values) are made consistently for all subsequent points whose coordinates are the same time series values. For example, suppose the time delay is I and the embedding dimension is 2. Then trajectories are perturbed so that the second coordinate of the ith point is the same as the first coordinate of the (i + l)st point. That is, when x, = (s, s,+1) is moved to the point 1, = ( ,, ,+ 1), we require that the first coordinate of 1 be , EW1l-,- X'11" 6. Results using experimental data self-inconsistency (2) We note that the attractor need not be chaotic where L(x,) =A~x, + b, w is a weighting factor, and the sum runs over all the points along the trajectory 17 . Eq. (2) can be slved using least squares, Heuristically, eq. (2) measures the selfinconsistency of the data, assuming that the linear approximations of the dynamics are accurate. See .3.We say the new sequence (x,) is more for this noise reduction procedure to be effective. Fig. 4a shows a phase portrait of noisy measurements of wavy vortex flow in a Couette-Taylor experiment (20]. This flow is periodic, so the attractor is a limit cycle (widened into a band because of the noise) and the power spectrum consists of one fundamental frequency and its 11 L, 1 +Ii, +I L, (I)11, self-consistent *7 #7In the results descnbed in this paper, the Eckmann-Ruelle points l;:.earization procedure is done using a collection of within a radius of 1-6% of each :,erence point, depending on the embedding dimension, the dimension of the attractor, and the number of attractor points. This results in collections of 50-200 pcnts per ball, which gives reasonably accurate map approxima,tions without making the computer program too slow. The weighting factor w is set to 1. harmonics above a noise floor. See fig. 4b. Figs. 4c, 4d show the same data after noise reduction. makes the limit The noise reduction procedure cycle much narrower, and the noise floor in the power spectrum is reduced by almost two orders of magnitude. However, no power is subtracted from any of the fundamental frequencies, and in fact some harmonics are revealed which previously were obscured by the noise. 189 E.J. Kostelich and J.A. Yorke/ Noise reduction ()3 () 2 I u 0 -2 -3 II 1e 3() 2 II -3. ! 0 1 .4 0)3 0.4 1 -2 * -3 0 I ! 0. 0. 0)5 Fig 4 Phase portraits and power spectra for measurements of wavy vortex flow in a Couette-Taylor experiment. (a). (b Phase portrait and power spectrum before noise reduction is apphed (c). (d) after noise reduction: (e), (f) after a low.pass hilter is applied to the onginal data The vertical axis in (b), (d) and (f) is the base-lO logarithm of the power spectral density, the honzontal axis is in multiples of the Nyqist frequency. These results are significantly different from those obtained by low-pass filtering. Figs. 4e, 4f show the phase portrait and power spectrum when the original data are passed through a 12th-order Butterworth filter with a cutoff frequency of 0.35. The dynamical noise reduction procedure is more effective than low-pass filtering since the noise appears to have a broad spectrum. However, the dynamical noise reduction method appears to subtract power from a mode whose fundamental frequency is approximately 0.3 times the Nyquist frequency. We do not know exactly 190 EJ. Kostelich and J.A. Yorke/ Noise reducton (bl I (a) 0 -2 .3 -4 (c)() -'4 0 8 16 24 12 404 Fig 5 Phase portraits and power spectra for measurements of weakly chaotic flow in a Couette-Taylor experiment (a). (b)Phase (d) after noise reduction The units for the power ,pectrum plot, portrait and power ,pectrum before noise reduction is applied. (c), are the same as those in ref. (51 why this occurs. However, this peak corresponds to the iotation frequency of the inner cylinder and may result from a defect in the Couette-Taylor apparatus [331. We do not consider this to be a serious problem. because the power associated with this mode is several orders of magnitude smaller than that of the wavy vortex flow. We emphasize that our objective is to find a simple dynamical system that is consistent with the data. It is possible for this method to elimnnate certain dynamical behavior from an attractor if those dynamics have very small amplitude, as fig. 4f shows. This situation is most likely to arise when there are not enough data to distinguish such dynamics from random noise. In the present example, the noise reduction procedure reveals the limit cycle behavior quite well". The results obtained by applying the method to chaotic data from the Couette-Taylor fluid flow experiment described in ref. [5] are shown in fig. 5. Fig. 5a shows a two-dimensional phase portrait of the raw time series at a Reynolds number R/Rc = 12.9, which corresponds to weakly chaotic flow [5]. The corresponding phase portrait from the filtered time series is shown in fig. 5b. Figs. 5c, 5d show 5 sWe have not attempted to find the smallest amplitude at which the noise reduction procedure can disunguish quasipenodic from penodic flow In general this will depend on the amount of data. the sampling rate. the embedding dimension, and other factors. I E.J. Kostelich andJ.A. Yorke/Noise reduction the power spectra for the coresponding time series" 9 . It is difficult to estimate how much aoise is removed from the data in this example on the basis of power spectra. One problem is that the transition from quasiperiodic to weakly chaotic fluid flow is marked by a sudden rise in the noise floor in the power spectrum (cf. fig. 3 in ref. [5]). Hence one carnot determine how much of the noise floor is c.ue to deterministic chaos and how much results from broad-band noise. The noise reduction procedure described here has the effect of reducing the power in the high-frequency components of the signal. One question therefore is whether reducing the high-frequency noise corre- sponds to discovering the true dynamics which have been masked by noise. We believe that the answer is yes, based on those cises where there is an underlying low-dimensional dynamical system. However, in chaotic processes some high-frequency components remain, because they are appropriate 191 stored, and a time series is generated by adding a uniformly distributed random number to each iterate. This simulates a time series with measurement noise, i.e., a time series where noise results from errors in measuring the signal, not from perturbations of the dynamics. We measure the improvement in the signal after processing by considering the pointwise error e,= x,,+i -f(x,, x,)II, i.e., the distance between the observed image and the predicted one. Let the mean error be 1/2 E t' =-k , the rmis value of the pointwise error over all A' points on the attractor. We define the noise reduction as R = I - Ef.,ied/Enomv, to the dynamics. We use eq. (3) to generate a time series as follows (with the standard parameter values a = 1.4, /f = 0.3). We choose an initial condition and discard where the mean errors are computed for the adjusted and original noisy time series, respectively. The quantity R is a measure of the self-consistency of the time series. (In other words. R measures how much better on the average the output attractor obeys eq. (3) as one hops from point to point.) When 1% noise is added to the input as described above, the noise reduction (measured with the actual map) is 79%Io . Nearly identical res,,ts are obtained when the input contains only 0.1% noise. In addition, noise levels can be reduced almost as much in cases where the noise is added to the dynamics. i.e., where the input is of the form {x,.: x,. =f(x, + 71,,X,_ + 7). , q,-I random). When the program is run on noiseless input, the mean error in the output is 0.025% the first 100 iterates. The next 32768 iterates are of the attractor extent, which suggests that errors *9 The time senes consists of 32 768 values, from which an attractor is reconstructed in four dimensions Linear maps are computed using 50-100 points in each ball Trajectones are Ut°The pointwise error is measured using eq (3) However. the attractor can be embedded in more than two dimensions fitted using sequences of 24 points, when performing the noise reduction 7. Numerical experiments on noise reduction One important question is how much noise this method removes from the data. The power spectra above suggest that the method eliminates most of the noise, but it is impossible to give a precise estimate for typical chaotic experimental data. However, the H~non map [191 provides a convenient way to quantify the noise reduction, because it can be written as a tinie delay map of the form x,+ I =(x,,x, 1) = 1 - ax" +1x,_ . (3) 1EJ. Kosteich and J.A 192 arising from small nonlinearities are negligible when the input contains enough points. 8. Simplicial approximations of dyaamical systems Recent work has shown that simplicial approximations of dynamical systems can reproduce the behavior of the original system to high accuracy 1361. (See also ref. [351 for a bilineat approach.) In particular. the fractal structure of the original attractors and basin boundaries is preserved over many scales. Such approximations can yield significant computational savings, especially when the original system consists of ordinary differential equations. This approach can be extended in a natural way to generate simplicial app .)ximations of the dynamics on attractors reconstructed from experimental data. Our objective here is to find an approximate dynamical system in a neighborhood of the attractor as follows, A simplex in an m-dimensional space is a triangle with #n + I vertices. Suppose the map is known at each point on a grid. Then there is a unique way to extend the map linearly to the interior of the simplex S whose vertices are grid points. Given a point P ia the interior of S. let { b, )'. be its corresponding barvcentric coordinates (see ref. [36] for an algorithm to compute them). Let 1(vo) be the mar at the ith vertex. The dynamical system at P i. rated by compi,in.g ,_. b,f(v. (4) -0 We apply this method to experimenal data by fin ling a linear approximation of the dynarrac , at each vertex v, with the least-squares method desc" -d above, .sing a collection of points in a si At ball around v,. The maps are stored and retrieved using a hashing algorithm similar to that described in ref. 136]. This yields a piecewise linear Yorke / Noise reductwn methods that previously were available only to theorists:-" . We illustrate the approach using a time series of 32768 values from the Hnon map with a = 1.2, /f = 0.3 uing eq. (3) and adding 0.1% noise as described above. The original attractor is shown in fig. 6a. We take a grid of points wbhch aie spaced at 1%intervals (,his and subsequent distances are expressed as a fraction of the original attractor extent). The trne series is embedded in two dimensions. and a linear approximation of the dymimics is computed at each grid point for which 50 or more attractor points can be collected with a ball of radius 0.03, the set of such grid points is shown in fig. 6b. We take an initial condition near the original attractor and show the first 3000 iterates using eq., (4) in fig. 6c. Although some defects are visible, the attractor produced by the approximate dynamical system looks almost identical to the original one. One application of simplicial approximations is the location of periodic saddles and the estimation of the largest eigenvalue of the corresponding Jacobian. That is. if x is a periodic point of period p, then we find the eigenvalue of DJ P(x) of largest modulus, where Df'( v) refers to the mairix of partial derivatives of the pth iterate of the map f evaluated at x. Given an initial guess for x. one can apply Newton's method using the maps computed at the grid points and eq. (4) to locate the saddle using the simplicial approximations. Likewise. eq. (3) can be used to locate the corresponding "exact" saddle. Saddle orbits up to period 8 have been computed in this way. In all cases. the saddle point for the simplicial approximation is within 2% of the corresponding saddle point for the H~non map. Table I shows the largest eigenvalues of the saddle orbits. (The columns labeled mi= 2 and m = 3 refer to the embedding dimension used to reconstruct the attractor.) In most cases, the =lus approach is less ambitious than that of Crutchfield approximation of the dynamics from a set of 181, experimental data which can be analyzed with the equations that creates the observed attractor who attempts to find a single set of nonlinear difference 193 EJ. Kostehch and J.A. Yorke/ Noise eduction (a) / ., (C) , f - 0 3 (b I I% . , / _____ ___ ___ _ • . (b) Fig. 6. i. Ur_ H~non attractor computed from eq. t3) with a ,, I 2. ft -0 3 (b) 1% grid on which hinear approximations of the -- .... .i) .. dynamics are computed from the available attractor points. (c) Attractor produced by the bimphcial approximations relative error is only a few percent, and in no case exceeds 25%. (The largest relative error is for the period 8 saddles, where one finds the eigenvalue of the product of 8 Jacobians computed from the lea5 squares.) "his meth,>d can be exten&d to experimental d it, sets. However, there are relatively stringent described in section 4.) The current computer implementation uses a large amount of disk space to store the linear map approximations at the grid points. We have constructed a simplicial approximation for an attractor obtained from a BelousovZhabotinskii chemical reaction [7, 301. The attrac- ,m.. irernmts on the data that can be handled: the time se, es must be long erl,)ugh to trace out many trajeclories near the principal unstable saddle orbits, and the noise Ivel must be low. (Presumably, ,.isy data can be preprocessed using the approach tor is reconstructed in three dimensions from a set of 32 768 measurements of bromide ion concentration. The phase portrait is shown in fig. 7a. Linear approximations of the dynamics are computed at each point of a grid consisting of 50 El. Kostehch and JA. Yorke/ Nom., reduction 194 Table I The largest eigenvalues of the Jacobian of the periodic orbits located using the simphcial approximalicri of the Hinon attractor, the attractor. Using initial guesses from some of the trajectories, we apply Newton's method to locate the saddle orbit shown in fig. 7b. Moreover, Period m-2 Exact m-3 we obtain estimates of the Jacobian Df of the map evaluated at a point on the saddle orbit. The 1 1.793 2.178 4.226 1.695 2.199 1.757 2.183 4051 eigenvalues of Df are estimated as X,- 1.14. X 2 - 0.102, and X 3 = - 1.53. These quantitative > 9.626 0 2 4 4.329 6 10.38 10.70 6 10.38 25.80 20.02 17.70 11.32 24.88 20.60 24,32 8 9 8 1212 30.25 20.38 21.70 intervals along each coordinate axis for which 50 within an points or more attractor a Ths poducs oin.be located ridofcan rdiu he 8% 8% radius of the grid point. This produces a database of 59 550 maps. We observe from graphi, cal evidence that many traje. tories approach what appears to be a period-3 saddle in the middle of results confirm that the orbit is a saddle since X, > X3 (Note that one expects X 0 for a flow generated from a set of diffetential equations.) 9. Conclusion Methods for approximating the dynamics of attractors reconstructed from experimental data provide powerful tools. Most of the same proce. dures that have been so important for theoretical insight. such as Poincari maps. unstable fixed points and their manifolds, basin boundarics, and the like, are now available to experimenters. at least in cases where the dynamics are low dimensional. There is little doubt that these tools will lead to breakthroughs in the understanding of a wide variety of physical systems. However. considerable effort is needed before we learn which kinds of systems will benefit most from these types of analyses. Significant improvements in technique will certainly extend the applicability of dynamical embedding methods, for example to higherdimensional attractors. i7- jI z In this appendix we outline a possible alterna- . ////' ,',regression I / i Fig 7. (a) The attractor reconstructed from a time series of bromide ion concentrauoas in a Belousov-Zhabounskil chemi- cal reacuon. (b) The penod-3 saddle orbit tive noise reduction method based on the theory when all the quantities in the of least squares are measured with error. In ordinary least squares, the variables in the problem fall into two classes: the independent variables, which are known exactly, and the dependent variables, which are observations assumed to be functions of the independent variables. The errors dependent variables are subject to random s th t areasuedinependent and enil that are assumed independent and identically dis- tributed (i.i.d.). I EJ.'AKoteIch and J.A. Yorkel Noe, reduction On an attractor reconstructed from experimental data. we assaime that the mapping which takes points in a sufficiently small '- 'I to their images is approximately linear. Howevt,. the locations of all the points are subject to small random errors because of the noise. Hence one cannot describe the points as independent variables and their images as dependent variables. The usual leastsquares method produces a biased estimate of the linear map, and this bias does not decrease if more observations are added [16. 231. The so-called "errors in variables" least-squares methods can be used to handle the latter problem, This approach can be used to obtain both an estimate of the linear map as well as estimates of the "true" values of each of the observations. At first this appears to be an underdetermined problem: from n pairs of observmptions one wants to compute the parameters of the functional relation between them as well as estimates of the n actual pairs " -. However, it is possible to solve this problem by making some assumptions about the errors [16, 231, In our case, we assume that the errors in the location of each point and its image are i.i.d. In particular, we let the covanance matrix of the errors in the variables be the identity matrix. This assumption is valid whenever the noise is indepen- dent of the dynamics, 13. We illustrate the procedure for the case where we are given a collection of n points (in R") and their images. Following Jefferys (211. we form a set of n equations of condition given by /,(x,) = x,,., - Ax, - b,mx,+, - L(x,), (5) where x, is the ith point, x,,, is its observed image, A4is an m X m matrix, and b is an m-vector. The goal is to find estimates of L (i.e.. A and " In the statistical hterature. the problem is said to be uidenufied 1Dyna.-ucal noise (iLe.. each poit is perturbed slightly before iteratun&, yields a covanance matnx which depends on the point However. as long as the dynamical noise small, is our assumpuons about the covanance mamx of the errors should not compromise the accuracy of the method. 195 b). together with perturbations 6. such that f,(x, + ,) . (x,,, +0Od,) - L(x, + 0) - 0 and such that the quadratic form so- tv-10 (6) is minimized. The superscript t denotes transpose and a is the covariance matrix of the observations (which we assume is the identity matrix here). This minimization problem can be solved using Lagrange multipliers (see refs. (21, 221 for a numerical algorithm). The solution gives A and b together with estimates x, -t- 0, of the "true" observations. It can be shown 1161 under fairly mild hypotheses that the estimates of L and the obser. vations are the best in the class of linear estimators. One way to approach noise reduction is to extend eq. (5) to include several iterations of the observed points. Given a collection of points in a ball, together with the next p iterates of each point, the method above is used to find a collection of linear maps Lt . L ...... L. approximating the dynamics. The method also finds estimates of the actual observations. In this approach. therefore, the calculation of the maps and the adjustment of the trajectones is done in one step. Moreover, each point and its image exactly satisfy a linear relationship. Of course, p cannot be too large, because nonlinear effects eventually will become significant when the dynamics are chaotic. On the other hand. eq. (5) provides a natural way to include quadratic or other nonlinear terms. We have written a computer program to imple-, ment this alternative noise reduction algorithm. So" far, the results of this approach have not been as good as those from the method described in the main part of the paper, but further refinement should improve them. Acknowledgements Dan Lathrop provided invaluable assistance in finding periodic orbits in the Hinon and BZ attractors. We thank Bill Jefferys for useful discus- I 196 I EJ Kosehch and J.A. Yorke/ Noise reduction sions and computer software for the errors in variables least-squares problem. Andy Fraser. RandiTab eand w npro all idred, Randy Tagg and Hquan HarryS Swinney all provided (131 P.R. Fenstermacher, H.L. Swinney and J.P, Gollub. J.Fluid Mech. 94 (1979) 103. (141 W. 737.Ftanceschini and L. Russo. J.Stat. Phys. 25 (1981) helpful suggestions, This research is supported by 11S) A. Fraser and H.L. Swinney. Phys. Rev. A 34 (1986) 1134. the Applied and Computational Mathematics Program of the Defense Advanced Research Projects Agency (DARPA-ACMP) ad by the Department of Energy Office of Basic Energy Sciences. W.A. Fuller, 161 York. 1987). Measurement Error Models (Wiley. New 1171 E.G. Gwinn and R.M. Westervelt. Phys, Rev. A 33 (1986) ,143. 181 S.M. Hammel. J.A. Yorke and C.Grebog, J.Complexity 3(1987) 136: Bull. Am. Math. Soc. 19 (1988) 465 1191 M. Hinon. Comm. Math. Phys. 50 (1976) 69. 1201 D Hirst, Ph.D. Dissertation. University of Texas iDecem. Reee111m III R. Badu. G.Bro.V, D.Denghett. M.Ravatu, S.Ciliberto 979 12 (I Lett. bO (1988) Rev. Phys..P.Behngerand and A Politi.121 .AknJ. FuidMc~h her 1987), (211 WH. Jll'erys, Astron. J. 85 (1980) 177 1221 W H. Joffervs, Astron. 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Ph.D. Thesis. University of Texas at Austin (1987) J.P. Crutchield and B. McNamara. Complex Systems I 181 (19871 417H D. Abarbanel. R Brown and J B. Kdtke, Prediction and system idenufication in chaoUc nonlinear systems: time senes with broadband spectra. prepnnt (January 1989). Dongarra. C.B. Moler, J.R. Bunch and G.W Stewart. (9) J.J. LIN-PACK user's Guide (Sciety for lndustnal and Applied Mathematics. Philadelphia, 1979). (101 J.-P. Eckmann. S.O. Kamphorsth D. Ruelle and S. Ciliberto. Phys. Rev. A 34 (1986) 4971 D. Ruelle. Rev. Mod. Phys. 57 (1985) [11I J -P. Eckmann and 61. 617. (12) J.D. Farmer and JJ.Sidorowich, Phys, Rev. Lett. 59 (1987) 845 NJ. 197.7 (301 J.-C. Roux. Phvssca 0 7 (1983)57: J.C. Roux. R.H. Simoyi and H.L. Swinney, Physica I) 8 (1983) 257. (311 M Sno and Y Sawada. Phys. Rev. Lett 55 (1995)1082. 1321 C Sparrow. The Lorenz Equations: Bifurcations, Chaos. and Strange Attractors iSpnnger. New York. 19842). 1331 R Tagg, pnvate communication. 134] F Takens. in: Dynamical Systems and Turbulence. Spnnger Lecture Notes in Mathematics, Vol. 898, eds. D.A. Rand and L.-S Young tSpnnger, Berlin. 1980) p 36 (351 B.H. Tongue. Physica D 28 (1987) 401 1361 F Varosi. C. Grebogi and J A. Yorke, Phys Lett A 124 (1987) 9 371 A Wolf, J B Swift. H.L. Swinney and J A PyiaD1 18)25 Physica D 16 (1985) 285. Vastano, I I BULLETIN iNew Senesi oF THE %MERICANMATHEMATICAL SOCIETY % oume .23. Number 21. October 1'990) ANTIMIONOTONICITY: CONCURRENT CREATION AND ANNIHILATION OF PERIODIC ORBITS 1. KLAN AND 3. A. YORKE -kBSTRACT. One-parameter families , of diffeoimorphisms of the Euclidean plane are known to have a complicated bifurcation pattern as ;. %'ariesnear certain values. namely where homochnic tangencies are created. We argue that the bifurcation pattern is much more irregular than previously reported. I I Our results contrast with the monotonicity result for the well- understood one-dimensional family g (.r = ;x( I - tu where it is known that periodic orbits are created and never annihilated as A. increases. We show that this monotonicitv in the creation ol periodic orbits never occurs tor any one-parameier tamilv of' C; area coniracting ditfeiorphisms of the Euclidean plane. excluding certain technical degenerate cases where I I our analysis breaks down. It has been shown that in each neighborhood of a parameter value at which a homoclinic tangency occurs. there are either infinitely many parameter values at which periodic orbits are created or infinitely many at which periodic orbits are annihilated. We %how that there are both infnitelv many values at which periodic orbits are created and tnfiniteiv many -it which periodic orbits are anniilated. We call this'phenomenon antimonotoniwuv. U 1.INTRODUCTION The orbit of point v under a ditfeomorphism of the plane fis the sequence {f(x)}. where for k > 0. J de.,iotes the k-fold 1 composition of f'. J- k denotes the k-fold composition of Cf and I' is the identity map. Let p be a periodic point with period n.- The stable manifold IW(p) of the point p is the set {xr 0c) = p} . Similarly, the unstable manifold 11""p) of' PIs J.( lim~ -0 , p} We assume that p Is a hyperbolic saddle. that is. the eigenvalues e, . e, otf Df"(p) are such that lim~ -0 le1 Ie< I < le, I. Since f is a diffeomrorphism of the plane, both W'(p) and W"(p) are curves. There exists a homoclinic tangency I Received by the editors November 2. 1988 and. in revised form. May 1,. 1989. 1980,tMathemancs S~ubject C'lassificationt 1985 Revision;i. Primary 54035,581F713. Partial support provided by DARPA/ACMP program and by AFOSR-81-02 17. *1(D 469 1990 American Mathematical Society 0273-0979/90 $1.00 + $525 per page ; KAN AD J. A. YORKE .a70) of p at q if iV'pj and IW'p, intersect tangentially at q. The homoclinic tangency of p at q for a one-parameter family j at ,.= o is called nondegenerate if IV(p and IVO(pj have quadratic contact at q and IV (p) has nonzero velocity transverse to W"(p) at q as ,. varies [R]. Any value 4, at which this occurs is called a nondegenerate tangency value. A one-parameter family of maps g, is called monotone increasing (decreasing)on an interval J of parameter values if there are no bifurcations for . E J in which periodic orbits are annihilated as ,. increases (decreases. respectively). We say f is antimonotone at ;0 if periodic orbits are both created and annihilated as ,. increases in each neighborhood of the parameter value ,-, The only smooth family for which monotonicity has been proved is the quadratic family , x,- = ,vi I - i tDouadv. Hubbard. Milnor. Thurston. Sullivan. see jMT]. By contrast we have the following theorem. .. Antimonotonicity Thf rem. Each dissipatve ( 3 planardiffeomorphism .izmdv is antinonotone at each nondegenerate honocliw tangency value. Note that this result says nothing about what happens near degenerate homoclinic tangency values, but we believe this situation is essentially the same as for the nondegenerate case. We sketch the proof for a model case. A paper detailing the proof of the general result is in preparation. If two curves are tangent at . = 4 and move apart. so that they do not intersect as A increases (decreases) beyond .,. then we say contact is broken at 4,) %contactis made at 4-. respectively). and we say ., is a contact-breakingvalue lcontact-making value, respectively). Bubble Lemma. If 4. is a nondegenerate tangencv value at which contact is made, then there are nondegeneratetangenci' values arbitrarilv close to ,, at which contact is broken (and vice versa). The theorem follows immediately from the Bubble Lemma because in each neighborhood of a contact-making nondegenerate tangency value, infinitely many periodic orbits are created (and near contact-breaking ones, infinitely many are annihilated) [N. GS]. Thus, in each neighborhood of a nondegenerate tangency, orbits are both created and annihilated, as is illustrated in Figure 1 for the example of the Henon family. II I : NTMOOTONICITY I -0089 . x i -Q084 -14702 w I III 1147062:7"' FIGURE I. SMALL BUBBLE IN HENON FAMILY ,- -).3t'. x. .I. \'. V - 5.000 PREITERATES. .X-COORDINATE OF 30.000 ITERATES PLOTTED PER /. V-\LUE. I II.PRELIMINARIES For each Cantor set C,: R Newhouse INJ defines a number in [0. c, called the thickness -',C, associated with C. \ "rmiddle- 0" Cantor set C, = I \ G., is constructed inductively as follows: I = [0. I1 and I,.0 and I, , are the left and right component of 1, G, . respectively. where '', is an open interval of length .111 in the middle of I . The thickness of C,, is iI - 01)120. Newhouse proves the following lemma. Thickness Lemma. Let F and H be (antor sets in R. with If hull(F) and hull(H) r F toth nonemptv, and r H). r F) I Then H n F is nonempty. I A Newhouse horseshoe famdy N;. is defined as follows. (See Figure 2 on page 472 for symbols. coordinates, and the role of the constants. and see Figure 3 on page 472 for the first iterate I. K-AN AND J. A. YORKE B 1/2+1 1/2- C 1/0 A C x FIGURE 2, COORDINATES FOR . II N(A) N(B) -- N(C) 0 ----- ( X FIGURE 3. FIRST ITERATE OF N,.. V). Define .V,(.. r = d,,. /Jr) for (X., V) C .1' (...) = I /(I i-)) for .v V)E B *N'. ", ) = (". -. + ' -(j I 1/2) for (X.Y) E C: and continue N, smoothly to the rest of R2 We choose ,t# < I so \N;. is dissipative (i.e. Idet D(N .)I < IJ throughout .4 u B, and we choose uk, /, S, -s such that N. is one-to-one on .4 u B u C,. This implies P > 2. Let A denote the maximal invariant subset of .4 u B; A is a Cantor set and is the product A, A of two Cantor sets. A, is the projection of A ' -NTIMONOTONICITY -'73 onto the x-axis and A, onto the v-axis. We assume that a and ,8are selected so that r(A) ,r(A) = .11 - )-Q(l - 2a) - i> I. A primary stable (unstable) segment is a line segment of the form [0, I]x {y} where v E A., ({x} x [0. 1] where x E A,. respectively). A primary unstable parabolais a parabolic arc of the form N.(x. [/2- e.I/2 + el) where X E A,. Newhouse and Robinson show in [N. R], that in effect, there exist parameter values ,. near homoclinic tangencies where for a proper choice of coordinates the map is similar to Figure 3. We are assuming that the map changes in a regular way as /".varies. thereby avoiding technical complications. III. PROOF OF BUBBLE LEMMA ASSUMING NEWHOUSE HORSESHOE FAMILIES OCCUR Let 4"0 be a nondegenerate tangency value, which we assume to be a contact-making tangency., We assume that on a small interval. arbitrarily near ;.,, there is a Newhouse horseshoe family. We rescale that small interval to be [0. 1]. The primary tangencies (the tangencies of primary parabolas with primary stable segments) are all conta making. We will show that arbitrarily near /. = 0. there is a nondegenerate tangency which is contact-breaking and is not primary. Theparabolicarcofthe (t.= l/2+t,/ "- -- t) -I -iform ,_ / 1- I for 0<,<(1-2//3) <fi-,. t. lies in a gap in the Cantor set of primary stable leaves as shown in .It" Figure 4. I I 4 FIGURE A 4. THE ARC ,OIt.5). 474 I. KAN AND J.k. YORKE Let "(cf) denote the set of parameters such that v't. :) lies or a primary parabola. For each , the vertices of the primary parabolas have y-coordinates at (-;. ,A,, so we see that F(¢) = -ToAu - - f-'" and the thickness of "(%)is equal to rtAU). The nth image of L'tm,) t' (t,,; under N;. is -)= -- -. ;' - .-1/ +I a-,l/2+ti2-" There is a c = ,, t = " at which the Y-coordinate has a stationary inflection point as shown in Figure 5b. and 1 and "satisfy 46ffn-V 11-I l1/2) = -31;',o'?-/3 -3I : ' and = Z,3 '461"-1 . Notice = I/2)( I - Zl / i- 3 - so for large n we have 0 < 7 < I - 2/p .'/I '' Clatm. For fixed , < . with sufficiently small, there exists a .E F( ) such that the nth iterate of the primary parabola containing (t. ,) has a tangency with a primary stable segment. This tangency is contact-breaking and is nondegenerate for .. . The first part of this claim follows from the fact that the local maximum v( , ;.) (see Figure 5a) of the Y-coordinate of ",, U.t .,. depends linearly on ;.. That is. E o) - A; t and so {y(4, ,.)iA E r( )} has thickness rA). By the Thickness Lemma, there exists some A.E (n) such that vIs. A Note that A is00") . Since . isin Frl) , there is a primary unstable parabola which contains v(t . so v,,It. 4,. , is contained in the unstable manifold of A and is tangent to a primary stable segment of A. As ;.varies near 0. the position of this primary unstable parabola is vtt, + A). Nondegeneracy and contact-breaking can be verified by considering the .i-coordinate of d(v,(t, +A ))/dA and noting that for sufficiently small - > 0 and large n this derivative is negative for t sufficiently close to 7.1 We have shown that there is a primary stable leaf S and a primary unstable parabola U so that the nth iterate of U has a I -,NTIMONOTONICITY - S infletioon/ I I I FIGURE I 5. THE INFLECTION VALUE Z. _ I FIGURE 6. CONTACT-BREAKING TANGENCY (I AT a = A. contact-breaking tangency with S (see Figure 0i. Since the stable and unstable manifold of the fixed point p at (0. 0) contain curves arbitrarily close to S and U. respectively, we see that p will have contact-breaking tangencies at parameter values arbitrarily near .. Finally, for n large, this , is near 0. i REFERENCES [GSI N. K. Gavrilov and L. P.Silnikov. On three.dinensionaldvnamical wms close to systems with structurally unstable homoclinic curve. i. 11. Math. USSR-Sb. 88(4) (1972), 467-485. ibid. 9l t 1973), 139-156. [MT] J. Milnor and W. P. Thurston. On iteratedmaps orthe ,nierval. D.V namical Systems: Proc. Univ. Maryland 1986-87. Lecture Notes in Math.. vol. 1342. Spnnger-Verlag, Berlin and New York. 1989. pp. 465-563. .76 [NJ (R] I KAN AND J. A. 'ORKE S. Newhouse. The abundance ot wild hiperoolic sets and nonsmooth stable vets fbr ditffeonorphisms. Inst. Hautes Etudes Sci. Pubi. Math. 50 (1978). 101-151. C. Robinson. Bifurcation to innitelv many sinks. Comm. Math. Phys. 90 11983). 433-459. DEPARTMENT OF MATHEMATICS. GEORGE MASON !'NIVERSITY. FAIRFAX. VIRGINIA 22030 AND NAVAL SURFACE WARFARE CENTER., WHITE OAK. %IARYL-%%D INSTITUTE FOR PHYSICAL SCIENCE AND rECIINOLOY. I'%IVFRSITY OF MARN- [.AND. C"OLLEGE PARK. MARYLAND 20742 Volume 145. number 2.3 3 PHYSICS LETTERS A 2 April 1990 CHAOTIC SCATTERING IN SEVERAL DIMENSIONS 3) Qi CHEN. Mingzhou DING I and Edward OTT : 3 Received 20 December 1989; revised manuscnpt received 30 January 1990: accepted for publication 30 January 1990 Communicated by AP. Fordy Laboratory for Plasma Research. University of Maryland. College Park, MD 20742. USA For chaotic scattering in two-degree-of.freedom (N, 2). time-independent. Hamiltonian systems. scattenng functions (i.e.. plots of the dependence of aphase space variable alter scattering versus aphase space variable before scattering) typically display singularities on a fractal set. For N> 2. however. sc3ttenng functions typically do not have fractal properties (even when the chaotic invanant set isfractal), unless the fractal dimension of the chaotic set is large enough. Anumerical investigation of this I phenomenon is presented for a scatterer consisting of four reflecting spheres at the vertices of aregular tetrahedron. Recentty, there nas been much interest in the phenomenon of chaotic scattering (see reviews I I I) due to its appearance in a variety of applications, including fluid mechanics, celestial mechanics, and, systems that these functions are typically singular on a Cantor set of values of the variable characterizing the state before scattering, Here we consider whether this situation persists in systems with more than two especially, molecular dynamics. In addition, the implications of classical chaotic scattering for the corresponding quantum scattering problem is a subject of active research (2]. Another line of study concerns the question of how chaotic scattering comes about and evolves as a system parameter is varied [ 31.In all of these past works, when specific systems or examples are investigated, they have almost always been effectively Hamiltonians with two degrees of freedom. Since many situations that will arise in practice can be expc,.ted to involve Hamiltonians with more than two degrees of freedoms, it is important to see whether new phenomena, not present in two-degree-of-fr'±edom systems, can be anticipated in these situations, In particular, let is consider plotting variables characterizing the st ite of the system after scattering as a function of a single variable characterizing the state of the system before scattering (with the other "before-scattering variables" held fixed). We call such plots "scattering functions". It is a striking hallmark of chaotic scattering in two-degree-of-freedom degrees of freedom. We find that the scattering function does not typically display fractal properties in N.degree-of-freedom chaotic scattering systems with N> 2, unless the Hausdorff dimension D, of the fractal chaotic invariant set exceeds a critical value. In particular, if the Hamiltonian is time reversible, then fractal behavior of scattering functions can typically be expected only if II Also at Department of Physics. 2 Also at Department of Electrical Engineenng and Physics. Dc > 2N- 3. (I) Since Dc is greater than or equal to one, eq. ( I ) is satisfied for two-degree-of-freedom chaotic scattering systems (N= 2). For N> 2, fractal behavior of the scattering function istypically always absent even though the chaotic invariant set itself is fractal, provided that I < <D2N- 3. (Because the chaotic set lies in the DE-dimensional energy surface (DE= 2N- I ), we always have D, < 2N- I.) Since Dc depends on system parameters, one expects that aqualitative change in the scattering function can be observed as a parameter of the system is varied through the critical value at which Dc= 2N- 3. Eq. (I) is derived below. We consider N-degree-of-freedom, time-indepen- dent, open Hamiltonian systems, such that the dy- 0375-9601/90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland) 93 Volume 145. number 2.3 PHYSICS LETTERS A 2 April 1990 namics is time reversible. That is, ifx=X(t),p=P(t) are solutions of Hamilton's equations (where x and p are the N-dimensional configuration and momentum vectors), then x=X(-t), p=- P(-t) are also solutions. The dynamics will be reversible if the Hamiltonian is an even function of p. For example, H--jp:+ V(x) (2) • )We ative (i.e., D(S) <I since D.-2 and D(S2 ) =I), then the probability that the randomly chosen line intersects the fractal set S,is zero. If the right hand side of (4) is positive (i.e., D(S,)> I ), then there is a positive probability that S,n S2 is not empty; and, furthermore, if S,r S2 is not empty, then D(St S2 ) is given by (4) with probability one. now apply (4) to the chaotic scattering situ- Let D,and D, denote the dimensions of the stable and unstable manifolds of the chaotic invariant set. Due to the assumed time reversibility of the dynamics. these dimensions must be equal, D,= D .. 3) ation. Since the intersection of the stable and unstable manifolds is the chaotic set, we see that (3) and (4) with D. = D,= 2N- I yield D,= N+ d, with (5) (Non-time.reversible dynamics occurs, for example, when magnetic fields are present and leads to Hamiltonians which are not even functions of p. In these cases, (3) need not hold.) We shall be interested in "iedimension of inter. sections of sets lying in the energy surface. As'background. we note the following. Let S,and S, denote two subsets of a D.-dimensional manifold, and let their dimensions be denoted D(S,) and D(S 2). If S, and S2 are smooth surfaces, then generically d, = (D, - 1). (6) D(SnS)= D(S,) + D(S)- D., (4) if the right hand side is nonnegative and SjnS2 is not empty. If it is negative, then S, and S, do not have a generic intersection. For example, two onedimensional lines in a three-dimensional space may intersect at a point, but .. iight perturbation of the position of the lines typically removes the intersection. Thus the original intersection is not "generic". We wish to apply (4) also to the case where S, is fractal and D(S,) is its Hausdorff dimension with noninteger value Si, For this purpose, we refer to the theorems in ref. [4 . As an example of these results, consider the case of a fractal set S,lying in a rectangular region of a plane (D. = 2 ). Now randomly choose a straight line S2 in the plane by first choosing a point with uniform probability distribution in the rectangle and then placing the line through this point at an angle chosen randomly with uniform probability in [0, 2n]. If the left hand side of (4) is neg,IFormula (4) applies ifS1 isa Souslin set and S2 is a smooth surface. A Souslin set isthe union of countable intersection of closed sets. See ref. 41. 94 We now observe that the fractal set of singular values for the scattering function corresponds' ints on the stable manifold of the chaotic s" .-he orbits originating from such points asymptote to the chaotic set. Orbits originating near these points will spend a long time "bouncing arouod" in the scat. terer before leaving the scattering region. that is, they stay close to the chaotic set for a long time and hence are sensitive to small perturbations of their initial conditions. Let d, denote the fractal dimension of the set of singular values of the variable in the scat. tering function which characterizes the orbit before scattering. Sweeping this single, before-scattering variable corresponds to moving along a curve in the Drimensional energy surface. Thus d, is the dimension of the intersection of the stable manifold of the chaotic set with a one-dimensional set, and (4) yields, d, = D,+ I -DE, or d,=d,+2-N, (7a) (7b) di = ID,+ 2- N, where in (7b) we have used (6). (Note that (7a) applies whether or not the Haniltonian is time reversible, while (6) and hence (7b) require time reversible dynamics.) If the right hand side of (7) is negative, then there is zero "probability" of intersection, and we will typically never observe fractal properties of the scattering function. Requiring d, > 0 in (7b) yields the previously stated condition for fractal behavior in the scattering function, eq. ( I). We emphasize that the critical value, D,= 2N- 3, Volume 145, number 2,3 3 PHYSICS LETTERS A for observation of fractal behavior in the scattering function results under the assumption that the scattering function is obtained by varying a single before-scattering variable holding all the others fixed. If instead, we choose to consider scattering functions which depend on n independent before-scattering variables with the others held fixed, then similar considerations can be applied. In this case, fractal behavior in the n-independent-variable scattering function is typically observable if D,> 2(N-n)- I (for time reversible systems), and the fractal dimension of the set on which the scattering function is singular is d. =d,+n+ I -N. In such cases we say that the chaotic scattering is an "n-dimensional ob. servable". Since, as a practical matter, it is much easier to examine a function of a single independent variable, we expect the one-dimensional observable case to be of most interest. We check the above qualitative features in a simpie system exhibiting chaotic scattering. It consists of a point particle of unit speed bouncing between four identical hard spheres. The centers of the spheres are located at the vertices of a regular tetrahedron (fig. I ) of unit edge length. The spheres are labeled by (0, I, 2. 3). The coordinates of their centers I (.v,, v, 1=O. 1, 2. 3) are: o), (.ro,. -U)--(0, 0. 01,). (x,,, : 1 )=( - O) 1, l//3, Z 0. r~ Y / -pact 1 i X Fig. I. The geometry of the scatterer, four reflecting hard spheres sitting at the vertices of a regular tetrahedron, (x 2, y 2, :2) = ( - 2April 1990 I/2,/3, 0) 1- (x 3 ,y 3, :s)= (0, l//0). Thus the bottom of the tetrahedron sits on the plane :=0. The radius of the spheres R is the only adjustable parameter in the system. and the spheres do not intersect as long as R < 1. There are an infinite number of trapped orbits. periodic or aperiodic, in our system. These orbits are all unstable since small displacem.nts from a trapped orbit are magnified exponentially by the defocusing effect of the spheres. All trapped orbits can be uniquely coded by a bi-infinite sequence (a,) of four symbols J0. I. 2. 31 in the following way. We introduce a discrete time as the time of collision of the particle with one of the four spheres. The symbol a, is set to k if the panicle collides with sphere k at time i. Obviously, the particle cannot hit the same sphere it collided with at the immediately previjus time. Therefore, when R is small enough, the sole constraints on the symbol sequence of trapptd orbits is a, ta,. 1. If the symbol sequence is periodic. the corresponding orbit is also periodic. For instance, the orbit bouncing between sphere one and sphere two is of period two, and its symbol sequence is 1.... 1, 2. 1. 2. ... = [1, 21, where the square bracket denotes the periodicity. There are a total of six period-two orbits: (0, , 0.21, 10, 31, [1. 21, (1, 31, [2.31. There is no penod-one orbit due to the constraint a,*a,_1 . The number of trapped periodic orbits grows exponentially with the period. The exponent is the topological entropy of the set of trapped orbits. For our system, when R is small enough, the topological entropy is log(3). To proceed, imagine the following situation. We choose a plane below the scattering tetrahedron of spheres,. = - K, K> R. We then consider trajectories originating from initial conditions (.re, Yo) on this plane and with initial velocity straight upward (i.e.. parallel to the z-axis). We refer to (x, yo) as the imparameters. For all initial conditions (x.r,yo' we define a nonnegative integer valued function T(xo, Yo) which we call the time delay function. Its value is given by the total number of collisions with the hard spheres experienced by the particle with impact parameters (xo, Yo). For almost all impact parameters, this function is finite, corresponding to a finite trapping time of the particle in the system. 95 Volume 145, number 2,3 PHYSICS LETTERS A 2 April 1990 1 a0.4 - 1 .0.40.4 Xo 0 0.4 XO Fig. 3.The intersection of the stable manifold with the hyper- 0 plane plane:= -K. p.puO. R=0.48. -0.3 -0.4 -0.1 Rc 0 0.1 0.2 0.3 112 R 0.4 X0 Fig. 2. (a) Hierarchical construction of the Cantor set structure of the stable manifold. R =0.4; (b) blowup of (a). Fil. 4. Schematic illustration of the dimension d. as a function of R. However, there are certain trajectories which remain in the system for an arbitrarily long time. Initial conditions (xo. Yo) for these trajectories are distributed on a Cantor set. This Cantor set is the intersection in the five-dimensional energy surface of the stable manifold of the trapped unstable set with the two-dimensional plane z= -K, p=p=O. The time delay function is singular on this Cantor set. To see the Cantor set structure of the stable man- detail. For some impact parameters, the particle will not hit any of the four hard spheres and will go straight off to infinity. Those initial conditions from which the particle hits one of the four spheres at least once are the vertical projection of the four spheres onto the plane of initial conditions. They are the four big circular disks in fig. 2. We denote this set from which orbits experience at least one bounce by C1. Inside each big disk, there are three small deformed disks, from which the particle hits the four spheres ifold, we consider the particle trajectories in more at least twice. These are images of the other three 96 I I Volume 145., number 2.3 PHYSICS LETTERS A spheres in the mirror of the first sphere. Thus we have a set C2 of nine small disks from which orbits bounce at least twice. Within each small disk, there are three smaller disks C3, from which the particle hits the hard spheres three or more times. The resulting set of this hierarchical disk organization, given by n-,, C,, is the Cantor set illustrated in fig. 3. Starting from any point in this set, the particle bounces between the four hard spheres forever, never escaping to infinity, The fractal dimension of this Cantor set is d, and 2April 1990 is related to the dimension of the stable manifold D, by D,= 3+d . It is reasonable to presume that d, is a monotonically increasing function of the radius R. When R is zero, there is no strange set on the plane of initial conditions, and hence d, is zero. For small R,the dimension d, increases sharply with R, d, -/ln(R-'' as can be shown by an argument similar to one given in ref. 13 ). On the other hand, if R>,1,5, the region 16 25 J 20 II 10 T T 10 ---n__ 0 , -0.2 0 0.2 0. I 0.4 0.6 e 0.8 1 1.2 0.64 0.56 0.58 e 0.6 0.62 0.64 25 ,, 20 h 15k 10F~!~ 5 0.59 0.595 0.6 0.605 0.61 Fig. 5. (a) The time delay as a function of the distance I alont the one-dimensional line cut in a case exhibiting chaotic scatter ing, R=0.48; (b) blowup of (a); (c) blowup of ko). 97 Volume 145, number 2,3 PHYSICS LETTERS A 2 April 1990 between the four spheres is closed to the outside (in this case, the spheres intersect since R > j ), and all the points in this closed region are trapped. Hence, all the points in the closed region are on the stable manifold (i.e., the chaotic set and its stable manifold are the same set). The dimension of the stable manifold in this case is equal to the dimension of the energy surface, D,=5, and thus d,=2. Therefore, if we vary R between 0 and I/,/3 the dimension d, in. at which d,= I, and the scattering will change qualitatively as R increases through R. Below R,, we will not see chaotic scattering from a one-dimensional cut in the plane of initial conditions. A question of prime interest in this context is whether R, < J. If it is, then we will be able to see chaotic scattering for typical one-dimensional cuts for R in a range of values (t< R< j ) iuch that the spheres do not intersect. We used a box counting algorithm to determine creases from 0 to 2. Thus there will be a value R- R the fractal dimension d,. We cover the Cantor set Ilk - 0 .0.2 0.2 0.6 0.4 0.8 1 1.2 1 0.54 0.56 0.58 0.6 VI 0.62 I 0.641 I C 0 - 0, I I Fig. 6. (a) The cosine of the angle to the :-axis made by the ex0.59 0.595 0.6 e 98 0.605 0.61 iting direction of the particle as a function of the distance I for the same one-dimensional cut as in fig. 5; (b) blowup of (a); (c) blowup of (b). Volume 145. number 2.3 2 April 1990 PHYSICS LETTERS A generated above by squares of edge length e. then in the limit e-.0. the number of squares N(E) needed I The exponent d, can be determined by a least-squares fit of N(W). When R=0.48. we found d,is approximately 1.4. J hus we verify the important result that R,< 1.See the schematic illustration in fig. 4. We also computed d,at a smaller R value. R=0.4. at which we obtain d, 1.07. Using a linear extrapolation from these two computed values of d,. we estimate & 03.0 R 0.38. 10 a 1 for the covering scales as We now describe some of our numerical results at K 01 I 001 oO, R=0.48. Since d,z 1.4> 1. we expect, with positive probability, to see chaotic scattering from a randomly chosen one-dimensional cut in the plane of 1000 es~aping set on this one-dimensional line should ',e equal to d, =d, - I = 0.4. We check this by generating one-dimensional random cuts in the plane. We pick a random point in the square centered at the point lo x=v=O, of edge length 2R. Then we draw a line at a random angle through this point. Restricting initial '=11 i 12 =13 L\ 0 initial conditions. The fractal dimension of the non- =10 _/s=o =14 , 02 I 1)6 ,8 10 b T K conditions to this line, we then plotted the "time de. lay" (i .. , the total number of bounces from spheres experienced by a particle) as a function of distance I along this line. Out of thirty such lines, we found nineteen cases exhibiting a fractal set of singularities of the time delay function. A typical form of the time delay function restricted to the one-dimensional line in cases where we observe chaotic scattenng is shown in fig. 5a. From the blowups plotted in figs. 5b and 5c, we conclude that the singularities in the time delay function are apparently distributed in a fracial set. Another way to confirm this is to examine the dependence of the scattering function giving the exiting particlh direction. Fig, 6a shows plots of the cosine of the aogle 0 to the :-axis made by the velocity of an exiting particle as a function of distanc. I along the same randomly chosen line as was used for fig. 5. In regions near singularities, this function oscillates wildly. Successive blowups of this function (figs. 6b and 6c) show qualitative similarity, again indicating fractal singularities. To determine the fractal dimension of the set of singularities on a one-dimensional line, we use the following algorithm (the usual box counting method i =10 01 t/ 0001 06 04 02 =12 =14 = 15____ _--__= 0 /-/ =1 =12 08 1.0 S Fig. 7 (a)The Hausdorff sum K'(s) as a function ofs for different level t; (b)the same plot for a diffe ni one-dimensonal cut yields an error comparable to the fractal dimension). The time delay function assigns naturally a level structure to the one-dimensional line. At level t, we measure the length of all the intervals where the time delay function is greater than or equal to i and denote them by I,. Then we form the Hausdorff sum K'(s)= X (I ) , (8) where the sum is taken over all intervals at level i. When i tends to infinity, this sum should give the Hausdorff s-dimensional measure [ 5]. Therefore, it 99 VomI 3I45. num to 2. PHYSK"S LETTERS A 2Apd 3 99 is ,-afinitewhen s is less than the Hausdorffdimensiondofthefrac- v- and is zerowhen sis greater than d,. Hence. , expect that for sufficiently- large fractal behavior in the scattering function. level i. the sums K'(s) for different levels will all intersect with each other at approximately the same point s=d,given by the Hausdorff dimension of the one-dimensional fractal set. supported by the Office of Naval Research (Phwics). by the Department of Energy (Basic Enerpy Sciences) and by the Advanced Research Projects Agemcy We thank Itmi Kan for discussion. This 'ork was For R=0.48. numerical calculation indeed shows that the sums K'(s) for large levels all intersect at approximately the same value, thus yielding an approximation to d,. Figs. 7a and 7b plot K'(s) as a function of s for different i for two one-dimensional line cuts of the plane of initial conditions. (Small i data are not shown here. since they do not reflect the fractal property of the singular set.) Within numerical errors. the intersection points are all centered at d, =0.4 ± 0.05. This value is also consistent with resuits obtained for other cuts exhibiting chaotic scattering and is also consistent with our box counting result d,- 1.4. When R=0.25, the fractal dimension d,is less than one. Consistent with this. from 100 random line cuts of the plane of initial conditions, we did not see any 100 iefeences I U. iB. Eckhadt. D 33 (9) t 39: Smilanskv. TPhlsaa The classical and quantum theeey oof chaotic sauag.Lectures at es Houchtm Sesmson Ut. Chaos and quantum pis. eds. M.-J. Giamnom. A. VofosandJ. ZionJmm tEke,,e..Amst 1990. to be Puished. Im. .Phy& Rey. Lett.60 (19U) -3777: [2 It- Blmd and U. Su P. Gasprd and S. Rice.J.Chem. Phy.L 90 11999) 2242.2255: p. C-itanovi and B. Eckhardt. Phvs Rev.Lett. 63 (1959) 823. (31 S.Weher. E On and C. Grebr.;L Phy. . cv. Lett 63 (1999) 919. Mattda. Acta %JatIh 152 (194) 77. Ann. Acad.Sci. [41pFewwcA 1 (19751 2-7. 51KJ.Falconer. The Igomcty of factal sets (Cambridge Univ. Prss. Cambridae. 1995). ! I PHYSICSILETERSA Voline 147. Number &9 I Cross-sections of chaotic attractors I Qi Chen and Edward Ott ' LabovamforrPlasnr a Rmaiwr 30JY 1990 Unn1wm u ofManianL Colge ParAL MD 20742. USA Rccei ed 26 March 1990: accepted for publication '8 May 1990 Communicated by A.P. Fordy We present an e'fictent algorithm for constructing cross-sections of chaotic attractors. The technique is particularly useful for studying the structure and fractal dimension of higher dimencional attractors. 3 One of the central topics in nonlinear dynamical systems theory is the study of the structure and organization of invariant sets under the -lynamics. In chaotic attractor is followed until it comes near the desired cross-section plane. Through a subsidiary calculation, a local approximation to the 'astable particular, the geometry of strange attractors [I I is of particular interest. For such studies, the visualization of the strange attractor is important for revealing structure as well as characterizing the attractor, This presents problems when higher dimensional attractors are encountered. For exampie, the projection of an attractor whose fractal dimension is greater than two to a plane yields a fuzzy blob. Questions such as whether the local structure of a typical higher dimensional strange attractor is the product of a continuum with a Cantor set 121 or is more complex than this cannot be answered by simply taking a projection of the attractor. In addition. numerical determination of the dimension of higher dimensional fractal sets by box-counting algorithms can require enormous memory storage and CPU time. If feasible. taking cross-sections of the attractor (i.e.. intersections of the attractor with a surface) might offer a way of both elucidating the geometry of the attractor and of estimating its dimension. In this regard, two procedures for taking a crosssection of a chaotic attractor were proposed by Lorenz 12 1. and the first of them was extended and further developed by Kostelich and Yorke [ 3 ]. This latter procedure is basically as follows. An orbit on the manifold through that point is found. Then the intersection of the approximate unstable manifold and the desired cross-section plane is determined, thus projecting the orbit point onto the cross-section plane. Assuming the attractor is smooth in the unstable direction (or directions), this intersection approximates to a point in the cross-section of the attractor. Repeating this procedure many times as an orbit is followed, a cross-section picture of the attractor is built up. In this note. we consider Lorenz's second procedure for taking numerical cross-section. Compared to the first procedure. this procedure can be easier to implement and yield faster computer computation. On the other hand. the method has certain limitations which will be discussed. Consider an N-dimensional invertible map, x.+ I= F(x,). Choose a compact volume V which contains the chaotic attractor. We shall find the cross-section of an m-dimensional hyperplane with the unstable manifolds of the invariant sets contained in V. This will typically include the attractor. By inverting the map, the attractor becomes a repellor. Consider a point x in V and examine its preimages F - I(x), F -2 (x) .... F - "(x). Let T(x) denote the smallest value of n such that F - "(x) is not in V. We call T(x) the inverse escape time from V.Under the inverse map, all points in the region V will finally escape except for those on the unstable manifolds of the invariant sets con- Also at Department of Electrical Engineenng and Department of Physics 450 0375-9601 /90/S 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland) Volume 147. number 8.9 PHYSICS LETEr A 30 July 1990 tained in V. This set, of course, includes the repellor of the inverse map originating from the chaotic attractor of the forward map. Points on the unstable manifolds of the invariant sets in V correspond to singular points of the inverse escape time function (T(x) =oo). (We assume that the inverse map has no attractors in V. For example, the inverse of a strictly contractive map (e.g., the HWnon map) can have no attractors.) Thus, if we start initial conditions from a hyperplane and collect all the singular points of the inverse escape time function of the map, we find the intersection of the hyperplane with the unstable manifolds of the invariant sets in V, and this typically includes the cross-section of the attractor with the hyperplane. In practice, we not not determine the singular points but rather we determine a succession of nested sets containing the singular points. We do this by computing x values for which T(.r) > N for successively la; ,er values of N. To obtain the intersection with the attractor, one should reject points which satisfy T(x) > N but do not lie approximately on the attractor. In principle, thiscan be done by calculating theofLyapunov exponents (or other ergodic quantities) F -'tfor each .r satisfyotr qthat ing T(x) >N along the orbit x, F -counting F-(-i)(x). For large N, these exponents will aproximate the negatives of the Lyapunov exponents of the forward map on the attractor, provided thatreon the attractor. If x does not x lies approximately he ttrator thn th Lypulie pprximtelyon lie approximately on the attractor, then the Lyapunov exponents for the inverse map starting from x sions agree with previous calculations which resolved only the attractor set. The dimension of the intersection set in the crosssection plane is related to the dimension of the unstable manifold set by a result of Mattila [6 ]. If the Hausdorff dimension D of a bounded fractal set lying in an N-dimensional space is greater than N- m, then a random cut by an m-dimensional hyperplane intersects the set with positive probability; if it does intersect the fractal set, the dimension d of the intersection set is related to D by D=d+ (N-rn) (m) will approximate those for another invariant set in V and will differ substantially from the exponents of the attractor. In this case the point x is rejected. It will not always be possible to apply this Lyapunov exponent test, because N must be sufficiently large to obtain reliable estimates of the Lyapunov exponents of the inverse map. Alternatively, one can omit the Lyapunov exponent test altogether. In this case, the set obtained may be larger than that for the attractor. Thus a calculation of the fractal dimension of this set yields an upper bound for the fractal dimension of the attractor. In our numerical examples, we have not applied the Lyapunov exponent test. Nonetheless, as shown below, for these examples, the method appears to yield very good approximations to the actual attractor, and the calculated dimen- the dimension takes values between 1.22 and 1.30. Fig. I shows the Hdnon attractor. It can be shown that the attractor is included in the square [ -2.0, 2.01 x I -2.0, 2.0]. This is the region V which we use for calculating the inverse escape time function. We take a horizontal one-dimensional cross-section through the point x=O, y=0 and calculate T(x) at regularly spaced intervals along this line. This is shown in fig. 2a. We see there is a natural Cantor set level structure in the inverse escape time function. At level 0, there is one interval from which it requires at least one backward iterate to escape the square; at level 1, there are two intervals from which it requires at least two backward iterates to escape the square; etc. The intersection of all these intervals is the cross-section of the Hnon attractor. Fig. 2b with probability one. Hence, by generating the crosssection of the attractor and measuring the dimension of the cross-section set, we determine the dimension of the strange attractor. To illustrate out algorithm, we first calculate onedimensional cross-sections of the Hdnon attractor. The Henon attractor is generated by the following map, x,+1 =a-x2+by., y,+ =x,. (2) At parameter values a= 1.4, b=0.3, H~non observed there exists a chaotic attractor. Numerical box techniques for the calculation of the dimension of a strange attractor were first applied by Russell et al. [4], who obtained a result for the diy rabre who fou at te tain sI wa sult was obtained by Grassberger who found that the capacity dimension is approximately 1.28 ± 0.01 [51. However, from different least squares fits of the slope, 451 I I I 30 July 1990 PHYSICS LETTERS A Volume 147, number 8,9 -4. I 7 y0 I T5 - -2 [ 3 1 -1 0 1 2 -2 0 2 0 I 2 X8 I Fig. 1.The Hnon attractor. shows the same function for the vertical cross-section through the same point x=O, y=O. To get the fractal dimension of these cross-section sets, we use the following procedure. We denote the lengths of the intervals at level i by I,. Then we form the Hausdorff sum K'(s)= t),(3) J where the sum is taken over all intervals at level i. When i tends to infinity, this sum is the Hausdorff s-dimensional measure [71. Therefore, it is infinite when s is less than the Hausdorff dimension d of the fractal set, and is zero when s is greater than d. Hence, we expect that for large i, the sums K'(s) versus s for different levels will intersect with each other at approximately the same point s=d given by the Hausdorf' dimension of the one-dimensional fractal set ". In fig. 3, we show results for the Hausdorff sums for different levels for a typical one-dimensional cut. The lines for this case have intersec'ions in the range Ithe to find sum (3) Hausdorffused of thepreviousy application numerical The fractal to study chahas been dimension 3 I and Yorke 191 guarantee otic scattering [8 ]. Results of Nusse that for hyperbolic horseshoes, an interval with succcis:ve nested increasing T(x) contains a point where T(x) = o. 452 11 T 5 1 -2 b Fig. 2.Inverse escape time function for the Henon map. (a) Hor- izontal cut through x=.0, y=0. (b) Vertical cut through x=O, y=o. d 0.24 to 0.30. Examining many different one-diand vertical cuts, we estimate mensional horizontal d to lie in the range 0.20 to 0 34. From formula (I1), the dimention of the Hdnon attractor is approxi- mately D; 1.20-1.34. The whole calculation for a Volume 147, number 8,9 PHYSICS LETTERS A 30 July 1990 cross-section plane, we collect those points from which after some chosen maximum number of it- 10.0 trates of the inverse map nmax, the point remains in -To Ki 1.0 -i= the hypercube region. Fig. 4 shows two two-dimensional cross-sections of the attractor using our algorithm (ym,,=0.5 and nm,,= 15). The pictures in fig. 4 appear to be identical to those in ref. (31. find the fractal dimension of the chaotic at1.0 2 0.8[- 0.11 0.2 0.0 0.6 0.4 0.8 0.61.1;- 0 1.0 X2 Fig. 3. The Hausdorff sum K' (s) as a function of, for different 0.4 levels ifor the one-dimensional vertical cut through x=z0.8. Y=o,. cut involved very little computer memory and took less than 5 seconds on the Cray XMP computer.:' . Our second example is the double rotor attractor 0.0 ' generated by the following four-dimensional, vol. ume-contracting map (101, 0.0 X1+l ",:=M, / (.I 2+lMt i0.2v+ M Y'n ) + (( c, /2 7 ) sin(2nx 'i+ +' lI 2~ (c2/2n)sin(2xx"2 5.8 \-6.602 M =(0.7496 M- 0.1203 a 0.2 0.4 x, 0.6 0.8 1.0 mod I 4) 4 [ Here x, x 2 take values from the unit interval [0, 1), and y, and v2 'ike values from the real line. At parameter valucs iven by M,( .",. . -6.602') - 12.40,' 0.1203) 00 Y 4 , -' . 02,-- 0.8699J' c, =0.3536, c,=0.5, Kostelich and Yorke [ 31 find that there is a chaotic attractor. Since the two x-directions of the double rotor map are compact, we choose for V the hypercube box given by max( lyil, 1Y21 ) <Y., Starting from a uniform distribution of initial points in the -0.4 0.0 __ 0.2 0.4 _ 0.6 0.8 1.0 b X2 Fig. 4. Cross-sections of the double rotor attractor. (a) Crosssectionaty;=0., y2=0. (b) Cross-sectionaty,=O,x,2 2/2 453 Volume 147, number 8,9 PHYSICS LETTERS A 30 July 1990 squares N(e) needed for the covering scales as 2 1s N(E) -E-d,. 21 The exponent d is determined by a least squares fit 22 of a straight line to a log-log plot of N(e). In fig. 5, we calculate the capacity dimension d for the crosssection sets of figs. 4a and 4b. The two values of D=d+2 determined from least squares fitting are 3.67 and 3.63. According to the estimates of ref. [3 ], the information dimension lies in the range 3.61 to N (2"r) 2' 3.68. Thus we find that the values of the capacity and 2' information dimensions (the latter must be smaller) are apparently quite close to each other. In conclusion, we have presented an efficient algorithm for calculating cross-sections of strange at- 201 4 a4 5 5 6 6 7 8 8 9 10 2* tractors. This method may be useful for the esti- mation of the fractal dimension of higher dimensional chaotic attractors. 2 tl ; - 0 We acknowledge helpful conversations with Ming- 212 N(2") 28s I zhou Ding and James Yorke. This work was supported by the Office of Naval Research (Physics), by the Department of Energy (Basic Energy Sciences) and by the Advanced Research Projects Agency. The computation was done at the National Energy Research Supercomputer Center. References 2 I (5) [II J.D. Farmer, E. Ot and J.A. Yorke, Physica D 7 (1983) 153. [21 E.N. Lorenz, Physica D 13 (1984) 90;,17 (1985) 279. [31 E. Kostelich and J.A. Yorke, Physica D 24 (1987) 263. 20 LO [41 D.A. Russell, J.D. Hanson and E. Ott, Phys. Rev. Lett. 45 3 b4 5 6 7 8 9 10 (1980) 1175. b[51 P.Grassberger, Phys. Lett. A 97 (1983) 224. (61 P. Mattila, Acta Math. 152 (1984) 77, Ann. Acad. Sc. Fenn. Fig. 5. (a) N(E)as afunctonoe for thecross-section set mnfig. A 1 (1975) 227. 4a. The least squares fit dimenson dff 1.67 _0.05. ivesacapacaty [71 K.J. Falconer, The geometry of fractal sets (Cambridge (b) The same plot for fig. 4b. The least squares fit gives Univ. Press, Cambndge, 1985). dffi1"63+0"05 [8Q. Chen, M. Ding and E. Ott, Phys. Lett. A"145 (1990) tractor, we used a box counting algorithm. We cover the resulting cross-section set with squares from a grid of edge length e. In the limit e-.0, the number of 454 154. 193 H.E. Nusse and J.A. Yorke, Physica D 36 (1989) 137. (103 C.Grebogi, E. Kostelich. E. Ott and J A. Yorke, Physica D 25 ( 1987) 347 I I I I Nonlinearity 4 (1991) 961-979. Printed in the UK Rigorous verification of trajectories for the computer simulation of dynamical systemst Tim Sauert and James A Yorke§ USDepartment of Mathematical Sciences, George Mason University, Fairfax, VA 22030. USA § Institute of Physical Science and Technology, University of Maryland, College Park. MD 20742. USA Received 30 July 1990, in final form 29 January 1991 Accepted by J D Farmer Abstract. We present a new techniq e for constructing a computer.assisted proof of the reliability of a long computer.generated trajectory of a dynamical system. Auxiliary calculations made along the noise-corrupted computer trajectory determine whether there exists a true trajectory which follows the computed trajectory closely for long times. A major application is to verify trajectories of chaotic differential equations and discrete systems. We apply the main results to computer simulations of the Hinon map and the forced damped pendulum. AMS classification scheme numbers: 58F13, 58FI5, 65G05, 65L70 I. Introduction Are numerical studies of chaotic systems reliable? More specifically, do computer trajectories 'correspond' to actual trajectories of the system under study? The answer is sometimes no. In other words, there is no guarantee that there exists a true trajectory that stays near a given computer-generated numerical trajectory. The question is especially pivotal for chaotic systems. Chaotic trajectories exhibit sensitive dependence on initial conditions: two trajectories with initial conditions that are extremely close tend to diverge exponentially from one another. At the same time, a great deal of phenomenological research on chaotic systems relies heavily on computer simulation. Therefore, the use of an ODE solver on a finite-precision computer to approximate a trajectory of a chaotic dynamical system leads to a fundamental paradox. Because of sensitive dependence on initial conditions, a small truncation or rounding error made at any step during the computation will tend to be greatly magnified by future evolution of the system. Under what conditions will the computed trajectory be close to a true trajectory of the model? Consideration of simple examples of nonlinear maps illustrate that there are critical points of trajectories where round-off error or other noise can introduce new behaviour. We discuss typical examples in section 2. At such 'glitches' the true trajectories all i 't Research supported by the Applied and Computational Mathematics Program of DARPA. 0951-7715/91/030961+19S03 50 © 1991 lOP Publishing Ltd and LMS Publishing Ltd 961 962 T Sauer and J A Yorke diverge from the numerical trajectory. In this case, there will be no true trajectory that stays near the numerical trajectory. In other cases, the numerical trajectory can be shadowed: some true trajectory remains close to the numerical trajectory. In the present work we state a result (theorem 3.3) which says that if certain quantities evaluated at points of the computer-generated trajectory, called a pseudotrajectory. are not too large, then there exists a true trajectory near the .computergenerated one. Rigorous upper bounds for these quantities can be generated by the computer as it produces the pseudo-trajectory. If these quantities satisfy the hypotheses of the theorem, which again can be rigorously checked by the computer, the result is a computer-assisted proof of'the existence of a true trajectory near the computergenerated pseudo-trajectory. For example, if the one-step errors in the pseudo-trajectory occur in the tenth decimal place, then the true trajectory that results from the theorem differs from the computer-generated trajectory in approximately the fifth decimal place. In particular. the initial point of the true trajectory can differ from the initial condition of the pseudo-trajectory at most in the fifth decimal place. A typical application of the theorem is to the forced damped pendulum y+a' +sin y- bcost. Setting the parameters a = 0.2 and b = 2.4, we prove the existence of an apparently chaotic trajectory with initial cor itions y(0) = 5(0) = 0 for time t ranging from t = 0 it0,lies within 10-' of an explicit to t = 101n. This trajectory, for all 0 4 t computer-generated (noisy) trajectory produced with a one-step error of 10-11. There are similar results for other initial conditions and other choices of a and b. To describe the theorem, we make a distinction between discrete and continuous models. Computational methods for approximating trajectories of systems of ordinary differential equations work by a series of small, discrete steps. We can therefore consider computer simulation of discrete systems and autonomous differential equations at the bame time if we define a dynamical system to be an invertible map f on R'. (We actually define dynamical system a little more generally, as a sequence of maps (f.) on R", tD also cover the non-autonomous differential equations case.) We will try to keep this distinction clear by using the word trajectory for continuous systems and orbit for discrete systems. Consider then a 6-pseudo-orbit of a discrete system f, which we can imagine having resulted from applying a one-step quadrature method with truncation error 6 to a system of differential equations on R1', m > 2. Assume that we have subspaces S. and U. at each point x. of the pseudo-orbit, which are self-consistent with tolerance 6. By this we mean that S. and U. are complementary subspaces of the tangent space R" at xn (see figure 1), that unit vectors in U. are mapped by f to within 6 of U.+ ,, and similarly for S. Define the positive number r,, to be an upper bound for the expansion rate of the linearization Df along Sn, and t,, to be an upper bound for the expansion rate of Df - I along U.. See section 3 for precise definitions. The quantities which need to be measured to assure the existence of a nearby true orbit are most easily expressed as recurrence relations. Set up a recurrence relation C, by beginning with Cc = 0, and recursively defining C, = csc 0 +r,_IC ,, where 6. is the angle between S. and U. Define Dn similarly: DN = 0, where N is the length of the pseudo-orbit, and D,, = csc 0 + tD+ , for n < N. Then as long as the quantities C. and D, are not too large for all n, there is a true orbit of f near the pseudo-orbit. More precisely: I I Rigorous verification of trajectories 963 U, 1l Sn Fiure 1. The splitting of the tangent space at the nth point of the pseudo-orbit. Theorem 3.3. Assume 6 < 1/20m2 and let B be a bound on the first and second partial derivatives of f and f -. If max{C.,D ,J mS/2 B2 ,/- for all n = O.....N. then there exists an orbit {w.} of f such that Ix. - wjI < v57 for n- 0,.... N. Note that we do not need to assume uniform contraction and expansion along the directions S. and U1. In other words, r. and t. do not need to be less than one for all n. The proof of the theorem is constructive, in the sense that it uses a procedure for refining noisy orbits originally given in [6]. The essential point of the proof is to show that under the conditions of the theorem, the iterated application of the refinement procedure, beginning with the pseudo-orbit, results in a sequence of refined pseudoorbits with decreasing noise level, and whose limit is a true orbit. In addition, the true orbit isnot too far from the original pseudo-orbit. The proof can also be considered a justification for using the refinement process computationally on the actual noisy orbit to reduce noise to near machine-precision, but that is a separate issue from the main question we are answering here. This direction is taken up in [7]. A true orbit that stays near the pseudo-orbit is said to shadow the pseudo-orbit. Several years ago, Anosov and Bowen proved shadowing results for hyperbolic maps on a differential manifold. The conclusion of Anosov [1] for a hyperbolic map says that, given any prescribed shadowing distance c (between the pseudo-orbit and true orbit) there exists a 6 > 0 so that any 6-pseudo-orbit can be £-shadowed by a true orbit. Bowen [2] showed that the same result holds if the map is required only to be hyperbolic on a basic set containing the orbit. Other proofs have been given, and one more is aconsequence of the present work. There are two factors that make the approach of Anosov and Bowen impractical for use in computer experiments. First, the 6 that isproduced can be orders of magnitude smaller than the machine epsilon of existing digital computers. Second, most interesting dynamical systems currently being studied are not hyperbolic. Theorem 3.3 does not assume that the dynamical system is hyperbolic. Our approach is to prove that as long as the system is sufficiently hyperbolic along the (finite length) numerical trajectory, then that piece of the numerical trajectory can be 964 T Sauer and J A Yorke shadowed by a true trajectory. On the other hand, when f is hyperbolic, C, and D', stay uniformly bounded for all iterates n, in whichi case arbitrarily long shadowing trajectories are constructed by the theorem for sufficiently small 6. Thus the shadowing theorem of Anosov and Bowen is a consequence of theorem 3.3, as is noted in [9]. In (5,6] a method is developed which creates computer-assisted proofs of the existence of finite length shadowing orbits on a case-by-case basis. In two dimensions, a small parallelogram is constructed near each point of the numerical orbit in such a way that there is a guarantee of a true orbit whose nth point lies in the nth parallelogram. They apply the method to one-dimensional maps and the two-dimensional Hinon and Ikeda maps, none of which are hyperbolic. These papers use auxiliary calculations in 96-bit precision to verify that there are true orbits near the pseudo-orbit, which was produced in 48-bit precision. The advantage of the present method over [6) is that the auxiliary calculations can now be done in the same precision in which the orbit was calculated. For the maps mentioned above, only 48-bit precision is needed to verify the existence of a pseudo-orbit produced in 48-bit precision. This fact is especially important when attempting to shadow differential equations. We found that the methods of [6] were not practical, at least for the differential equations we tried. For example, in orwer to produce long shadowable pseudo. trajectories for the forced damped pendulum. we needed to use a one-step error of no more than 10-19, which already requires 96-bit precision. In this case, there is no extra precision available for the auxiliary calculations of [6). Thus the new method, superior even for maps, is evidently essential for shadowing differential equations. The improvement is largely gained by sublimating the refinement process, done explicitly in a computer-aided proof in (6), into the proof of theorem 3.3. It is proved here that under the hypotheses of the theorem, the refinement process, when iterated, theoretically converges to a true trajectory. The main result of this paper was announced in (9), in a slightly less streamlined form. Other work along these lines for the one-dimensional case is reported in [3). In the next section, it is shown by example that shadowing can fail for some pseudo-trajectories. The details of the main theorem (theorem 3.3) are presented in section 3. Section 4 consists of a number of remarks relevant to the implementation of the computer algorithm based on theorem 3.3. Examples are given in section 5, and section 6 contains the proof of the main theorem. 2 Why shadowing works What makes it possible to find a true orbit near a pseudo-orbit in the presence of sensitive dependence on initial conditions? The short answer is hyperbolicity along the pseudo-orbit. Even for a non-hyperbolic dynamical system, as long as the pseudo-orbit avoids areas of phase space that lack hyperbolicity, it may be possible to find a nearby true orbit. Of course, on typical ergodic chaotic attractors, this avoidance is only done as a matter of degree. Roughly speaking, the pseudo-orbit must stay far away from non-hyperbolic areas compared with the size of the errors being made. Our method essentially relies on measuring how successful the trajectory is in staying hyperbolic. As a simple example, imagine a map which contracts distances. Assume that the distance between any two points x and y is decreased by a factor of K by the map f, where 0 < K < 1. Thus If'(x) - f"(y)l < K"Jx - yl. It follows that any pseudo-orbit I 965 Rigorous verification of trajectories can be shadowed by the true orbit beginning at its own initial condition. All distances are contracted, including errors that are made along the pseudo-orbit. To be more precise, let a 6-pseudo-orbit be denoted by {X,X If(xo) -. II< 6,and further If2 (xo) - x.1 4 1 ,.... ,xN.. Then If2(Xo) -f( 1 )I+If(x1 ) -x 21 KIf(x0 ) - x1j +If(xl) - X21 < (K + 1)6. Continuing in this way, If"(xo) - x.i < (K - I+ K1-2 +... + 1)4, and we can see that the true orbit {x0,f(x 0 ),..... N(x0 )) shadows the pseudo-orbit within 6/(l - K). Although this hyperbolic map is not sensitive to initial conditions, it is an instructive example. C-nsider next a diffieomorphism which expands distances, so that If (x) - f"(y)l > K"Ix - yI for K > I. This map is sensitive to initial conditions, yet any pseudo-orbit {xo,x1..,...xs) can easily be shadowed. The inverse of the map contracts distances, so the true orbit (f-N(xN),f-N+i(xN).... xN) will shadow the pseudo-orbit within 41(1 - I/K). A general hyperbolic dynamical system is a combination of the above tivo examples. At each point, some directions are expanding and the rest are contracting. To construct a true orbit, one needs to use information from the beginning of :he pseudo-orbit in the contracting directions and from the end of the pseudo-orbit in the expanding directions. This idea is the basis of theorem 3.3. On the other hand, not every pseudo-orbit can be shadowed. This is not a failure of any particular shadowing procedure. The simplest examples of nonlinear maps provide cases of pseudo-orbits for which there is no corresponding true orbit nearby. Consider the one-dimensional logistic map f(x) = I - 2x2, shown in figures 2 and 3. The interval I = [-I, 11 maps onto itself under f and so is an invariant set. True orbits which begin in I remain in I for all time. Figue 2. A pseudo-orbit of f(x) = 1-2x2 which cannot be shadowed. The initial condition is the dot at the origin. An error of size 6 is made in computing f(0), which causes the orbit to eventually approach -o. Now consider the 6-pseudo-orbit which begins with x0 = 0, x, = l + 4, and which from then on is computed without error. Then x, = f(xi) < -1,and the pseudo-orbit 966 T Sauer and J A Yorke I I Fitwre 3. For the map 1x) 1- .2x. an initial condition in the open interval Li"lcngth V86 around zero can be attracted to -x if an error of size 6 is made. diverges to -x. See figure 2. Clearly, there .io true orbit of the system f which shadows the pseudo-orbit by a distance of less than I. Any true orbit within I unit of .- 0 must stay witain I for all time. In this simple case, points escape from true behaviour near the critical point, or fold, of the map. Informally, we call such a divergence from legal behaviour a glitch. In general, the logistic map f(x) = a(l - x") - I will have pseudo-orbits that cannot be shadowed not only for a = 2 as above, but when the parameter a is less than and within 6 of 2. where 6 is the noise level of the process. (This corresponds to the critical value of the fold in figure 2 being between I - 6 and 1.) Thus the occurrence of a glitch is a robust phenomenon. The same phenomenon occurs in higher-dimensional chaotic dynamical systems, because of the folds caused by homoclinic tangencies and near-tangencies of stable and unstable manifolds. How often should we expect glitches? The answer should depend on the noise level 6. In the logistic map example fix) = I -2x-, there isan interval of length v,8 around for which it is possible for an error of size 6 to cause a glitch. This is illustrated ,ure 3. Any initial condition in the designated interval around 0 is susceptible to .1g mapped to a value greater than I, and therefore mapped out of 1, towards -)0. , computer-generated orbit of that type does not correspond to any true orbit. If we assume that this interval of length 03- is sampled by the dynamical systew. approximately in proportion to its length, we expect a glitch to occur on the order of every l/v'3 steps. Numerical studies in [6] on two-dimensional maps and thc studies of differential equations undertaken for this work roughly support this scaling. I I I I I 3. Shadowing theorem The theorem can be used to shadow diffeomorphisms or differential equations. To include both cases, we will consider a dynamical system to be a sequence f 0 ... fN of C2-diffeomorphisms on R" for some positive integer N. When attempting to shadow a discrete map f, we will use f, = f for all n. For a non-autonomous differential equation t = F(t, x), we would let f, be the map on phase space which takes an initial point x at time t to the point on the trajectory time at time t + hn , where h, is the current step size of the ODE solver. If we assume, for simplicity, I I I I I I 967 Rigorous vei fication of trajectories that the differential equation is being solved with a constant step size h, then h. = h for all n. In this case, the ODE solver induces a map called the time-h map of the system. In the case of an autonomous differential equation, the induced time-h map will be the same for all t.On the other hand, if the differential equation is non-autonomous, the time-h map towillinclude dependboth on the t. The following and definition of an orbit cases. of a dynamical system is made autonomous non-autonomous Definition 3.1. Let N be a positive integer, and let f, : R" -. R" be a C2diffeomorphism for each 0 4 n < N. The finite sequence {y}, n = 0,..., N of points i'. R" is called an orbit of the dynamical system {f.}, n 0,..., N - I if f,,(y.) - y.,+, !or n = 0..... N - 1.An orbit is sometimes referred to as a true orbit to contrast with the notion of pseudo.orbit. The finite sequence {x.J is called a 6-pseudo-orbit of {ff. if If,(x,) - x,,Ill <6 for n = 0..... N - I. The 6-pseudo-orbit fx,g is r.-shadowed by the orbit y of the dynamical system {fJ if Ix. -y, < c for n = 0,..... Here, as below, we use the Euclidean norm: I - 1/2 1 for a vector v = (v,... v). We also need to define the concept of moving frame from the point of view of computer simulation. Im The moving frames we will require will be numerical approximations S.and U. to the stable tangent space and the unstable tangent space at x., if they exist, and the next best thing, if they do not. Let N and k be positive integers. For each n = 0...., N -- I, let J,be a non-singular x mmatrix. For each n = 0., N let ,, v () be a set of k vectors in R", and define 4,, to be the m Y k matrix with columns Definition 3.2. The set 1,, "..... g,}N .ois 7,,. called a 6-pseudo-frame for the dynamical system {J.I if for all 0 < n < N, I., The entries of the k x k matrix ATA. - lk are no larger than 6 in absolute value; 2. J,,t....Jnv, are each within 6 of range(A,, ,). Informally, we call property I of the definition almost-orthogonality, and property 2 consistency. The usefulness of this definition for computer-assisted proofs lies in the fact that a 6-pseudo-frame consisting of machine-representable numbers can be constructed using standard computational procedures. Assume that we begin with a set of k vectors {to 1 ,. vk} in R"' which form an orthonormal set. (That is, the vectors in the set are mutually orthogonal unit vectors.) Assume further that the components of the vectors vol,.... vok are machine-representable floating point numbers. Then we use the Gram-Schmidt orthogonalization procedure on the set (J0 (vo) .. J0 (vok)}, and define (v, ..... v1k} to be the machine-stored vectors that result from this finite-precision computation. (In some cases we found thai a more stable form of orthogonalization [4] improved this step.) Continuing in this way for 0 < <n N we define a 6-pseudo-frame for a small number 6, such that each vector v,, in the frame is machine representable. T Sauer and J A Yorke 968 We can now describe the main theorem. For each 0 < n < N, let f,,' R' - R. be a C 2-diffeomorphism. Let {x,,}.o be a 6-pseudo-orbit of the dynamical system {f.). Define J,, = Df(x) to be the matrix of first partial derivatives of f. Let B, (respectively B,) be an upper bound for the absolute values of the first (respectively, second) partial derivatives of the component functions of f, and f.-; on the union of balls of radius 61/2 centred at x. for n = 0,...,.N Set B = max{2,B1,B 2}. For positive integers k+l = m,let {sj,. ,Skob (resectively,{u,,,.. . , } be a0 -pseudo-frame for {J; '} (respectively, {Jj}) such that {S. .. ,,. .. u~ } spans R" for each n. 1 Define the subspaces S, = span{s., ... SM), U, = span{u,,,.. .,u.}, and define 0 to be the angle between S,, and U. Let r, and t, be numbers satisfying IJ.yl < rjlyI IJ,':l < t,,I:l for y 1ES for z e Un+i. Define Co = 0.C = csc0,, + rnC.,- for n > 0. Similarly, define Dj = 0,Dn = escOn,+ tDnI for n < N. Theorem 3.3. Let {x,}. o0 be a 6-pseudo-orbit for the dynamical system {f,} on R m'n > 2, and assume that 6 < 1/20m . If max{C.,D.} < for all n = 0. n= 0..... N. N, then there exists an orbit {wv} of {f,,} such that Ix,, - w,, < V6 for Theorem 3.3 gives an alternative approach to Bowen's shadowing lemma (2]. Let f R'" - R'" be a C 2-dff'eomorphism. A compact invariant set A is called hyperbolic if there is a continuous splitting of the tangent space TR = El Eu for x E A, and positive constants A.< 1,C > 0 such that 1. Df (x)(E ,) = E' 2. Df(x)(Eu) = 3. IDf"(x)(v) < CA-IvI tbr v E E1 4. IDf-"(x)(v) < CA-vi for v E Eu for allx EAandforall > 0. Theorem 3.4. [2]. Assume A is a hyperbolic set for f. For every e > 0 there is a 6 > 0 so that every 6-pseudo-orbit in A can be c-shadowed. Theorem 3.4 is a direct consequence of theorem 3.3 (see [9]). 4. Computer-assisted shadowing In this section we describe a computer algorithm which uses the above theorem 3.3 to verify the existence of true orbits of a dynamical system near the pseudo-orbit determined by a numerical computation. Along with the pseudo-orbit being computed, there are some auxiliary calculations to be made to check that the hypotheses of the theorem are satisfied. Namely, it is necessary to find upper bounds for the constants B, cscO, r,,,t, and finally C,, and D,. We next describe these auxiliary calculations, which if successful provide a computer-assisted proof of the existence of a true orbit. P:oous rerio. i 5 of ir-jecniaes 969 4.1. Conurucuon of sutable and ummable fraes I... The algorithm wods best when J-pseudo-frame the 's . and . are chosen to approximately encompass the tb and unstable directions, respectively, for the dynamical system If. at the particular map f,. One way uO_of vectors to accomplish this is as follows. Begin with an orthonormal set ,. in R" chosen arbitraily. Inductively define the orthonormal set Iu, +,.. uj} to be the computed results of applying the technique of Gram-Schmidt orthogonalization. followed by normalization. to the set Df,(x,)u,..... Df .(,,ud!. Because of computer will be only approximate, which is not important these6computations round-off, v a 6-pseudo-frae It is straightforward to fin a is"=,"..",,,f" For which ,6 for which both parts of definition 3.2 are satisfied. Part I is easily checked with the computed !.--- - and depends on the residual error of the Gram-Schmidt orthogonalization. In most cases of following a trajectory of a system of ordinary differential equations, the , will be determied by part 2 of definition 3.2. which depends on the error bound of the ODE solver being used to follow the tangent vectors along the pseudo-orbit. Begin with an arbitrary € is defined analogously. The frame ,. Svk, fl.m R". Give.. . for n N. apply orthonormal set . (x3 _.)s,,k. The stored values of Gram-Schmidt to the set {Dxf;t(x _t,. the resulting computation are I .s....._, by definition. The calculation of csc O., where 0, is the angle between S. and U,. is simple if the dimension m is small, but for higher dimensions the following scheme may be helpful. Define A. to be the m x m matrix whose columns are {s,_ ... s,u,, ..... u}, and let B. = A-'. Let B. be the m x m matrix whose top k rows are the same as those of B. and whose bottom 1 rows are filled with zeros. Let B, be the m x m matrix whose top k rows are the filled with zeros and whose bottom I rows are the same as those of B,. Note that B,, = B, + B. . Now define S,, = A,,B, and U,, = .4,Bu. It is clear that S, and L,, are projections onto S,, and U,. respectively, and that S + U,, = A,,(Bs +Bu) = 1 Further. S,, and U,, are the unique m x m matrices with these properties. It is a standard fact that csc 0 = IS,[ = IU., where as usual we use the Euclidean matrix norm. This scheme provides a computationally stable method for computing a strict upper bound on csc O,,. which is necessary for bounding the C,, and D,. 4.2. Calculation of r,, and t, We have dcfined r,, to be a positive number that bounds the growth of f,, in the direction S,, at x,. That is. r,, satisfies IJ,,yJ < r,,Jyl for vectors Y in S,. Such a number is impossible to find by measuring JJ,y on a general basis of S,. This is the reason that almost-orthogonal frames are needed in the calculation. Lemma 4.1. using A = J, and W = S, shows how to find an upper bound on r,, solely using information about the action of J, on the almost-orthogonal basis of S,. Analogously, t, can be found by using lemma 4.1 with A = J;-, on the subspace U,,+,. Lemma 4.1. Let A be an m x m matrix and W a subspace of R' with basis Let W be the m x k matrix with columns .... Wk}. Then max ,Ew.1,,=i AvI << AW VI - WJVTJW - 1 ivw. 970 TSauer and J AYorke when the right-hand side exists. Lemma 4.1 is proved in section 5. 43. Calcudaion of C.and D. Computing C. and D, appears simple once cscQ0, r. and t. are known. There are two more details, however, that greatly reduce the data requirements of this task. In applications of this algorithm, it is typical for N. the number of points in the pseudoorbit- to be of the order of several million. On the other hand, we have previously suggested that the computation of the stable frame s,,.sJ' (and therefore 0.) be done by beginning with a random frame at n = N. and applying J.-' to create frames N - I.....0. To avoid the problem of storing all frames simultaneously. we iuggest buildine.... . in pieces of length N, < N. For example- we found N, = 5000 to be reasonable. The idea is to find each block of 5000 nearly-orthogonal bases by stopping after each b -k of 5000 points in the pseudo-orbit, finding the next 1000 points, and then applying J, ' 6000 times to a random starting orthogonal basis to produce stable directions, and then go on to the next block of 5000. In all cases we ha.e tried, the stable frame produced this way satisfied definition 3.2 within the prescribed 6. The second problem is deciding whether the recurrence relation D, stays within the bound of the theorem, given that it is defined beginning at the end of the trajectory. The following simple lemma shows how to verify the bound on Dn in forward time. In short, a new recurrence relation E, is defined which is computed in forward time. The lemma shows how to tell by computing E, whether D. violates a given bound. Lemma 4.2. Let D., = 0: Dn = a, + bD., be a recurrence relation for n = 0,.... N and let .4 be a real number. Define another recurrence relation E0 = A; E.+1 = min{(E. - a,)/b,A for n = 0,...,N. If E, > 0 for n = O....,N, then D.<< .4forn=. .... N. 4.4. Calculation of B The calculation of B, the upper bound on the magnitudes of the first and second partial derivatives of the f., is normally trivial if we are given the explicit map. In more interesting cases, we are following the (possibly time-dependent) flow of a system of differential equations, and need bounds on the derivatives of the time-h map for step size h.It is this map which is being approximated by the numerical ODE solver. To this end. consider the first-order system = F(t,y) where v is a vector in R"' and t denotes the independent variable. Define g(t,s,Z) to be the value of the solution with initial condition y(s) = : at time t.Then the time-h map of the differential equation which maps the value at time to to the value at time to + h is given by y). fin.h(Y) = g(to + h, to, The following lemma establishes upper bounds on the partial derivatives of the in component functions of f = f,,.h = (f . f 971 Rigorous verification of trajectories Lemma 4.3. 1. Define E, = m Then m 2. Define E. = max <e h O E I0FL. ,I2 f , -, IhE,eI 'CmI E' I. Then max 1Y- IYY '-' eIy ey C.YJkI The proof is an exercise in using the Gronwall inequality (see for example lemma 4.1 of [8]) on the first and second variational equations of the system. 4.5. Quadrature method To apply theorem 3.3 to a differential equation such as the forced damped pendulum. we need a quadrature method which has high accuracy. and which has an explicit error formula. The former is necessary to allow application of the theorem with a reasonably small 6 (and therefore a long shadowing time). The latter is necessary to assure that we have a rigorous bound on 6. The simplest method that satisfies these two criteria is the Taylor method. The formula for the one-step error is explicit, being essentially the Taylor remainder. However, the major diffict.lty with implementation of the Taylor methoes in general is that they require explicit differentiation of the right-hand side of the differential equation. Thus, applying the seventh-order Taylor method to the differential equation (I) j + a,+siny = bcost • evidently requires differentiating the differential equation five times. The formulae fill a few pages. Fortunately, there is a trick which allows application of the Taylor method as an ODE solver without doing the symbolic calculation of higher derivatives of the differential equation. We illustrate the trick in terms of equation (l).Set z sin y and z, = cos y. Then = (cosvyp= :, z,=(-sinyAy = -z ,. calculate the higher Now given of a point (y, :) inFirst phase space time t, we show derivatives ; at time t. of all, we atcan calculate z,z, how fromtothe definitions and V from equation (1). Then, for i > 1, we recursively calculate ) z,0 = 1 - I-- (11=0 yj+2) =_-ay(+1 - I ) (If . ) I ,I+ b(cos )"1 using the differential equation and the product rule of Leibniz. The higher derivatives of y at time t are therefore known, so we can apply the Taylor method of arbitrary order with no symbolic calculation beforehand. A similar trick applies to the variational equation of (1). We applied the seventh-order Taylor method to follow solutions of both the differential equation and the variational equation. The latter is necessary for calculating a rigorous 6-pseudo-frame for the computer-generated trajectory. 972 T Sauer and J A Yorke 5. Examples As a first example, consider the Hinon map f(x,y) = (a - x2 + by, x) of the plane. For parameter values a = 1.4, b - 0.3, this map has an apparently chaotic orbit. Using the method described above, a computer-generated 6-pseudo-orbit with initial condition (0,0) and 6 = 10-' 4 was found to have a true orbit within 10- 7 for over one million iterates. Similar statements apply for other initial conditions, and for other parameter values. The pseudo-orbits generated by our computer satisfied lxi < 2, IYl < 2 in every case. In this range, the magnitudes of the first partial derivatives of f = (f1,f2) and the easily-computed inverse f-I = (g, g 2 ) are bounded above by 4. The magnitudes of the second partial derivatives are bounded by 2. Therefore we used in = 2, B = 4 in the hypotheses of theorem 3.3. This map was originally shadowed in [6], and similar results were reported. In that paper, a different approach was taken, which uses 96-b:t arithmetic (machine-epsilon = 10- 28) to verify shadowing of a 6-pseudo-orbit c Icuh' in 48-bit arithmetic, i.e. with 6 = 10- 14. The method of the present paper does not require such higher precision for this map. This point becomes especially relevant when systems are studied that are inherently more difficult to shadow. Consider the forced damped pendulum, which satisfies the . .,Jerentiai equation ; +av +siny = bcost. To achieve good shadowing results for this differential equation we needed to generate a 6-pseudo-trajectory with 6 = 10-18. We accomplish this by using a seventh-order one-step quadrature method with an explicit truncation error formula, using a step size of h = ir/1000. The implementation details of the quadrature method are given in section 4.5. The fact that the quadrature error formula is explicit is critical. Without it we could not get a rigorous bound on 6. For the forced damped pendulum with parameters a = 0.2 and b = 2.4, there is an apparfntly chaotic trajectory with initial conditions y(O) = y(0) = 0. Using theorem 3.3, we proved the existence of a true trajectory within 10- 9 of the computer-generated trajectory for time t ranging from 0 to 104ir. This trajectory corresponds to 107 time steps of the ODE solver. Again, there are similar results for other initial conditions, and other values of a and b The mapN f,, used ii rem 3.3 were the time-h maps of the non-autonomous differential equation, where = t/1000. The derivation of B for the forced damped 'duluri uses lemma 4.3. Write the pendulum equation as a first-order system. Then in lemma 4 3 is F(t,y 1,y2) = (' 2, -- sin v, - av, + b cos t). It is easy to check that the first and second partial derivatives of F with respect to Yi and Y2 are bounded in absolute value by 1, so that E, = E2 = 1. Lemma 4.3 says that B = max{2,e 2 h,4he6h}. Since h = 10- 8, we use B = 2,m = 2 in the hypotheses of theorem 3.3. Note that the inverse of a time-h map is a time-'mirnus h' map, so that the same B works for f '. 973 Rigorous verification of trajectories 6. Proof of theorem The convention in this section, as in the entire paper, is that all vector and matrix norms are 12 (Euclidean) norms. The norm of an m x m matrix A is defined interms of the vector norm, as follows: AI = VERI.lrl=l max IAvJ. It follows from the definitions that JAI = V1/ai(TA, where o(B) denotes the maximum absolute value of the eigenvalues of the symmetric matrix B. Lemma 61. If A is an m x m matrix whose entries are at most 6 in absolute value, 3 3 then JAI < m6. Proof. 141 < IAIF, where l.412 =-Lemma 6.2. IfW isan m x k matrix and x = WY,then < .,/ I - w IYI V1l- IWTi 3 I 3 3I ,=, a-. See [4]. -I I 'when the right-hand side exists. Proof. 2 Iyl= V,T WT Wy + IX1 - = yT(! - wTW)y + Ix12 IyI(I - WTW)yI + Ixl2 ~< 1 - wT1WVyI 2 + 1X12. < Proof o1 lemma 4.1. Let x E W. Then x = Wy, and Ax = A WY. By lemma 6.2, max and '%I=IXEW AxI = I II V max IAWYI 2 = max yERk lyI= max 1I1 ER'.!1 =1 r 117 .4Wv2 v.(AW)TAWy =a((AW)TAW). yERIIhl=1 Let (v,, Vk}__o be a 6-pseudo-frame for the matrix J, where < 3/(4k). Let An be. the matrix with columns {v,, . vk}. For each v E range Ao there is a w E range A, such that dt, - wI < 2vA6lvl. Lemma 6.3. 6 I Proof. Let v = 'J iv- = c, vo,; that is. v = Aoc. Define w = A c ==l cU,. Then wI = Zk ci(J I °- Ch) C Ik 6<vk< V1 I 26 v 'klvl 6vAOIv - where the last line follows from lemma 6.2. 0 U I 974 T Sauer and J A Yorke The next two lemmas refer to a C 2-map f which maps a convex subset S of R to R"'.. Define B, (respectively, B,) to be an upper bound on the magnitude of all first (respectively, second) partial derivatives of all component functions of f on S. Assume that x and x + It lie in S. i IU Lemma 6.4. 1. If(x + h) - f(x)l < m /BlhI. 2. 1Df (x +t)- Df (x) I < im/"mmB,Ih 1. Prool. For a scalar function g, Ig(x + It)- g(x)l < max 211 , I/hf vimax = I OX 1111OX) Applying this to each entry of the vector f, and the matrix Df, respectively, one gets the stated estimates. i 1 Lemma 6.5. m\/'mB2IhlI2 If(x + i)- J(x) - Df(x)hl <m 2 Proof. Each component g of f satisfies Ig(x +/t) -g(v) - Df (v)hl < from which the result follows easily. mh'B 2 0 Now assume that ... Snk}n() and u,, U}n=O are 6-pseudo- frames for the dynamical system on Rm,. where k + I = m. Let B be an upper bound for the magnitude of all entries of the J,,. Let S, and U, be the subspaces spanned by the moving frames and let S,, and U,, be the projections onto the subspaces such that S,, + U, = 1. U I Lemma 6 6. I. For u ( U,, (a) IS,,J,,ul < 21ml"26 iSn+1i uI, and (b) IS,,_lJnl1uI < 2m'i" B6IS,_JIluI. 2. For s E S,, (a) IUn1Jn_1si 2mn'26JU,+IIsi, and (b) iJL +,1J sl < 2m3 '2B61U,-, 1 sIi. I 975 Rigorous verification of trajectories N isa-pseudo-frame for {J.}N-' we use the fact that {u,,,..., Proof. To prove 1, Ifwe are given u E U., there isw E U,+, such that lJu - wl < 2vJ61ul, by lemma 6.3. IS,.+ 1Jul = IS.N+Jnu - S.+ w + Sn+w 1 = ISn+iJnu - Sn+IwI <2V 6S,,+,llu!. Secondly, we use the fact that {un,, . _ ut1}N is a mB-pseudo-frame for j-}N< 2v'imBIlul. Given u E Un, there isw E U,,-.. such that JJ 1u IS._-J',ul = ISn._IJ -,u - Sn._ Iv +S,,-_Il = ISnldJn_!u - S._tw[ < Sn_1llJA-_,u - w1 I4 . 2ma/2 B61Sn_,lluJ. . Part 2 issimilar. + R" is a C2-diffeomorphism. Let Proof oj theorem 3.3. For each 0 < n < N, f.:R" -(x.)= 0 be a 6-pseudo-orbit of the dynamical system {f.}. Define J,= Df(x.). Let BI (respectively, B,) be an upper bound for the absolute value of the first (respectively, second) partial derivatives of the component functions of f.and fi on the union of balls of radius 61/2 centred at x,for n = 0,..., N. Set B = max{2,BIB,). For positive unI,. ...unt} 0 ) be a 6-pseudo-frame (respectively, {= integers k+1 = m, let . SfkunI.u} spans RI for each n. for JJ;"' (respectively, {Jn}) such that {Sn| . 0 Unt}, and define n to be the angle Snk}, Un = span{un1 . Define Sn = span s. between S,, and Un. Let S. (respectively, Un) be the (unique) projection onto S. (respectively, U.) such that Sn + U, = . Recall that ISI = IUni = csc On. Let r. and in be numbers satisfying IJhy for y E Sn rnlyl for: E Un+ IJn': < tn1': Define x0 = x, y = :v= 0 for . = 0,1,2,..., and define Y', = Sn(f(x,_-d - x,' +J.-l,,-) = U,,f (x,+1 ) - x,+ J,T-,+n) . i1+1 += V1.+ z' for n = 1... N. (2) for n =0....N - 1. (3) for n = 0,..., N. (4) The sequence {x'} =0 isthe result of applying the refinement technique i times to the Define p by 6 = m/ 2 B2. Let CO = 0, Cn = ISnI+rnICn-I original pseudo-orbit ( for n > 0. Similarly, let D. =0,Dn = IUI +tDn+ I for n < N. 976 T Sauer and J A Yorke Assume that C,, < 6P- 112 and D,, < V-1/ 2 for n = 0,..., N. Then for Lemma 6.7. n=0,..., Nandi> O: + 1/2 < 2-'6C,, 4 2-'pp+ l/2 (a) IY'I < 2-'6D, < 2-i6 (b)lz'l 2 + 1 (c) lx'+1 - X"I < 21-,6p / 2 (d) Ix', ' - x°l < 46P+11 < 40B < Proof. Statement (d) follows from (c). Statements (a), (b) and (c) are proved by double induction on i and n. If i = 0: (a) 1Yl =0. and for n > 0, ly~~l1Sl6+r._i bmll < IS,,16 +r,,_16C,,_1 < C.b. (b) I-O< D.6, by reasoning similar to (a). < 5(C, + DN) < 626P-I/z = 26 + tz.l (c) Ix"- Vol '<ly.°I + 1/2 . Now we assume that (a) holds for i - 1, and prove it for case i. We induct on n. The n = 0 case is trivial, since lyll = 0. Assume that (a) holds for the case i, n - I and prove it for i,n: xn YA -_l f,.(x' -n +J-I1) +f Y. =" = S.- = S.f.- 1(x .- 1) -f..-I(x - 1- - (x+.-I)-i -) 1 - Df-(x'-,)(X' 1 ') - x Sn-l,,-In-I where we have used the facts that S,(:z'') = 0 andI n +A - , 1(4 1 = by the definition of y' '-- We will bound the Euclidean norm of each of the four terms of the last sum separately. xj- l1 _)- A. ln-1 t S"Y ( _ -,, " - I( Df n,,- 1,x 'n-I ' - xI-11))I< -1 n- -2-B I n'3/2" m3/ 2 n-1 - 32 2 -2 4 ISI2- ' 32B,2'm / 6 p -2 -, 4< ISJI2 since m > 2, B >S2 implies that 6-2P > m5B4 = 32Bm31 2 (m/ 12B3)/32 > 32Bm3 /2. I 977 Rigorous verification of trajectories * 2. IS.(Dfn. 1(x'-1) - i+Z!:,)I -J)(Y nz-1~ J.-t)- IS"Im3/ 2B2zx-1t X Ixi-jI i-:I -< - n-1-+Zn-,1 2 < ISnIm3 B246P+/ 326B62pm IS.162' 3<ISI62 2-'6" 2 3/ 2 - '- 2 since 6-2P > 32B2m/ 2. * 3. n-1Slv 1 IS.J._i-._t[, 3 <26P-1/2 v, 62-'+16p+1/2 3 = 16624 v <2-'-6 2 26 since 6- 2P > 16v/4i. 1 4. 1J-11-- 1 IUAJ-14-11 ISA-14 < r._tlyy,_- I + 2m3/ 2B6UI2-U'4 32 > 8m B. since 6-2P < r, -tIYn-II +86 ,ly' 2 i/2 3 Pm /2B 2-o-26 26 + 2 _I Adding up the four bounds we have "1In'1 Iy.I 4< 41S,162 - ,- 2 +r I < S.1b2-' +r,_12-'6C.,_1 =C,62-'. This proves (a) for the case i. The proof of (b) is similar, except that we use descending induction on n. The n = N initial case of (b) is trivial, since I.I I = 0. Finally, (c) is a 0 simple consequence of (a) and (b). Lemma 6.7 shows that for each n, y, -- Ozn - 0, and that {x'},' is a Cauchy sequence. Therefore xn converges to some wn in R'. The sequence Nw~}' 0 is the limit of the refinement process applied to (x}_.0. Moreover, lemma 6.7 (d) implies that Iw - xI < 61/2 We will complete this section by showing that f.(w.) = wn+1 for n = 0,...,N - 1, so that the {w.} represents a true shadowing orbit of {x.}. According to equations (2) and (3), S+ 1(fn(wn) - wn d = 0 (5) T Sauer and J A Yorke 978 and U"U. "'(w ,+ ) - w) = 0 (6) for n = 0... N - 1. Furthermore, we have If"(wO) - w,+1< lfn(w n) - fn(x,)l + If (xn) - x,+Il + Ix,.+ - W,+11 < mBlw - x,1 l +6 +46 p "2 < (mB, + )46P+'/ 2 +6 /: 6 6i2 +m5/2B2 (6'/ + 4(mB+ )) 61/<< 14(4+1,) 78=o=+ 25/z222--T '' < 61/2, A similar calculation shows that if; - w I < 61/2. I Secondly, corollary to this calculation are the facts that IfI,(w ,) - ',,+,I < 6"i / and R'-IOw,.+i0 - X,, I < 6 1i 2 Thereibre J.,,(w,) and f,,'(wn+i) are within the balls around x,,,x,,, respectively, for which the lemmas 6.4-6.5 concerning growth bounds on f. apply. Lemma 6.8. The sequence {wv}' o is an orbit of the dynamical system {fj}. That is, = w,+, for n = 0,..., N - !. f,,w$) Proof Equation (6) says that U,(w. f;'(w +i)) =0. Since S.+U. = 1, w.-f;I(w,+ 1 ) belongs to the subspace S. We evaluate IS.+,(J,(w - f-'(w,+1 )))l in two ways. First, using the fact that S,, + U,,+l = I, i)1 ->S, .fC-i(wn., Iw,, > I,(w,, ';7i(w~l+i))I > - IU~+ 1Jg 1 (wn - f'(w~l+))lI w, - f,-i(w.i)I - 2m 3i 2B6tU,,+Illw,, -f f'I(w+ 1 ) 1 -- 2Bm321vn+u16) lw(mB where the last inequality uses lemma 6.6. '(w,+I)I I I 979 Rigorous verification of trajectories I On the other hand, = f=(w.) - w.+, - (f.(wn) - w.+, - Df.(w.)(w -f J.(w. -y-f'(w.t '(w.+l)) + (Jn - Df,(wn))(wn - f !(w,+l)). 3 Since S,+t(f.nw) wn.+') = 0 by - equation (5), we have IS"+ I(V . - f,-' (w,+ 0)))1 < 4m/ 2BlIS"+IIIw" - f'I(w.+1)I2 + m31 B2 IS,.+jIx - w"Iw, =m'/2B,1S,,+,IlIlw,.- f- j (w,,+1) + 46P+1/2)w,-f-Iw+) Putting the two inequalities together. we have that either w case we are done. or else -f = f.-I(w,.,. ), (w,+1) in which 1 t -~2Bm3/21U 1+15 < m3/2BiS.+1l(1lw,, _f,-(w,,+1)I +46P+1/2 M1 I8 I **< Bm3 /26 (26"/ ' ) + 6P where we use the bound 6P-1/2 on IS,+, and IU,,+,I. This inequality implies that 3 6- ' < MS/2B2 -+1 +7,OM+ 4 , ) < m 51 2 B2 I 2B2 . Therefore since m > 2,B > 2. This contradicts the assumption 6-P = mS5, 0 , N - I. +) for n =0. w, =f'(wn References (I] Anosov D V 1967 Geodesic flows and closed Riemannian manifolds with negative curvature Proc. Steklov Inst. Math. 90 121 Bowen R 1975 w-limit sets for Axiom A ditffeomorphisms J. Diff. Eq. 18 333-9 131 Chow S-N and Palmer K 1990 On the numerical computation of orbits of dynamical systems: the one-dimensional case Preprint (41 Golub G and Van Loan C 1989 Matrix Computations 2nd edn (Baltimore. MD The Johns Hopkins (51 (6 [71 University Press) Grebogi C, Hammel S and Yorke J 1987 Do numerical orbits of chaotic dynamical processes represent true orbits? J. Complexity 3 (1987) 136-45 Grebogi C, Hammel S and Yorke J 1988 Numerical orbits of chaotic processes represent true orbits Bull. Am. Math. Soc. 19 465-70 Hammel S 1990 A noise reduction method for chaotic systems Phys Lett. 148A 421-8 Hartman P 1964 Ordinary Differential Equations (New York. Wiley) Sauer T and Yorke J 1990 Shadowing trajectories in dynamical systems Computer Aided Proofs in Analysis ed K Meyer and D Schmidt (Berlin: Spnnger) pp 229-34 [8] [91 I Ergod. Th. & Dynam. Sys. (1991), 11, 189-208 Printed in Great Britain 1 * 3 Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic setst HELENA E. NUSSEt* AND JAMES A. YORKE*§ University of Mavland, College Park Maryiand 20742, USA I (Received I February 1989 and revised October 1989) Abstract. In dynamical systems examples are common in waich there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such regions. In such dynamical systems one will observe chaotic transients. An important problem isthe 'Dynamical Restraint Problem': given a region that contains a chaotic set but contains no attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time. We present two procedures ('PIM triple procedures') for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time. 1. Introduction Studying dynamical systems, one often observes transient chaotic behaviour, apparently due to the presence of horseshoes. For example, for suitably chosen parameter values, the H6non map has an attracting period orbit with period 5 and also a non-attracting chaotic set, and one observes that the duration of the transient chaotic behaviour of many trajectories is rather short before they settle down on the period 5 attractor. Other famous examples with chaotic transients are: the Hinon map for large parameter values where almost all trajectories go to infinity and there is a bounded non-attracting invariant set; the forced damped pendulum; and the Lorenz equations for values of the Rayleigh number below the standard values that have a chaotic attractor. Transient chaos is also present whenever there is a fractal 3 boundary separating the basins of two or more attractors. Let M be a smooth n-dimensional manifold without boundary, and let F be a C3.diffeomorphism from M to itself. We denote by p the distance function on M. t Research in pan supported by AFOSR, by DARPA under the Applied & Computational Mathematics Program, and the Netherlands Organization for the Advancement of Pure Research (N.W.O.). t Institute for Physical Science and Technology, University of Maryland. * Rijksuniversiteat Groningen, Fac. Economische Wetenschappen, WSN.gebouw, Postbus 800, NL-9700 AV Groningen, The Netherlands. § Department of Mathematics, University of Maryland. 190 H. E. Nusse and J. A. Yorke A region R isan open and bounded set in M. We say a region R is a transientregion if it contains no attractor. We will be studying these regions in cases where the trajectory through almost every initial point eventually leaves the region. We investigate special trajectories that remain in such a transient region for all positive time. For example, the horseshoe is usually pictured mapping a rectangle to a horseshoe shape; the rectangle is a transient region. The great majority of the trajectories of the horseshoe map will leave the region after a few iterates. We are looking for numerical procedures for finding chaotic trajectories that stay in the transient region as long as we wish to compute them for t - 0. The main problem that we would like to address is: The dynamic restraint problem. Find a (nonperiodic) orbit numerically that remains in a specified transient region for an arbitrarily long period of time. The above problem explicitly concerns numerical (i.e. computer) procedures of finite precision. It leads to the following problem where it is assumed all computation. can be made exactly. The static restraint problem. Find an initial point whose orbit stays in a specified transient region for an arbitrarily long period of time. We will establish a procedure (the PIM triple procedure) for finding points whose orbits will stay in specified regions in M for dynamical systems in ideal cases that are uniformly saddle-hyperbolic systems. The unstable manifold of each nonwandering point in the transient region is assumed to be one dimensional. Let R be a transient region for F. The stable set S(R) of F is (xE R: F'(x)E R for n =0, 1,2.... }; the unstable set U(R) of F is tx rR: F-"(x)E R for n = 0, 1,2.... ). The set of points x for which F"(.) is in R for all integers n is called the invariant set lnv (R) of F in R, that is, lnv (R)= S(R) n U(R). A component of S(R) (resp., U(R)), which contains a point of lnv (R) is called a stable (resp., unstable) segment. We call Inv (R) a chaotic saddle when it includes a Cantor set. We assume that for the transient region ' the set Inv (R) is nonempty. We will refer to R\S(R), the complement of the stable set S(R) in the transient region R, as the transient set. We will say that a point p in S(R) is accessible from the transient set R\S(R) if there is a continuous curve K ending at p so that K\(p) is the transient set R\S(R). For uses in dynamics, see [GOY] and (AY]. We wouiu like to address the following problem: Accessible static restraint problem. Given a segment J that intersects the stable set S(R) transversally, describe a procedure for finding a point (in J n S(R)) which is ible (from R\S(R)). ac %c will establish a procedure (the Accessible PIM triple procedure) for finding such accessible points in M for the same class of dynamical systems as above. Both our procedures are based on our presumed ability to specify an initial point p and compute the time TR(p) its trajectory takes to escape from R. In the PIM (Proper Interior Maximum) triple procedure, we seek out triples of points a, c, and b on a curve segment with c the 'interior' point, that is, c is between a and b.The I Chaotic saddle hyperbolic sets 191 triples are selected with an 'interior maximum' of the escape time, which means TR(c) > TR(a) and TR(c) > TR(b). We then look for new triples that lie in the a, b segment but are closer together and so are 'proper'. The most challenging cases are those in which the average escape tin. is short so that the transient trajectories of typical points in R do not come close to the unstable chaotic set. The organisation of the paper is as follows. In § 2 we present the PIM triple procedure and the Accessible PIM triple procedure; the main results for the validity of these procedures for hyperbolic systems are stated precisely in § 3. § 4 is devoted to the proofs of the results in § 3. In § 5, we will discuss the associated numerical procedures (including the shadowing of the numerical orbits by real orbits of the dynamical system). Finally in § 6, we will explain why the PIM triple methods also can be used for basin boundaries, we will describe how the results carry over to higher dimensional systems; and we also will argue that it is sufficient to assume that F isof class C2. 1 3 2. The procedures Let the manifold M, the diffeomorphism F, and the transient region R be as in the introduction. The escape rime TR(x) of a point x in M for R is defined by Ts, W f=min (n =0: F"(x)z R} T lo ifF"(x)ER for alln=0. For the example of the horseshoe map, the escape time function T, has the following properties: (I) TR(x) = cofor x on a Cantor set of stable segments; (2) if a, c, and b are three points on a segment L of an unstable segment J so that: (i)c is between a and b and (ii) TR(c)> max {TR(a), TR(b)}, then the segment [a, b] c J from a to b intersects the stable set S(R. These properties pla) a crucial role in the PIM triple procedures, and lead to the following definitions. Let J be an unstable segment in R. Then J is homeomorphic to an interval, and we may assume it has the ordering of an interval. The notation (a, c, b) for a triple means that a, c, and b lie on J and c is between a and b. Let L c J bt a 'segment', that is, a connected subset of J. Assume L intersects the stable set S(R) transversally, and let (a, c, b) be a triple on L. Since L is homeomorphic to an interval it has an ordering. We assume that the ordering on J (and hence on L) is such that a < c < b; and for points x and *y in J we write [x,y]s for the segment on J joining x and y. The triple (a, c, b) is called an Interior Maximum triple if TR(c)> max {TR(a), TR(b)}; and (a, c,b) is called a Proper Interior Maximum (PIM) triple on L, if (a, c,b) is an Interior Maximum triple and at least one of the points a and b is in t,,e interior of L. For each e > 0, an e-refinement of {a, b} (w.r.t. J) is a finite set of points a = go< gl< ...< gN= b in [a, b]s such that (E12). p(Ca, b]s) <-p([gk, gk,]S E p([a,b]s) H. E Nusse and . A Yorke 192 for all k 0 s< k <- N- 1. and an E-refinemem of (a. c. b) is an e-refinement of fa, b} as above so that c=g, for some k. I -k!-N-I The outline of the PIM triple procedure is the following. Let R be an appropriately chosen transient region for F and let L be a segment on an unstable segmeni J (intersectinf the stable set transversally). Let E>0 be sufficiently small. Given a PIM triple (a., c., b,) in L. starting with n =0. choose some -refinement P. of the triple (a.. c., b,) in [a.. bIi, select any three not necessarily consecutive points from P. which constitute a new PIM triple (a,.,, c,.,, b..,) on [a.. b.],. The new triple must be "proper; proper here means [a.,. b,., ], is a proper subset of [a,., b. ],. The condition guaranteeing the existence of such a PIM triple when E is sufficiently smail. will be described in § 3. Note that. according the definition of PIM triple. p([a..,. b...,b) is (I -0.5e)p([a.. b.], ) Thus the nested sequence of the intervals {[a,, b.].)... converges to a point which we will call a PIM limit poain. The E above can be chosen small enough that it is independent of n. We will show that under reasonable conditions the orbit of the PIM limit point stays in the transient region X. The choice of the PIM triple i3 typically not unique and different choices will result in different PIM limit poin's. This 'static' problem's solution is not directly implementable on a computer because computations are made with finite precision. but it lead to a practical solution of the dyna.;,ic restraint problem as discussed in § 5. The idea of the Accessible PIM triple procedure is like the PIM triple procedure except that the PIM triples (a,, c,. b,) are selected more precisely so that [a., a,.,]s does not intersect the stable set S(R) for all n =-N for some NE N. The difficulty here is that we o-ily compute the escape times of the grid points and yet we must be sure that [a,, a,,,], contains no points of S( R). We must guarantee the procedure will succeed if E> 0 is small enough, where e is fixed, depending only on the diffeomorphism and region. Our objective is to describe the Accessible PIM triple procedure that selects in a unique way a nested sequence of PIM triple intervals on J which leads to an accessible point in S(R) on J. The accessible point p in J S(R) that we will find, will be accessible using the curve [r, p]s for some r in J,so we say p will be 'accessed from the left', that is from the side containing r. We could alternatively have chosen to approach from the right and we would expect to find a different point. Given an e/3-refinement P. = {x,: 0- i s N(e)} on J of a PIM triple (a,, c., b.) in J with a. = xo< x, <-. <xN,) = b,. Assuming that P. includes a PIM triple. c,,, b, ) in P, in the following way: then we choose the next PIM triple (a,, () Select h,+, to be the leftmost point in P. such that it is the right point of a PIM triple in P.: (2) Select c,, to be the adjacent point to the left of b,, in P.; (3) The systematic choice of a,+, in P. is the following: Let M. be the minimum value of {TR(x,): x, E P,,x, < c,, ).We write: a +, is the rightmost point of {x, E P.: x, < c,,, TR(x,) = Mn}; a,+, is the adjacent point to the right of ao,. in P,; I I 3 Chaotic saddle hvperbolic sets 193 3Case a' ., is the rightmost point of Ix,E P.: x-c.-,, TR(Y)= T(a;.,)}. ii). If either M. < T,(a.) or M. > min { TR(x,): x, e P.), then choose a.., = oa.., ; otherwise, Case (ii). If M. = TR(a.) and P. is not an c-refinement of (a., c..,, b..,), then choose a.., = a.; otherwise, Case (iii). If M.= TR,.a.) and P. is an E-refinement of (a., c..,. b..,), and if a., > a. or a.., = c,.,, then choose a,,.. = a,; otherwise, 3 Case liv). If M., TR(a.) and P. is an c-refinement of (a., c..,, be.,), and if a..,=a. and a', < c..,, then choose a,,.. = a,., Repeatedly applying the Accessible PIM triple procedure leads to an accessible point on S(R). To understand rule 3. notice that rules I and 2 imply that the graph of TR is rather simple on P. n [a., c.., Jnamely., TR is monotonic increasing on P. between ao., and c.,,, and TR is non-increasing on P. between a. and a.., .These properties follow from the fact that b.,, was chosen as far left as possible. We will show that after the first few iterates TR(a.) =min { T(x,): x,E P}. 3. Results In § 2 we presented the idea of the procedure for finding a point whose orbit stays in the transient region. In this description, we assumed that there exists an E> 0 for which every E-refinement of a PIM triple includes a new PIM triple. Furthermore, the associated curve segment from a.., to b,,, has a length at most (I - L/2) times the length of the previous one (from a. to b). We will justify these concepts. Let the manifold M and diffeomorphism F be as in the introduction. A subset A of M is hyperbolic if it is closed and F-invariant and the tangent bundle TM splits into dF-invariant subbundles E' and E" on which dF is uniformly contracting and uniformly expanding respectively. A hyperbolic set A,is called a saddle-hyperbolic set if dim F' - 1 and dim E" - I. We will call a region R a saddle-hyperbolic transient region if R satisfies all the following conditions: (AI) R is a transient region; (A2) Hyperbolicity property: nv (R) is a nonempty saddle-hyperbolic set; (A3) Boundar, property: U(R)cn8R is mapped outside the closure of R; (A4) Intersection property': each nontrivial component y of U(R) is an unstable segment, that is, y intersects Inv (R); note that such a segment y must intersect S(R) transversally. We assume throughout that dim E" = I. For the sake of simplicity, we Issume that n = 2; the more difficult case n a 3 will be discussed in § 6. For a saddle-hyperbolic transient region R and E>0, the properties (A1) and (A2) imply that the escape time of almos' every point on an unstable segment is finite. (A result due to Bowen and Ruelle [BR] shows that S(R) has Lebesgue measure zero.) Hence, one may assume that such a refinement does not intersect the stable set S(R). I 194 H. E Nusse and J. A. Yorke If R is a saddle-hyperbolic transient region, then the escape time map T restricted to an unstable segment J c U(R) has the following two properties, which follow from Proposition 1 and the T-Jump Lemma below. (i) All the points in a chosen segment [a, b]., on J will escape from R if and only if no e-refinement of {a. b} includes a PIM triple; (ii) TR is locally constant on an open subset of full measure of J. and if TR(x) < 00 and x is a point of discontinuity of Tl,, then liminfTR(y)=TR(x) and limsup TR(y)= TR(x)+l. Application. The objective of the paper is to present procedures which enable us to obtain numerical trajectories lying on chaotic saddles, and to justify that these procedures work in ideal cases. The examples of interest will rarely satisfy all our hypotheses, and yet we observe that frequently we can successfully use the procedures to obtain pictures of Inv (R) by plotting the numerical trajectory. Consider the following example. Let t..- difteomorphism F acting on the plane be given by F(x, y) = (A-x 2 + M. Jx). It is well known that the map F is equivalent under a linear change of variables with the Hinon map. We choose the parameter values M = 0.3, and A = 3 in figure I(a), A =4.2 in figure I(b) (and figure 2) and A =2.0 in figure 1(c). Then a result due to Devaney and Nitecki [DN] implies that B = {(x, y): -3 < x < 3, -3 < y < 3} includes the nonwandering set of F, so we select B for the transient region. When A = 4.2, the nonwandering set is a uniformly hyperbolic chaotic saddle. We start the numerical procedure with the horizontal line segment with y = I extending from the left side of B to the right side. By using the PIM triple procedure' we obtain a numerical trajectory consisting of tiny intervals. The result is presented in figure 1. When A = 4.2, the region B is a saddle-hyperbolic transient region: the results due to Devaney and Nitecki [DN] imply that B satisfies the conditions AI)-(A4). When A = 3 we do not know if condition (A2) will hold, and for A = 2.0 we have a non-fully developed horseshoe. In figure 2 we present the sets U(B) and S( B) for A = 4.2 (the chaotic saddle is the intersection S(B) n U(B)), and the accessible fixed point on the chaotic saddle is indicated by an arrow. The reader is referred to [NY] for other applications such as the Lorenz equations, the pulsed rotor map, and the forced pendulum equation. Rather than state one or two theorems the results seem best stated a progression of ideas: (1) PIM triples exist, (2) refinement of PIM triples incluac PIM triples, and (3) the resulting sequence of PIM triples conveige to a desirable point. The special case of accessible PIM triple sequences must be discussed separately. From now on, we will assume that R is a saddle-hyperbolic transient region for F with dim E" = 1, and that J c U(R) denotes an unstable segment. That implies that both ends of J are in the boundary of the transient region R. We know by the Intersection assumption that J intersects the stable set S(R). Clearly, this property Mw 195 Chaotic saddle hyperbolic sets .1 5 , • -- (a) z (a) 1 1 *.'. It , F [ ... (b) FIGURE 1. (a) Numencal trajectory obtained by the PIM tnple procedure for the Hinon map in the transient region -3 < x, y < 3, and parameter values A = 3, M = 0.3 (b) Numencal trajectory obtained procedure for the Henon map in the transient region -3 < x, y < 3, and parameter by the PIM triple values A =4 2, M =0.3. (c) Numencal trajectory obtained by the PIM triple procedure for the HWnon map in the transient region -3 < x, y < 3, and paiameter values A = 2 0, M = 0 3. H. E. Nusse and J. A. Yorke 196 [" I FIGURE i-continued. will not hold for each subsegment L of J. since J n~S(R) is nowhere dense in J andI one can choose the segment L lying entirely in the complement of J n S(R). Our first restilt 'PIM Existence Proposition* characterizes the segments intersecting the stable set S(R).I PROPOSITION 1, (PIM existence.) Let L statements are eq~uivalent: = [a, b]j be a segment in 1. The following f" (i) there exists E > 0 such that everv r-refinement of (a, b} includes a PIMI triple;I (fi) L contains a point of Inv (R) in its interior. In the PIM Existence Proposition the segment L can be chosen so that it intersects " R) only at points extremely close to one of the end points of L and so E must1 extremely small, so e depends on the choice of L. However, the PIM Refinement Proposition, stated below, shows that a single e (depending on F and R) can be used, once we have found our first P1 M triple. One might expect that our assumptionsI of uniform hyperbolicity would imply that the uniformity of E would be an easy corollary. In fact, the existence of an E for each PIM triple is much easier than E can be chosen uniformly, and this uniformity is essential for the PIM triple procedures. In principle it can be difficult to find the first PIM triple if the initial interval L is chosen badly. PROPOSITION 2. (PIM refinement.) There exists E> 0 (depending on F and R) such1 I Chaotic saddle hyperbolic sets 197 FIGURE 2. The stable and the unstable manifold for the fixed point at approx. (I 729, 1 729) for the Hinon map in the transient region -3 < x, ' < 3, and parameter values A = 4.2, M = 0.3 The accessible fixed point on the chaotic saddle is indicated by an arrow. that there is a PIM triple in each e-refinement in J of each Interior Maximum triple in J.for every unstable segment J c U( R). The next result deals with the convergence of the sequence of nested PIM triple segments [a,+,, b,+,]j c [an, b.]j on J, in other words, the PIM triple procedure is valid. A sequence of PIM triples {(a., cn, b.)}.n 0 on J is called a PIM triple sequence if (a, , c. , b.,,) is in an e-refinement of the Interior Maximum triple (a., C,,, bn) for all n. We say {(an, cn, b.)}..-, is the accessible PIM triple sequence if (an, C., b.) is selected using the Accessible PIM triple procedure for all n. 3. (PIM convergence.) Let E > 0 be as in Proposition 2. Every sequence of nested segments {[an,, .]j},,, that is associated with the PIM triple sequence {(a., c,, bn)}.n,, on J, converges to a point on S(R). PROPOSITION The next result is the key in proving that the 'Accessible PIM triple procedure' is valid. PROPOSITION 4 (Accessible PIM Refinement.) Let E > 0 be as in Proposition 2. Let {(an, cn, bn)) .0 be an Accessible PIM triple sequence on J. Then there exists integer N >-0 such that [an, an,,+]j does not intersect S(R) for every n : N. Recall that a nested sequence of PIM triple intervals obtained from e-refinements will converge to a PIM limit point on S(R). Note that the PIM limit point of the PIM triple intervals associated with PIM triples in Proposition 4 is an accessible 198 H. E. Nusse and J. A. Yorke point on S(R). The next result implies that the Accessible PIM triple procedure is valid. 5. (Accessible PIM convergence.) Let e > 0 be as in Proposition 2. If the PIM triple sequence {(a., c., b,)} . 0 in Proposition 3 is an accessible PIM triple sequence, then the sequence of nested segments {[a,. b.]j},o on J, converges to an accessible point on S(R). PROPOSITION I 4. Proofs 4.1. Preliminaries We assume that R is a saddle-hyperbolic region for the diffeomorphism F By Smale's 'Spectral Decomposition Theorem' [S] we know that we can decompose the nonwandering set fi into a finite collection of disjoint closed invariant subsets and on each of these subsets F has a dense orbit; these maximal invariant subsets of 0i appearing in the decomposition are called the basic sets (see e.g. [Ni] and [GH] for the definitions and several properties of uniformly hyperbolic systems). From now on, let r denote basic set of F From the definition of lnv (R) it follows immediately that either ra Inv (R) or r Inv (R) = 0. This implies that we can decompose lnv (R) into finitely many basic sets. Note that r n lnv (R) = 0' does not imply 'n R = 0', and r n R #0 ' does not imply 'rnlnv (R)* 0'. Recall that for Z El the stable manifold WS(=) (resp. unstable manifold W"(z)) of z is the set of points x forwhich p(F"(z), F"(x)) - 0 (resp. p(F"(z), F-"(x)) - 0) as n-oX; the local stable manifold Wlo,(z) (resp. the local unstable manifold Wl'o,(=)) of z of size)3 is the set of points x in W(z) (resp. W(z)) so that p(F"(z), F"(x)) -3 (resp. p(F".z), F-"(x)) - 1) for all integers n a 0, where 0 > 0. When the stable or unstable manifold is a curve, we writes orWl(z) and W;oc(z) for the u. where a is either two components of W1o0(z)\{z), We will call r a trivial basic set if r consists of one periodic orbit, and we call r a nontrivial basic set if r includes more than one periodic orbit. Assume that r is nontrivial, we call U periodic if there exists m EN such that F'"has no dense orbit on r, and we call r nonperiodic if it is not periodic. The following results 4.1, 4.2, and 4.4 are reformulated from [NP] and [PT]. 4.1. There existsfinite sets P, P', and P" of periodicpoints, P = P' U P", such that for all x lnv (R): (1) If x is not a limit point of both W" 3 (x) n!Q and W"c(x) n fl, then x is in Ws(p) p E P". for some (2) If x is not a limit point of both W o(x) n f1 and W j-(x) rn fl, then x E W"(p) for some p E P'. PROPOSITION Proof For a proof, see Newhouse and Pali. [NP]. 4.2. Let P' and P" be as in Proposition 4.1. Let U be a nontrivial nonperiodic basic set in Inv (R). Then there exist finitely many disjoint regions R, being diffeomorphic images of the square B = [-l, lx [-1, 1], say R,= g,(B), 1 - i:5 PROPOSITION Nfor some NEN, such that: (1) rUnR, 0 0 for all i; (2) rc U , R.; (3) F(05R,)c = g,({(x, y): Ix= l,-l-y1})resp. U, 1 a,RandF-'(aR,)cU.R,.1 aRwhere aR, I I Chaotic saddle hyperbolic sets = 8=R,=g,({(x,y): -1- x-5 1, yI the unstable set W"(P'). 199 1}) are segments in the stable set W 5(P") resp. Proof For a proof, see Palis and Takens [PT]. Remark. The intersection of r with the union of the regions in Proposition 4.2 is a Markov partition for F, see Bowen [B] for the notion of Markov partition. PROPOSrTON 4.3. Let P" be as in Proposition 4.1. Then we have x E S(R) is accessible ifand only if x e W'(p) for some p E P'. Proof Apply the Propositions 4.1 and 4.2. From now on, let z e F anv (R) be fixed, and let I"c W"(z) be a segment such that I" crosses each region Rk at least once, where Rk, 1 - k -sN, is as in Proposition 4.2. Palis and Takens [PT] have shown that there exist finitely many disjoint regions denoted RM(IU) in UN., R, that have the same properties as the R,'s such that I' crosses each R(I") exactly once, 1 t<-jsN(Iu), for some N(Iu) EN. Therefore, we will assume that I' crosses each region R, from Proposition 4.2 precisely once. There exist a CI' a stable foliation l' on a neighborhood U, of F for some a > 0, and it is no restriction to assume that every region R, is contained in U1., 1: i:5 N; see [PT]. Let 7:R - Wu(z) be a C' parametrization, and define a projection 7r:F-* U.NI R,r I' by taking in each region R,, 1 si-5 N, the projection along the local stable manifolds into the intersection I" with that region. This projection can be extended from r to the union of the regions R,, by projecting along the leaves of the foliation ?'.This extension will also be denoted by 7r. We obtain (see [PT]) the following result that says that for some iterate Al, the map F can be viewed as expansive along unstable segments. PROPOSITION 4.4. There exist a positive integer M and a C' ' map ,P:U ' . T _(JI R,) -R defined by (x) = T-- o oFo(x) such that Iq'(x)l > 1,for some a >0. From now on, let I,,., N be N disjoint compact intervals on the real line, and we write Y=.)JN- I, Let f: Y- R be a C ~'map, for some a>0, with the following properties: (1) f is C'*+ on an open neighborhood U of Y; (2) Y" Interior(f(Y)); (3) there exists numbers A,> 1 such that If'(x)I -A, for every x E1, 1-j- N; (4) the transition matrix A ,. is primitive, that is, there is an integer Q>0 so that all the entries of (A y.j)Q are positive; where A¢y. is defined by A,, I(j, m)---I if f(lj) nI 0, and Ay.j(j, m) = 0 if f(Il) n , = 0, 1 75j, m - N. Note that condition (2) implies that either 1,rf(Im) = 0 or 1,n Int (f(Im)) = 1, for all l -j,m - N. The escape time Ty(x) of x E Y underf is the minimum value n with the property f (x) Isnot in Y We define for every integer k -1: Ak={XE Y: Ty(x)>-k} Dk = xE Y: T,(x)=k}. H. E. Nusse and J. A. Yorke 200 In particular, A, = Y. Hence, for each integer k a 1 we have Ak+, is the set of points in Ak whose escape time from Y is at least k; hence Ak.1 is the set of points in Y that stay in Y under f k. The points in Y which will stay in Y under all iterates will be denoted by A ={x E Y: Ty(x) = o} For every integer k a I we have: (a) Ak = Ak+, u Dk ; (b) Y=Ak+,Ul k., D, that is, Y is the union of the set of points Ak+, whose escape time from Y is at least k + 1, and the set of points D, whose escape time from Y is j, where Il--j - k. Denote the length of an interval L by LI. 8 GEOMETRIC LEMMA 1.There exists f > 0 such thatj r every integer k : 1,the following holds: (i) Every component of Ak contains components of Dk and of Ak+,; (i For each component D of Dk n A, one has IDI/IAI - 3f, and each component U of Ak+, - A, satisfies IUI/IAI 8, with A an arbitrarily chosen component of Ak. Proof of the Geometric Lemma L For each integer i 1, we write R, for the sum of the entries on the ith row of Ayj, 1:- i:5 N. Assumption (4) implies R, is at least I for all i, and the sum of the R,'s is greater than N. Proof of (i). Let k a I be a given integer. First, we assume k = 1. Let L be a given component of At = Y, say L = I, for some j, I sj s N., The assumptions (1)-(4) imply f(L) contains R,+1 components of D1. Since L = {x E L: Tv(x) a 2}u {xE L: Ty(x) = 1), we have that L contains R, components of A,. Now we assume k > 1. Let A be a given component of Ak. By the definition of Ak and the assumptions on J, we havej'-'(A) is a component of A,, sayJ'-'(A) = 1, for some j, I !-j - N. Therefore, A contains R, + I components of Dk and R, components of Ak+. Proof of (ii). We are looking for 8 > 0 such that for each integer k - 1 and for each component A in Ak, we have A(D)/A(A) - 35,and A(U)/IA (A):>- 6f, for each component D of Dk o A, and each component U of Ak,+ n A. From (i) and the assumptions on f we obtain that for each k 2 1, the number of components of Ak and that of Dk is finite. Let, for each integer k !1, N(Ak) be the number of components of Ak, and let N(Dk) be the number of components of Dk. We write, tor each k - 1, the sets Ak and Dk as the union of their components in the following way: N (A,) Ak =U N(V~ A,,, Dk = Dk ,, For each k - I and each component A in Ak, we define IVI/IAI, Sk(A) = min V where the minimum is taken over all components V of the sets = m i k(A), we define A Dk and Ak+,; and Chaotic saddle hyperbolic sets 201 where the minimum is taken over all components A of the set Ak. We are done if there exists 8 f > 0 so that Sk a 8f for all k Let k > I be a given integer. Let A be a given component of Ak, and let D be an arbitrary component of either Dk or A, such that A includes D. From the foregoing, we can fix an integer n(k), I :s n(k)<- N(Ak), such that A = Ak.wk,, and an integer re(k), 1 <-mik) 5 N(Dk) if D is a component of Elk, and 1!-5 r(k)!E- N(Ak ,) if D is a component of Ak~,, such that D= Set for each integer i, 2si:- k: Dk.,,,k. Applying the mean value theorem, we tan find for every integer i, 2 ! i s k. real numbers a, in A,.,, and d, in D,,,,, such that Ij'(a,)l .A, ,,, ' iA,-1,11-1 J, If (d,)i IO....... = ID_1,,,11I. This leads to: ID,. ,A,I/IAA,.i,,,I= (iI If a,)/J'(ld,)I)' (ID,.m,,,/IA,.e.,,. (I) From now on, we can mimick the proof of Lemma 5.5 in [Nul, and we obtain: lim H If(a,)iJf(d,)I>0. (2) The results (1) and (2) imply that there exists y .0 such that ID,.,,I/fAnIAklk V. Therefore, , :- y for each k 2. We conclude: IDI/IAI >-8, for every component A in Ak, for every component D with Dc A, where D is either a component of Dk or D is a component of Ak,, for 6, = min {5, y}. This completes the proof of the Geometric Lemma I. 4.2. Prools of the PIM propositions Let J c U(R) denote an unstable segment. Recall that both end points are on the boundary of the transient region R, and that J intersects the stable set S(R). We define for every integer k : 1: Ak(J) ={xEJ: TR(x) -k} Dk(J) ={xEJ: TR(x) = k}. In particular, .4,(J) = J. Hence, for each integer k 2t I we have Ak.,(J) is the set of points in A(J) whose escape time from R is at least k+ 1; hence, Ak.,(J) is the set of points in J that stay in R under F'. The points in J which will stay in R under all iterates will be denoted by A,-(J). For every integer k a I we have: A (a) A(J)=AkA+(J)UDk(J) (b) J=AkA,(J)u U D,(J), that is, J is the union of the set of points Ak,,(J) the escape time of which from R is at least k + 1, and the set of points D,(J) the escape time of which from R is j, where lj:5. We write D,(J)=U t, Dk(J). Note that A (J)=n'qk.oAk(J), and J = A-,(J) u D(J). In the lemma below we will state that, if the value of the escape time map TR changes then it changes in steps of 1. I 202 H. E. Nusse and J. A. Yorke For every x = DA(J) there exists an e >0 such that for each y E J with p([x, y]j)<e one has ITR(x)- Tr(y)1:5 1. T-JUMP LEMMA. Proof of the T-Jump Lemma. Let xE D,(J) be given. We will write D '(J)= U'., Int(Dk(J)), where Int(Dk(J)) means the interior of Dk(J) for each k-2 1. First, consider the case where x e D'(J). Then, by the definitions, TR is constant on the component of D~'(J) including x. Consequently, there exists an E > 0 so that TR(y)= TR(x) for all Y in J with p([Xy]j)< F. Now we consider the case where x E D,(J)\D,'(J).Let M a 0 be the integer for which FM(x)E Bndy (Rl, where Bndy (R) means the boundary of R. From the fact that each point in Bndy ( R) is mapped outside R it follows that TR(x) = N + I. We obtain that there exists E > 0 so that for each y E J with p([x, y].j) < - either TR(y) = M or TI(y) = A + 1. We conclude: there exists e > 0 so that for every y E J with p([x, yvs) < e either TR(x) = TR(y) or ITR(x) - TR(y)j = I. This completes the proof of the T-Jump Lemma. Denote the length of a segment L c J by p(L). GEOMETRIC LEMMA II. There exists 8 > 0 such that Jor ever, J in U(R), and .for each integer k - I, the following holds: (1) Each component of Ak(J) contains components of Dd(J) and A , (J); (2) Let A be an arbitrarily chosen component of Ak(J). For each component D of Dk(J) nA, one has p(D)/p(A) -8, and each component U of Ak,,J)nA, satisfies p( U)/p(A) - S. Prool oJ Geometric Lemma I!. Let J E U(R). Proof of 1). For k = i, the assumptions (AI)-(A4) imply that AI(J) = J contains at least two components of DJ), and it contains at least one component of A_(J). Now assume k > 1, and let A be a component of Ak(J)., By the definition of Ak(J) and the assumptions on F,we have F&-'(A) is a component of U(R). Hence, A contains at least two components of Dk(J) and at least one component of Ak ,(J). Proof of (2). Let U be a neighborhood of Inv (R) on which a C' ' stable foliation R exists, for some a > 0. The case that a basic set is nontrivial periodic is similar to that of a nonperiodic basic set but the notation is more complicated. Therefore, we assume that every basic set in Inv (R) is either nontrivial nonperiodic or trivial. For each nontrivial nonperiodic basic set F let I" and the regions R,(F), 1:5 i S N(F), be as in Proposition 4.2, and let U, be an open neighborhood of F such that (1) U., R,(F)c U, c U, (2) the set r-'(I" r Ur) and its closure consist both of N(F) components, and (3) the map p in Proposition 4.4 may be extended to T 1 (J "n U, ). For each trivial basic set F, let U, be an open neighborhood of F in U such that Urn U% is empty, fo, each basic set A in Inv (R)\I'. Select an integer K ->I such that the union of the Ur's include AK(J); the existence of K is guaranteed by the fact that Ak(J)- W'(lnv (R))nJ as k-coo. From the assumptions on F we obtain that the number of components of both Ak(J) and Dk(J) is finite for al k For I I I Chaotic saddle hyperbolic sets 203 every k > I and each basic set r in Inv (R), we define 6k(J) = minA minv p(V)/p(A) and sk(J; F) = minA {minv p( V)/p(A):" A r is nonempty}, where the minimum is taken over all components V of the sets Dk(J) and Ak.,(J), and all components A of the set Ak(J) such that VC A. Obviously, 8 k(J) s (J; F), for all k Let F be a basic set in Inv (R). Write Ur(R) = {J e U(R): J0F is nonempty. We first show: there exists 31.>0 such that for each JE Ur(R), for all k- > 1, and for every component A of Ak(J) that intersects F, one has every component D of Dk(J) r) A satisfies p(D)/p(A) a 8,., and each component U of Ak ,(J) r A satisfies p( U)/p(A) a: 8.. The case that I' is a periodic orbit is left to the reader. We assume that I' is a nontrivial nonperiodic basic set. Applying Proposition 4.4 and Geometric Lemma I we obtain that there exists 5,,(J;l')>0 such that 8,(J')-&,(J;F) for all k>K. We write 8,(J;F)= minIk.K8k(J;F), then 6k(J; ')->8p(J;l ") for all I --k sK. Now we define s(J; F) = min {q(J, F), s,(J; ')J >0 and get p(V)/p(A)a u5(J; F) for every component A of Ak(J) and every component V with Vc A, where V is either in Dk(J) or in Ak.,(J). Now, we define 61. = inf {6J; F): J e U.(R)}. Since U, (R) is compact we obtain 61. = min {(J; r): J E Uj,(R)} > 0. Finally, since F was arbitrarily given, we define 6 = min {8,: F basic set in Inv (R)), and conclude 6 k(J) ->6 for all k a I. This completes the proof of Geometric Lemma II. Proof of Proposition 1. Let L be as in the proposition. (i) (ii)': We assume that there exists e >0 such that every F-refinement of (a, b} includes a PIM triple. Ifthe interior of L does not include a point of Inv (R), then A (J) 0 L is empty, and thus no e-refinement of {a, b} includes a PIM triple. Hence, the interior of L contains a boundary point of Dk (J) for some integer k a 0. Therefore, the interior of L intersects A,(J). '(ii) = i)': Now we assume that the interior of L contains a point q of A,(J)o 1', for some basic set I' in Inv (R). Select integer M- I, such that L contains a component A of Am (J) that includes q. Let S > 0 be as in the Geometric Lemma II. Now we select E = 82 . p(A). From the Geometric Lemma 11 we know that A contains at least two components of D (J) whose length of each of them is at least 8. p(A), and A contains one or more components of A 4 1,, whose length of each of them is at least 8. p(A). We obtain that each E-refinement of a and b includes a PIM triple in A. This completes the proof of Proposition 1. From now on, we fix 6 as in Geometric Lemma 11 and e = 82. Proof of Proposition 2. Let (a, c, b) be an Interior Maximum triple in J. First, we assume that TR(a):- TR(b)< TR(c). Case 1. Assume k=min .... b TR(x)< T(a). Let D be the component of D&(J) containing at least one point of [a, b]j, for which TR(y) = k for all v in D. Then Dc int ([a, b]j)c A, where A is the component of Ak(J) for which Dc A. Since p([a, b] ) < p(A), applying the Geometric Lemma II gives p(D)/p([a, b]j) a 8. Then, for every (3-refinement P of (a,c, b), with 0< P.s3 we have PqrD#0. We obtain: for each p E P, r- D either (p, c, b) or (a, c, p) is a PIM triple in Pq. 204 H. E. Nusse and . A. YorkeI Case 2. Assume mi.. TRW?_ TR (a) and TR (c) Z-TR(a)+ 2=m +1.Then, by the T-Jump Lemma, there exists a component D of Dm.(J) in the interval [a, c],. Since (a, b]j lies in a component A of A,,-i(J), the Geometric Lemma 11 impliesI p(D)/p((a, b]i) Z-8. Hence, every (3-refinement of (a, c, tb) includes a point p of D, so (p, c,b) is a PIM triple, where 0 < P3 . Case 3. Assume TR(c) = TRt(a) + I = m and that Case I does not apply. This impliesI TR (b) =TR (a). Set p3= 82; let Pp be a 13-refinement of (a,c,b), say PO = fx,: 0sis N(P)) cJ with a = x0 <x < ..*< xhN~ =b and xk =c for some ~ N(P3)- 1.I From the Geometric Lemma 11 we get that (a, b]i contains a component D of D.,.iV), and p(D)/p(fa, b~j) a-6*. We obtain that every 1-refinement of (a, c,b) includes a PI M triple for each 0<1 P S. The case TR (b) t. TR(a) < TR(c) is similar. The conclusion is: For E = S we have: every e-refinement of a PIM triple in J includes a PIM triple. This completes the _~ proof of Propositio'n 2.I Proof of Proposition 3. Left to the reader. Before we will prove Proposition 4, we will present a monotonicity property for the escape time map as well as an auxiliary observability result for Accessible PIMI triple sequences. MONOTONICITY LEMMA. Let a and cbe two points on J, and let Pc: a,c]j be aI (3-refinement of a and c, saY P = (x,: 0!5 i~s N(P3)) and a = x0 < x, <, < xNp C. where (3> 0. Assume that TR is monotonic on P (that is. TR(x&, I) a TR(xk), 0!5 V NO) - 1), and TR(c) > Tt(a). Write mn= min fTR(x): XE(a, Cj} .I < 8, Dm(J)n (~a,c~j consists oj one component, and it Then, JOr ever), (, 0 < 13 includes a Proof of the Monotonicit)' Lemma. Let (3,a, c, P, TN, and m be as in the Lemma. By the definition of mn, we know that [ a, cI., is contained in a component A of A,,MJ. Assume that 0 <1P < 8. Suppose that TR(a) > mn. Then there is a component D of D,,AJ) such that Dc [a, clu. (Note that neither a nor c is contained in D.) From Geometric Lemma 11 we know that p(D)/p([a, ci) 2:p(D)/p(A) a 8 >1P; this implies Prn D 0. But this contradicts the assumption TR is monotonic on P. Hence, m = TR(a).I Suppose Dm(J)cr-, (a, c]j consists of two components, say D and D'. We will assume D' includes a. The Geometric Lemma 11 implies there exists a component U of Am,,(J) between D and D' such that p(U)/p([a,ciO) p(U)/p(A) 2 3> . We obtain that P includes a PI M triple (a, c', b') with C'E P r) U and b'E P n DI (since both p( D)/p([a, C].,) >1P and p( U)/p([a, C].,) >1P), which coniradicts the monotonicity of TR on P. This completes the proof of the Monotonicity Lemma.I Let Pcai be an e/3 -refinement of an Interior Maximum triple (ao, co, b) in , and assume TR(x) 2:TR(aO) for evenv, X EP. Let (ao,cj, bj) be the PIM triple in P, in which b, and cl are selected as in the Accessible PIM triple OBSERVABILITY LEMMA. procedure, and let a,) and a: be defined as in the Accessible PIM triple procedure. Chaotic saddle hyperbolic sets 205 I fP is an e-refinement of (an, cl, b,), then (il If a'> ao then [ao, a']. does not intersect S(R); otherwise, (ii) if a I= a0 then TR(b,)> TR(ao), a, < cl and [ao, afl] does not intersect S(R). Proof of Observability Lemma. Let P, (an, cl, b,), a' and a be as in the Lemma, and assume P n [ao, b], is an e-refinement of (an, cl, b,). Note that from this latter assumption it follows that Pr)ao, c,].j is a -refinement of {ao, c'} for some 0 03<8. Let m =min{TR(x): xe [ao, b,]s}. The assumptions 'TR(x,)a TR(ao) for all x,E P', 'Pn[ ao, bl]j is an E-refinement of (an, c,, b,)', and the Geometric Lemma 11 imply that m = TR(ao). Proof of (i). Assume that a'l'> an. By the Monotonicity Lemma we obtain that TR(x) = TR (ao) for all x E [ an, a' ]s ; hence, [ an, a'l] does not intersect S(R). Proof of (ii). Assume that a'= a0 . Suppose TR(b,) = TR(aO) = m. From the Geometric Lemma II and the assumptions we get that the interval [aO, b]s contains one component A of A,.,i(J), and p(A)/p([a,,, b,]j)> 8. Applying the Geometric Lemma II again, we get that there are at least 2 components U, and U. of D..,(J) and at least one component U3 of A..2(J) in A, and for each k, l<--k-3, p(U)/p((a, b,]s) =(p(Uk)/p(A))(p(A)/p([a., b,]s)>8 2=r. Hence, each U", I < k - 3, contains at least one point of P This implies b, is not the leftmost point in P that is the right point in a PIM triple, which contradicts the assumption. Conclusion: TR(bj)> TR(ao). The facts "(ao, cl, bl) is a PIM triple' and 'TR (ao)< TR(bl)' imply TR(cl) TR (a,,) + 2. We obtain from the Geometric Lemma II that there is a component D of D ,,(J) in [a,,, cj]j such that p(D)/p([a), b,],)- :S. Using the T-Jump Lemma, we obtain that there is a point qE Dn P with TR(q) TR(a,,)+ I and for all x in P between a,, and q one has TR(ao)- TR(x)- TR(aj)+ 1. It follows that the point a exists. Applying the Monotonicity Lemma we obtain m = TR(ao) <- 7R (x) TR(a ) = m + I for a!l xE [aO, a. ],; hence, [a,,, a] does not intersect S(R). This completes the proof of the Observability Lemma. Proof of Proposition 4. Let e be as in Proposition 2, and let {(a,, cn, bn)}n.;u be an Accessible PIM triple sequence in J, that is, (an, co, b) is an Interior Maximum triple and for n a 1, (a., c,, b,) is obtained by the Accessible PIM triple procedure. For n a0, let P. be an e/3-refinement of the Interior Maximum triple (a., c., b.), and recall that M, = min tTR(x,): x, E P., x, < c.,}. Further, we write m.=min{TR(x,): x, P.,). Note that the Geometric Lemma 11 implies m= min { TR(x): XE [a., b.]s}. We will show that there exists an integer N 0 such that for every integer n -N: TR(a)=AM.;IT(a.,l)-TR(a.)l1; and [an,a,,b] does not intersect S(R). From the T-Jump Lemma, the Geometric Lemma II, and the assumption that {(a., c., b.)}.,, is obtained by the Accessible PIM triple procedure we obtain for each n a0, the following properties: (1) if TR (a.,)>M, then TR(a,,.:)=M,,; (2) if TR(b,,)=m, then TR(b, , )- TR (b )+ 1; H. E Nusse and J. A. Yorke 206 t3) if T(b.)=m. and M.> m, then m,a:_n.f+l (4) if T(b.)> m. and M.> n, then Tt(b..,)>-m.. These properties imply that there exists a minimal integer N -:0 such that TR(x,) z M, = m., - T(aN ) for each x. E P. Case 1. P,. is 11o E- -finement of (a,. c..., fb.). Since a.., = a., we have t i) T(v.) > It.., = m.v+, = T(a,,) for each x,E Pv,, and (2) (aN . .j], does not intersect S(R). Obviously. TR(x) = TR(a.v) for all x in [aN.a..-,1. rst. assume that % Case 2. P.. is an E-refinement of (a By the Monotonicity Lemma. and the Observability Lemma we obtain for a., = TR(a,) for all x ta.,.a.,_,j, a'v.,: 11) TR(X)= (2) TR(,) 2:%, ., = m.ni, = TR(a_... for each x,E P..,. and (3) [a... aN.,,], does not intersect St R). Now assurne that a%,., = a. Applying the Monotonicity Lemma. and the Observability Lemma yield. for a ., = a-.,; (I) TR(x) = TR(aN) for every x E [a,. a.,J, (2) -,tx,) M. m. , = T. 'a..,)= T(a.,,)+I for each x,E P.-,, and (3) [aN, aN ,], does not intersect S(R). By induction, one obtains the desired result. This completes the proof of Proposition 4. Proof of Proposition 5. Left to the reader. 5. Discussion of the numeri al procedures Now we will return to the "dynamic' question addressed ir. the beginning, namely, how can you numerically follow a trajectory on an invarian: set for an arbitrarily long period of time? A line segment [a, b] straddles the stable manifold of a point P i^ [a. b] intersects this manifold transversally. In tl.e cases we zwudy, P will be replaced by chaotic saddles (nontrivial basic sets) and [a, b] will straddle a subset of S(R). Furthermore, in practice [a, b] will be very short and will be extremely cl-hse to the invariant set lnv (R). The numerical procedure goes as follows: (1) Choose 1-,ith some experimenting) efine and choose PIM a straight line segment 1; (2) Apply PIM triple proced triple int.rval) repeatedly until the length of the PIM triple interval is less than some distance o (e.g. a = 10-8); call this interval 1= PIM,,(I); (3) For a straight li'ne se.nent L with end points a and b, we write PIM,7 (L) to denote either [a, b] if 1[a, bil < o- or the resulting interval when some PIM triple procedure is applied . ,itil an interval of length less ihan a is reached. Note that this operator depends oniy on the end points of L. The basic process then is itera-:1g PIM,(F(L)). While F(L) is an interval, only Fla) and F(b) are relevant. Thus we obtain I.., = PM,(F(I,)), a sequence of straight line segments. We thus obtain a traJectory of tiny straight line segments I, and to the precision of the computer (..bout 10") we typically have I,, c: F(/.), and selecting any point x. from I,, perhaps the midpoint, we have that Ix 1 - F(x,)j is small, typically of the order of oa. Since computers can never be expected to produce true trajectories I I I I Chaotic saddle htverbolic sets 207 (except in trivial cases such as fixed points), we may say {x,.}.o is a numerical trajectory. We call the sequence of intervals {I.}o a saddle straddle trajector because the interval straddles a piece of the stable set S(R) of a chaotic saddle set. It typically approximates (after a few iterates) a basic set in the invariant set (which is a chaotic saddle) in the interesting cases. Furthermore, a saddle trajectory approximates the trajectory of a point in the Static Restraint Problem. Despite the complexity of the construction, we will refer to x., as the iterate' of x,,. Remark In -ractice we find that every L-refinement of two points {a, b), with r= 1/30, includes several PIM triples. In computing the sequence of PIM triples (a,, c., b.) defined by the Accessible PIM triple procedure, once either case 3iii or 3(iv) holds, and if c is more than r - lb - al from a and b. then it can be shown that every E-refinement of the end points {a. b} of a PIM triple (a. c, b) includes a PIM triple; in the computer program we do not use c at all. For the examples in this paper and in [NY] we find that the Accessible PIM triple procedure leads to accessible fixed points or periodic points, which is in agreement with the fact that all the accessible points for two dimensional hyperbolic systems are on the stable manifolds of finitely many periodic points. In this paper we have shown that our procedures are valid in ideal situations. We find it works well in practice even in less than ideal cases. From the examples in fNY], we have seen that the PIM triple procedure works quite well for a variety of dynamical systems. It is important to ask if such straddle trajectories represent true trajectories of the system. In other words, does there exist a true trajectory of the system that shadows (i.e., stays close to) the numerical trajectory obtained by the PIM triple procedure? When a map is sufficiently hyperbolic on the invariant set in question, Bowen [B] obtained a result saying that each noisy trajectory in the nonwandering set can be shadowed by a true trajectory if the perturbation is small; see [B] for the precise statement. We will say that Inv (R) satisfies the 'no cycle condition' if for every family of basic sets F , k...F,,in Inv (R) such that the stable set of has a nonempty intersection with the unstable set of F,. ,, for all I <-i< M, the stable manifold of rk,., does not intersect the unstable manifold of r,,,. Assuming lnv(R) satisfies the 'no cycle condition* and 6 is sufficiently small, we can show that every saddle straddle trajectory of a two dimensional uniformly hyperbolic system with a chaotic sadole obtained by the PIM triple procedure, can be shadowed by a true trajectory for as long as the saddle straddle trajectory can be computed. Ir(, I 6. Concluding remarks 6.1. Higher dimensional systems. One of the ingredients in the analysis of the validity foliation of the PIM triple procedures in dimension two, is the existence of a C 9;' on a neighborhood of a basic set. The existence of such a foliation for the two dimensional case, is guaranteed by a result due to De Melo [M]. Unfortunately. the existence of a foliation 7-'on a neighborhood of a basic set in higher dimensions is not known, see e.g. [PT]. 208 H. E NVusse and I. A. Yorke Let from now on, the dimension n =-3. Let F be an Axiom A diffeomorphism, let R be a saddle-hyperbolic transient region for which dim E" = 1, and assume that for each basic set F in Inv (R) there exists a C ~ stable foliation AV' on a neighborhood of 17, for some a > 0. Then the Propositions 1, 2. 3. 4. and 5 are still valid. The proof is almost the same, except instead of Propositions 4.1 and 4.2 one should use the properties of Markov partitions of basic sets. see Bowen [B]. 6.2. Order of Tiferentiabilit of the Daffeomorphism. We assumed that the difteomorphism F is C". This assumption implied the existence of a C' - expanding map. for some a > 0, in Proposition 4.4. If F is of class C2. then it is known that such an expanding map is C'. WVe would like to point out, that the Haider exponent a is only used to obtain 12) in the proof of the Geometric Lemma 1. Fortunately, we can prove the Geometric Lemma I tin particulk: property IN) for the C'-map ,p of Proposition 4.4 by combining the techniques of the proof of Proposition 6 in [Ne] and Lemma 5.5 in [No). Thus in fact, it is sufficient to assume F is C2 to guarantee the main results of the paper. REFERENCES [AV] 1B] [BR] [DNI [coy] [GNOVI [GH] [m) (NP] [Ne] [Nil [Nul K. T. Alligood & J1.A. Yorke. Accessible saddles on fractal basin boundanies. Preprnt 1989. R. Bowen. Equilibrium States and the Ergodic Theory _f Anosov Diffeomorphisms. Lecture Notes in Mathematcs 470, Springer Verlag: Berlin. 1975 R. Bowen & D. Ruelle. The ergodic theory of Axiom A flows. Intent. Math. 29 (1975). 181-202. R. Devaney & Z. Nitecki. Shift automorphisms in the Hinon mapping. Commun. Math Ph vs. 67 (1979). 137-146. C. Grcbogi. E. Ott & J A. Yorke. Basin boundary metamorphoses: changes in accessihle boundary orbits. Phrsica 24D (1987), 243-262.I C. Grebogi. H. E. Nusse. E. Ott & J. A. Yorke. Bassic sets: sets determine the dimension of basin boundaries In: Dvnamical Sistems., ed. J. C. A~lexander. Proceedings oj the Universityof Man/and 1986-87. Lecture Notes anMath. 1342. pp. 220-250 Spnnger.Verlag. Berlin. Heidelberg, New York, Lonlon. Pans. Tokyo, 1988. J. Guckenheimer & P. Holmes. Nonlinear Oscillations. Dynamical Svttems, and Btfurcationf of Vector Fields. Applied Mathematical Sciences 42, Springer Verlag: New York, 1983 W. de Melo. Structural stability of diffeoimorphisms on two-manifolds. Inrent. Math 21 (1973). 233-246.I S. Newhouse & J. Palis Hyperbolic nonwandenng sets on two-dimensional manifolds. In Dynamical Systems. pp.293-301, ed. M. NI. Peixoto. Academic Press New York and London. 1973. S. E. Newhouse The abundance of wild hyperbolic sets and non-smooth stable sets forI diffeomorphisms. Pub. Math. I H.ES. 50 (1979), 101-15 1. Z. Nitecki. Differentiable Dynamics. %IIT Press. Cambridge. 1971 H. E Nusse. Asymptotically periodic behaviour in the dynamics of chaotic mappings SlAM J AppI. Math 47 (1987), 498- 513. [NV] (Pr] IS] IV) H. E. Nusse & J. A. Yorke. A procedure for finding numerical trajectonies on chaotic saddles. Physica D36 (1989). 137-156. J. Palis & F. Takens. Homoclinic bifurcations and hyperbolic dynamics. 16' Coldquio Brasileiro Matematdica, IMPA, 1987.I S. Smale. Differentiable dynamical systems. Bull. Amer. Math. Soc. 73 (1967), 747-817. J. A. Yorke. DYNAMICS. A Program for IBM PC Clones. 1987, 1988. Reenms Eram JUL*.as. to :iTATWXi4L Pinw 164 *'5. %moi. 4. NoveUSU iV91 Embeology run, Sauer.' James A. Yorke.-- and Martin Casdagli" Receired Afarit). 1991 I Mlathematicai formuviations oi the cmoeddint methods commonlh umeo for the .econstruction . t attractors tram data Neries aire *J~sculscd. F.-becioina -neorems. casea tin previous ~%r(r hte iuFT'. -.'C csao6ihea for compact suosets .i oi Euctidean bvace R' If it i% an integer iurger ian twice the box-counting dimension of 4, then almost every map irom R. to R". in the %cnseof prevalence. n, one-to-one on. A. and moreover is an embeddine on smooth manifolds contained within 4. If .4 is a chaotic attractor of a typical dynamical system. Lhen the same is true for almost every delav-c vrdiate it? 1 from R, to R*. Thewe results are extended its two other directions. Similar results areproedin the more general case ot reconstructions wtich ub ov 4vea~c ofdelaycoriae.Scn.ifrainvgieonh sf-trclinstthat exists %hen it is less than or cqual to twice the hox-counting dimension of .4. KEY WORDS: rmbeddine. ..haotic tittractor: tttractor %cconstruction. probability one: prevalene. box-counting dimension: delay coordinates 1. INTRODUCTION In this work we give theoretical Justification ofl data embedding techniques used by experimentalists to reconstruct dynamical information from time series. 'We focus on cases in which trajectories of the s~ystem under study are asymptotic to a compact attractor. We state conditions that ensure that the map from the attractor into reconstruction space is an embedding. meaning that it is one-to-one and preserves differential inforrnati r-O~F approach integrates and expands on previous results on c~aibedi by 3Whitnev,9 and Takens.' 1 27 Department of Mathematical Sciences. George Mason University. Fairfax, Virginia 22030. Institute of Physical Science and Technology, University of Maryland. College Park. Maryland 20742. Santa Fe Institute. Santa Fe. New Mexico 87501. Current address. Tech Partners. 4 Stamford Forum. 8th Floor. Stamford. Connecticut 06901. 579 W22-4715 91 1100.0579106 50.0 t' 1991Plenum ilublisiin; Corporation Sauer et ai. 5a Whitney showed that a enenic smooth map F from a d-dimensionai smooth compact manifold .11 to R- " 'is actually a diffeomorphism on . That is. .11 and F(MI are diffeomorphic. We generaiize this in two ways: first. :y replacing "generic' %th "probability-one" tin a prescribed sense I. and second. by replacing the manifold If by a compact invariant set . contained in R" that may have nonmnteger box-counting dimension iboxdimi. In that case. we show that almost every smooth map from a neighborhood of A to R" is one-to-one as lone as I 1 n > 2 boxdimu.4 ) We also show that almost every smooth map is an embedding on compact 'iding subsets of smooth manifolds within .1. This ,u-gests that 'but ..s techniques can be used to compute positive Lyapunov ex; .ot necessarily negative L'apunov exponents u. The ?o)itixc .j.ouno% -xponents are usually carried by smooth unstable manifoids on arctors. We give precise definitions of one-to-one. embedding. -nd aimost e,erv in the next section. Takens di..it with a restricted class of maps called dela -coordinate maps. A delay-coordinatt map is constructed from a time series of a single observed quantity from an experiment. Because of this. a typical delaycoordinate map is much more likely to be accessible to an experimentalist than a typ:ca! map T.kens:i" showed that if the dynamical system and the observed quantity are generic. then the dela. -coordinate map from a * d-dimensional smooth compact manifold .11 to R " ' is a diffeomorphism on Ml. Our results generalize those of Takens"'" in the ,aime two Was as for Whitnev, theorem. Namei'. -.%e replace ,!Qneric %ith probabdity-,ne .and the manifold 11 by a possibly iractal ct. Thus. tor a compact u,,ariant subset .1 of R", under mild conditions on the dynamical s~stem. almost every delay-coordinate map F from RA to R" is one-to-one on . provided will be that n> 2 boxdimi.-I). Also. any manifold structure within preserved in F(A). These results hold for lower box-counting dimension (see Section 4) if boxdim does not exist. The ambient space R can be replaced by a k-dimensional smooth manifold in the general :ase. In addition. ,.ve have made explicit the hypotheses on the dynamical ,%stem (discrete or continuousi that are needed to ensure that the delay-coordinate map gives an embedding. In particular. oniv C' smoothness is needed. For flows, the delay must be chosen so that there are no periodic orbits whose period is exactly equal to the time delay used or twice the delay. (A finite number of periodic orbits of a flow whose periods are p times the delay are allowed for p > 3.) Further. we explain what happens I I I I IU I I I logy boxdiml A). In that case we put bounds on the case that n < -• on of the self-intersection set. which is the set on which the oneproperty fails. Finally. we give more general versions of the delayate theorem which deals with filtered delay coordinates, which tore versatile and useful applications of embedding methods. ere are no analogues of these results where the box-counting on is replaced by Hausdorff dimension (see Theorem 4.7 and the on that follows 1. In an Appendix to this work written by I. Kan. %sare descnbed of compact subsets of R*. for any positive integer Iihave Hausdorff dimension d = 0. and which are difficult to project e-to-one way. The requirement n > 2d discussed above translates in e to > 0.However.ecery projection of such a set to R'. i < k. fails u ne-to-one. Section 2 we explain the new %ersion of the Whitney and Takens ig theorems. In Section .3 we discuss filtered delav coordinates. 4 contains proofs of the results. W TO EMBED MANIFOLDS AND FRACTAL SETS U -actal Whitney Embedding Prevalence Theorem 3ltici |ume ,Pis a flow on Euclidean space R*. generated. for example. by ,nomous system of k differential equations. Assume further that all ries are asymptotic to an attractor .4. The study of long-time ,rof the system will involve the study of the set .1. a typical scientific experiment, the phase ,pace R' cannot be lv seen. The experimenter tries to infer properties of the system by measurements. Since each state of the dynamical s stem is uniquely I by a point in phase space. a measured quantity is a I.unction Irom pace to the real number line. If it independent quantities Q . Q. measured simultaneously, then with each point in phase space is ted a point in Euclidean space R". We can then talk about the F(state = (Q . Q,,) naps Rk to R". r example. suppose all trajectories in phase space R' are attracted riodic cycle. Thus. .1 is topologically a circle lying in R". Imagine o available measurement quantities Q, and Q2 are plotted in the Then there is a measurement map F from A to R' given by )= (Q1, Q2). To what extent are the properties of the hidden )r A preserved in the observable "reconstruction space" R2? I 582 Sauer et al. The answer depends on how the circle is mapped to R2 under F Consider the case where R* = R3 and Q and ( ire simply the two coordinate functions x, and x,. In Fie. ' a. the relative position of the points is preserved upon projection. and we may view F(A) as a faithful reconstruction of the attractor A. If distinct points on the attractor .4 map under F to distinct points on FA). we say that F is one-to-one on A. In the case of Fig. lb. on the other hand. two different states of the dynamical system have been identified together in RA ). In the reconstruction space. which is all the experimenter actually sees, the two distinct states cannot be distinguished. and information has been lost. The one-to-one property is useful because the state of a deterministic dynamical system. and thus its future evolution, is completely specified by a point in phase space. Suppose that at a given state Y one observes the %alue F(.' in the reconstruction space. and that this is followed I sec later )y a particular event. If F is one-to-one. each appearance ' the measurements represented by F(vi ,ill be followed I ec later by the .,ame event. This is because there is a one-to-one correspondence betmeen the attractor points in phase ,pace and their images in reconstruction .,pace There is predictive power ,n finding a one-to-one map. Perhaps the measurements F(.W would not be repeated precisely. Yet F is reasonable. similar measurements %ill predict similar events. if th, This approach to prediction and noise reduction of data is bcing pursued by a number of research groups. Although most of the interest lies in the case that .1 is an attractor of a dynamical system. the main question can be posed in more generalit. Let A be a compact subset of Euclidean space R'. and let F map R' to another Euclidean space R". Under \%hat conditions cara \e be assured that i'embedded" n R" by typical maps F' 4 is The precise definition of embeddiniz involves differential structure \ one-to-one map is a map that does not collapse points, that is. no ' points are mapped to the same image point. .\n embedding is a map that does not collapse points or tangent directions. Thus. to define embeddin,-g. we need to be working on a compact set .1 that has well-defined tangent spaces. Let I he a compact smooth differentiable manifold. (Here. as in the remainder of the paper. the word .vnooth %%ill be used to mean continuouly differentiable, or C'.) A smooth map F on .1 is an inrinersuon if the derivative map DF(.x) (represented by the Jacobian matrix of F at k) is one-to-one at every point x of .-I. Since DF(Rx) is a linear map. this i! equivalent to DF(x) having full rank on the tangent space. This can happen whether or not F is one-to-one. Under an immersion, no differential structure is lost in going from A to F(A). I I I I I I I I I 1 I U U 583 Embedoilogy An ernheddine of A is a smooth diffeomorphism from A onto its image F(A . that is. a smooth one-to-one map which has a smooth inverse. For a compact manifold .4. the map F is an embedding if and only if F is a oneto-one immersion. Figure la shows an example of an embedding of a circle into the plane. Figure lb shows an immersion that is not one-to-one. and Fig. Ic shows a one-to-one map that fails to be an immersion. Whether or not a typical map from .4 to R" is an embedding of A depends on the set .-. and on what we mean by "typical.- \ celebrated result o t this type is the embedding genericity theorem of Whitney.", which says that if .4 is a smooth manifold of dimension c. then the set of maps into R-'-' that are embeddings of .4 is an open and dense set in the C'-topologv ,of maps. The iact that the set of embeddings is open in the bet of smooth maps means that uien each embeddina. aroitrari.%:mail perturbations wll still he emoeuaings. Fle lact that the ,c[ ot emueddings h iOlt in the ,ct oi maps means that every imooth map. whether it is an emoedding or not. is arbitrarilv near an embedding. One would like to conclude from Whitney s I ,FF 1 A a F // 'C" Fig. I I FA) 2 ia) An embedding F of the smooth manifold .4 into R (b) An immersion that fails to be one-to-one. (c) A one.to-one map that fails to be an immersion. 584 Sauer et al. theorem that n = 2d + I simultaneous measurements are typically sufficient to reconstruct a d-dimensional state manifold .4 in the measurement space R". However. open dense subsets, even of Euclidean space. can be thin in terms of probability. There are standard examples. many from recent studies in dynamics. of open dense sets that have arbitrarily small Lebesgue measure, and therefore arbitrarily small probability of being realized. A well-known example is the phenomenon of Arnold tongues. Consider the family of circle dilfeomorphisms g,,,.(x)=x+w+ksinx mod 2-. where 0 < wj - 2.: and 0 < k < I are parameters. For each k we can define the setI Stab k i= )<,. < " : ., has a stable periodic orbit' For )<k < 1. the ,et Stab~k) is a countable union of disjoint open intervals of positive length. and is an open dense subset of [0. ]. However. the total lengt' i Lebcsguc measur-) of the open dense et Stablk) approaches zero as k -0. For small k. the pr,-baulitv of landing in this open dense set is %ery small. See ref. 3 for more details. With such examples in mind. an experimentalist would like to make a stronger statement than that the Net of embeddings is an open and dense set of smooth maps. Instead. one would like to know that the particular map that results from analyzing the experimental data is an embedding with prohabilit v one. The problem with such a statement is that the space of all smooth maps is infinite-dimensional. The notion of probability one on infinitedimensional spaces do,- -ot have an obvious generalization Irom finitedimensional ,paces. I is no measure on a Banach ,pace that corresponds to Lebesgue measure on finite-dimensional subspaces. Nonetheless. we would lilke to make sense of "almost every" map having some property, such as being an embedding. Following ref. 24. we propose the following definition of prevalence. Definition 2.1. A Borel subset S of a normed linear ,pace I' is prevalent if there is a finite-dimensional subspace E of 1",,uch that for each r in V. v+ e belongs to S for iLebesgue) almost every L'in E. We give the distinguished subspace E the nickname of prohe Npatc. The fact that S is prevalent means that if we start at any point in the ambient space V and explore along the finite-dimensional space of directions specified by E, then almost every point encountered will lie in S. I I I Embedology 585 Notice that any space containing a probe space for S is itself a probe space for S. In other words, if E' is any finite-dimensional space containing E. then perturbations of any element of V by elements of E' will be in S with probability one. This is a simple consequence of the Fubini theorem.' From this fact it is easy to see that a prevalent subset of a finitedimensional vector space is simply a set whose complement has zero measure. Also. the union or intersection of a finite number of prevalent sets is again prevalent. We will often use the notion of prevalence to describe subsets of functions. It follows from the definition that prevalent implies dense in the Ck-topology for any k. More generally, prevalent implies dense in any normed linear space. When a condition holds for a prevalent set of functions, it is usually illuminating to determine the smallest, or most efficient. probe subspace E. This corresponds to the minimal amount of perturbation that must he available to the ,ystem in order for the condition to hold ,.ith uirtual certainty. As stated above, for subsets of finite-dimensional the term prevalent is synonomous with "almost every," in the sense ,paces )f outside a set of measure zeru. When there is no possibility of confusion. we %.il say that "almost every"' map satisfies a property when the set of such maps is prevalent, even in the infinite-dimensional case. For example. consider convergent Fourier series in one variable, which form an infinite-dimcnsional vector space with basis ',e ' . In the ,ense of prevalence. almost ever' Fourier series has nonzero integral on EO. 2.]. The probe space E in this case is the one-dimensional space of constant functions. If E' is a vector space of Fourier series which contains the constant functions. then for every Fourier ,cries i. the integral of !' will le nonzcro tor almost every e in E' With this definition, we introduce a prevalence version of the Whitney embedding theorem. Theorem 2.2 (Whitney Embedding Prevalence Theorem). Let .Abe a compact smooth manifold of dimension d contained in R' \lmost every smooth map R' - R2 " is an embedding of .1. I In particular, given any smooth map F. not only are there maps arbitrarily near F that are embeddings. but in the .ense of prevalence. almost all of the maps near F are embeddings. The probe space E of Definition 2.1 is the k( 2d + I )-dimensional space of linear maps from R' to R2 , '. This theorem, which is proved in Section 4. gives a stengthening of the traditional statement of the Whitnev embedding theorem. It is quite interesting that Whitney later proved the different result that under the same circumstances. there exists an embedding into R2d. (This Sauer et al. 586 could be called the Whitney embedding existence theorem) However, an existence theorem is of little help to an experimentalist. who needs information about maps near the particular one that happens to be available.I Knowledgze that an embeddir: exists sheds little information on the particular F under study. The example of Fig. l b shows that the dimension -'.+ I of Theorem -1.2 is the best possible. The map F is not one-to-one on the twisted circle .4. thus does not embed .4 into R-. Further. no nearby map even in the ("-topology) embeds Ai.On the other liand. if a given map of the circle .1 into R 3 was not one-to-one. there would necessarily be a prevalent set of nearby maps that arc embcddings. The first main a021 of this section was .o express Whitney's embedding theorem (and Takens' theorem: see belowi in this probabilistic sense. Thc .second is to extend WVhitney's theorem to sets .1I that are not manifoldsI Here \%e usc tile fractal dimcnsion known as hox-countinu dimension. he hox-,.ounting i or capacit%i dimension oi a compact 'sct I :nl A" i be the set of all point,, defined as follo%%s. For a positi~e number .let within . of .1. i.e.. -1, = GeR"' denote the 'i-dimensional outer dinL'/sWio for some it A:. Let %oil , olume of .1, . Then 'ie ho.tounhim,. ~-i al. : ' of A is boxdim.A ) =it - 1im log voll .l -41 logz; if the limit cxists. if' not. the upper (rcspccti%-ely. loweril ox-Countiig dimension ciin be defined by replacing the limit by the lim inif iresp., lim sup). WVhen the box-counting dimension exists, the appro\imate \caling law ohl~ holds. where d = hoxdjm(. I). There arc ,everal equivalent definitions of box-counting dimension. For example. R" cuin be divided into .-cubcs by a grid based, say. at pointwhose coordinates are L.-multiples o1 integers. Let .V(o be the number oI boxes that intersect I Then boxdim(.1 =lim -o V l0! og -102gr with similar provisions for upper and lower box-counting dimension. ThleI scaling in this case is NWI Embedology 587 Even if we know the box-counting dimension of an attractor A. Theorem 2.2 gives no estimate on the lowest dimension f6r which almost every map is an embedding. Suppose we know that .4 is the invariact set of a flow on R"° . and that the box-counting dimension of .4 is 1.4. In the absence of any knowledge about the containment of .4 in a smooth manifold of dimension less than 100, the use of Theorem 2.2 to get a oneto-one reconstruction requires the use of maps into R oi In fact. the smallest smooth manifold that contains the 1.4-dimensional attractor may indeed have dimension 100. But as the next result shows. one can do much better: almost every reconstruction map into RI will be one-to-one on A. I Theorem 2.3 Fractal Whitney Embedding Prevalence Theoremi. Let .I be a compact subset of R " ot box-counting dimension d. and let z be an integer greater than Zd. For almost every smooth map /- R" - R". I. 1is one-to-one on .1 2. F is an immersion on each compact subset C of a smooth manifold ccntained in ..I., The proof of the one-to-one half o the fra, '.al Whitney embedding prevalence theorem may be sketched as follows. Choose any bounded finite-dimensional space E of smooth maps F so that varying F by elements of E results in perturbing F(.') - F(v) throughout R" for each pair v; v in A. For example. the probe space E can be taken to be the space of linear maps from R" to R". Then the probability (measured in E) that the perturbed F(v) and F( v lie within ;: is on the order of i:". Similarly. if B and B, are r-boxes on .1. the probability that F(B ) and F(B, intersect is on the order of ;". Here we assume that there i, a hound on the magnification oi F. ,is when F i,, a ,mooth map near the comtact ,ct I The ,,ct I can be covered by essentially i. " boxes ot size :.. ano the number ol pairs of boxes is proportional to o: -Y1 The probability that no distinct pair of boxes collide in the image F(A is proportional to k : -'l c=;.* " If Sthe n > 2d. this probability of choosing a perturbation of F that fails to be oneto-one is negligible for small i.. More precise details of the proof. as well as immersion part. are !n Section 4 2.2. Fractal Delav Embedding Prevalence Theorem Despite the beauty of Whitnevs embedding theorem. it is rare for a scientist to be able to measure a large number of independent quantities simultaneously. In fact. it is a rather subtle problem to decide whether two different simultaneous measurements are indeed independent. These problems can be sidestepped to some degree by introducing the use of 588 Sauer et al. delav coordinates. In this approach. only one measurable quantity i needed. In a typical experiment, the single measurable quantity is sam pied at intervals 7' time units apart. The resulting list of samples :Q is called a time series. Think of the measurable quantity as an observation function it on the state space R" on which the dynamical system 0 is acting. EachI reading Q, = iix)is the result of evaluatine the observation function it at the current state ., Definition 2.4. If (P is a flow onl a manifold Al. T is a positi'beI number (called the delavi and hi: At - R is a smooth function. define the dlaY-coordinatc' iiap FRh. '1P. T): Al - R" byI Rh/. 0P. T)(xY) = ti Mx). Itu0 1jn. /it0 if MO vi To start with a Nimple e,\ample. Ict I be a periodic orbit oi the fio\I P We iound above that in the aosence ot dynamics. three mfuereenUnft coordinates are rcquircd to embed I in reconstruction space. ,r more precisely, that Almost cverv mooth map F= i ,)romf neighborhood of .4 to RI is an embeddin - on .1. Now the situation is dif.ecnt. InstezA' of three functions 1. I, /, that must be independent. t .here is a single function It. and the corresponding map Fl/i. A. T) pictured in Fig. 2. We want to know that for almo-,t c~erV function It from . Ito the real numbers R. the delay-coordinate map1 RhI. AP T) from .1 into R" is an embeddine. It should be stressed that this does not follow from Theorems 2.2 and 2.3. The maps F/i. 'P. F) form a restricted subset of all maps: whether they contain enough \ariation it) perturb away self-crossinmzs of .4 needs to be determined. In fact. the Leneral Fig. 2. The attractor on the left is mapped using delay coordinates into the reconstruction space on the right.I I Embedology 5 589 answer is that they do not contain enough variation. Extra hypotheses on the dynamical system ( are required to ensure that almost every h ieads to an embedding of .-. To see the need for extra hypotheses. consider the case the, . is a periodic orbit of a continuous dynamical system whose period is equal to the sampling interval T. Topologically. .4 is a circle. In this case. F(h. . T) cannot be one-to-one for an', observation function h. Let x be a point on the topological circle I.Since the period of .1 is T. hWx)= IitP ':= . . ,- r(X)). so that F= F(h/.(P. T) maps x to the diagonal line Ix, ..... : .'v,= . =.,, in R". A circle cannot be mapped continuously to a line (in this case, the diagonal line in R") in a one-to-one fashion. See Fig. 3. The one-to-one property also fails when .4 is a periodic orbit of period Define the function = .oxxi .)l on .1. The lunction ,a i either identicailv zero or it is nonzero tor some x on .. in \%hich c'ahe it has the opposite ,ign at the image point 'P. txi. and changes )ign on .1. In any case. dx) has a root x,, on .A.Since the period of .I Is 2T. we have h(Xu) = 11(0 1 ()) = Il0 ;rtCi)) -. Then F(h, '. T) maps \,, and IP ,ix,) to the same point in A If x0 and 1P - I x,.) are distinct, this says that F is not one-to-one. If \,, = -.1.xn, then the orbit actually has period T. ard F fails to be one-to-one as above. In the presence of periodic orbits of period :T. Fh. 1P. T) cannot be one-to-one for an ohservation function h. On the other hand. when .I is a periodic orbit of period 3T. or an' period not .'qual to T or 2T. there is no such problem. In this case the dela.-coordinate map of a periodic orbit 1 into R" i, an embedding for .mot every observation [unction )! a Ione as the reconstruction dimen,ion i,, at least three. Uhe ,iatcment :or more ,-'-nerai attra.tor,, I i,, follows. 3 ", 3-T. I , F' I? R Fig. 3. A two-to-one map from a topological circle to the real line. 59 SaUW of &I. Theorem 2.5 4Fractal Delay Embedding Prevalence Theoremii..:t * be a flow on an open subset U of R". and let A be a compact subset of U of box-countine dimension . Let n > 2d be an integer, and let T> 0. Assume that A contains at most a finite number of equilibria, no periodic orbits of 0 of period T or 2T. at most -;.teiy many periodic orbits oi period 3T. 4T.- nT.and that the lineanzauons of those periodic orbits have distinct eienvalues. Then for almost everv smooth function h on U. the delay coordinate map F(h.4. TI: U-*R" is: 1. One-to-one on .4. 2. An immerrion on each compact subset C of a smooth manifold contained in A. Where Takens - :l ;howed that the delay-coordinate maps generically ,m the C-topology ,gveembeddines of smooth manifolds ot dimension ,. -e -substitute comoact -ets of box-countine dimersion ... .nd ::inacc genenc with prevalent. The assumption of Theorem 2.5 that there are no periodic orbits ,i period T or 2T can be satisfied by choosing the nme delay r to h suffiLiently small. In fact. ia we assume that the recto, field on . -atisfics a Lipschitz condition. that is. .= 'xl. where 1'(xI - i1y; < L.x- ,. then it is known' 1 ' that each periodic orbit must have period at least - L. Hence. if T< ,n:L. there will be no periodic orbits of period T or "7. Theorem .5 assumes u>_,l to avoid clf-intersecton .f the reconstructed imnage of A. To see that this requirement cannot be reiaxcd in general. consider the case d= 1. it 2d= 2 shown 1i Fig. 4a. Let the observation function h be the coordinate function v,. and consider the delay coordinate map R' -- R2 defined by F(x,.',P T" i v,ix. *,1 P viii In the situation illus, ..... in Fie. 4a. 1,4' ,jh))<.,((P ,ia))< v,(a)= vIh). and \,0 ,c)) < v,tP ;it)) < ic )= \,(. Setting F= F(x,. 't.T). this means that in the reconstruction ,pace R2. F(a) lies , directly above Fh). and Ffd) lies directly , e F(I See Fig. 4b. The map F is continuous on the tralectory. ,o there i, a continuous path. parametrized by \-,. connecting Fa and Fl( i. There i, also ,uch a path connecting F(h) and F(d). According to Fig. 4b. there must be a %alue of v, in between where the curves meet. and two different points on the circle map together under F Otherwise said. somewhere in between thete .,, an ,c coordinate such that the upper and lower parts of the trajectory advance the same amount in the x, direction during the time imerval T. and thus have identical delay coordinates. The map Ph. P.T) is not an embedding. I If thee observation function or flow is verturbed'a smnall amount. the saMe NTheorem 2.5 Is a special case of a sae ntabout diffeomorphisms. Blefore statinga that version. we redefine delay coordinate maps for diffeomorphisms. 3and U Definition 2.6. if g is adiffeomorphism of an opensubset U of R' h: L' - R is a function, define the dedaY twordinaze i"zap Ffh. -1to:U' R by (DtaJ 37 t a _ _ _ _ _ _ _ d): _ _ __T_ F(d) F(a) F(b) I 3 b Fig. 4 (a) A trajectory of a flow that cannot be mapped using two delay coordinates ina one-to-one way. (b) The point at which the paths cross corresponds to a et of delay coor- dinates shared by two points on the trajectoi' Sauer et al. We get the previous theorem by substituting g = statemenlt. r in the folLwing Theorem 2.7. Let g be a diffeomorphism on an open subset L of R' . and let .4 be a compact subset of U. boxdimi A i = d. and let i > _'d be an integer. Assume that for every positive integer p < In. the set .1. of periodic points of period p satisfies boxdim.tAi< P-2. and that the linearization Dg- for each of these orbits has distinct cigenvalues. Then for almost every smooth function h on U. the delay coordinate map F(h. g): U- R is: 1. One-to-one on .4. 2. An -immersion on each compact ,ubset C of a smooth manifold contained in .4. Remark 2.8. Fhe probe space ir this prevaient Ect can be taker it, 'e any set I... ot poiynomials in h .araoles %%nich includes .ill roi.nomials of total degrec up to 2n. Given any mooth function ,:. on I for almost all choices of x = (c, ....2f from R'. the function .',= k),-,-ZA" x,Ih, satisfies properties I and 2. Remark 2.9. The proof of Theorem 2.7 is easily extended to the more general case where the reconstruction map F consists o a mixture of lagged observations. The more general result ,ays that F = hI(.v '(.\'l)'- I ,0 ' satisfies the conclusions of Theorem 2.7 as long as ,, + -,i., 2,1 and the corre;ponding conditions on the periodic points are ,atisfied. Those conditions arc that boxdimi . p 2 for ;, - max n. The reconstruction of ch. :tractors using independent coordinates from a time series was advocateu ill 980 by Packard ct ,d.'' The delaycoordinate map is attributed in that work to a communication MIth D. Ruelle. The metho .ctuailv illustrated in ref.. 21 is somewhat different: namely, it is to use the value u, of the time series and its time deriva'es i,. ,.as independent coordinates. In 1981. Takens ' published the first mathematical result, on the delay-coordinate map. \round the .,Ame time. Roux and Swinne\ " exhibited plots of delay-coordinate reconstructions of experimental data from the Belousov-Zhabotinski reaction. In 1985. Eckmann and Ruelle'" took the idea one ,tep further and suggested examining not only the delay coordinates of a point. but also the relation between the delay coordinates of a point and the next point which occurs T time units later. In principle, one can then approximate not only 593 Embedogy 3 I the attractor. but the attractor together with its dynamics. Since ref. 9 it has become common practice to gath.;r points that are close in reconstruction space. and use their next images to construct a low-order parametric model which approximates the dynamics in a small region. This idea has begun to be used for prediction and noise reduction applications. See. for example. refs. 1. 6. 12. 13. 15. 16. 18. and 28. 1 2.3. Self-Intersection In the case that the reconstruction dimension it is not greater than twice the box-counting dimension d of the set A. the map F in the fractal Whitney embedding prevalence theorem (Theorem 2.3) will often not be an embedding. However. if d < n.most of .4 will still be embedded. In the case that .I is a smooth manifold of dimension d. almost every F will be an embedding outside a ,ubset of A of dimension at most "'- ,1. If d < n. then 2.1 - it d. and so this exceptional subset will ha~e positive codimension in A. If A is simply a compact set of box-counting dimension ,.then the situation is slightly different. We will call the pai: x. v of points 6-distant if the distance between them is at least 6. Then we define the 6-distant selfintersection set of F to be the subset of .4consisting of all .asuch that there is a 6-distant point Y with F(x) = Fly); that is. (F. j)= 'xe.4, F(x) = F(.) for some ive A.k -.II> 6 Then the result is that for everv ,J > 0. the lower dimension of the i-distant self-intersection ,et f (F.6) is at box-countine most 2 - itfor almost every F. \ precise statement i,, given bv the next theorem. 3 3 Theorem 2.10 (Self-Intersection Theoremi. Let A be a compact subset of R' of box-counting dimension d. let it < 2d be an integer, and let 6> 0. For almost every smooth map F: R - R": I. The 6-distant self-intersection set r(F.6) of F has lower boxcounting dimension at most 21 -it. 2. F is an immersion on each compact subset C of an in-manifold contained in A except on a subset of C of dimension at most 2m -it- I. For example, consider mapping a circle to the real line. In this case d= in = n = 1, and Theorem 2.10 says that a prevalent set of F are immersions outside a zero-dimensional set. This is clear from Fig. 3. where the zero-dimensional set consists of a pair of points. The map is at least 2 to 1outside this set. and hence nowhere an embedding. 594 Sauer efak. On the other hand. settig d n: I and it2 in the theorem we see that a prevalent set of maps F from the circle to the piane are immiersions. and are embeddines outside a zero-dimensionai subset. Thus. the maps shown in Figs. l a and lb are of the prevalent type. immersions w..hich are one-to-one except for at most a discrete izero-dimensionail set of poitits. Figure Ic. on the other hand. is nonprevalent. Almost any map near F will perturb away the cusp. There is also a ,ef-intersection version of the fractal delay embeddinea prevaence theorem I Theorem 2.5) which one ets by making the obvious changes. Thus, if it < :.d. then for each 6 > o there exists a subset 1 (F. )I. whose box-counting dimension is at most 2'- it. on which the delaycoordinate map fails to be one-to-one. Note that the result is independent of 6 >0. If .11 Is a closed subsed of an in-manifold contained in .4I. then there is a subset E, of .11 of dimension at most 1n - it - I on %khich thle map fails to be an immersion. 2.4. How Many Delay Coordinates Do You Need? When usinie a delay coordinate map (or filtered delav coordinate mnap. described in the next sectic' ) to examine the image R(AI in R" of I ,et .1 in R*. the choice of it depends on the objective of the investigation. Different choices of 11suffice for the different goals of prediction. Calculation of dimension and Lyapunov exponents. and the dctermination A thle stability of periodic orbits. To compute the dimension of .1. all that is required is that 12.1 dimP.4) = dim .I wvhether the dimension beiniz used is box-counting. [Iauhdortf. iniorrnatiofl. or correlation dimension. The latter two depend on a probability Jcnsit% on .4 and F(A). It is hhown in ref. 24 that for the case of Hausdortf diimension. the equality (2.1l holds for alrnozt every' measurable map F,. in the sense of prevalence. as lone as it >, dim. I . The probe space of perturba1 tions for this result ib the space of all linear transformations from R' to R" \v4attila' ' proved that equality (:.I) holds for almost every orthogonal projection F It is rewhat surprising that there are exampi-s for which 12.1 Idoes not hold lor anyv map F wvhen box-counting dimension is used. Q~en under the hypothesis ni > boxdim A I. An example of this type is given in ref. _'. However. in most cases of compact sets which arise in dynamical ,vstems. we expect Hausdorff dimension to equal box-counting dimension. In pr .ical situations. if attempts to measure boxdimt.) result in ariation would answers dependent on it. where nt > boxdim A4). then the % U 'IEmbedoiogy 595 seem to be a numerical artifact, since there is no theoretioa justification for which of the values of n greater than boxdimIA) gives the more accurate result. The usual technique is to increase n until the observed dimension of boxaim F(.A) reaches a plateau. and to use this -esult. The resulting number might be called the piateau dimension. While the plateau dimension may indeed give the best numerical estimate of the dmension of A. there does not seem to be theoretical or numerical justification of this bias. and the question needs further investiation. Notice that n > boxdimi.4 does o'it guarantee that almost every F is one-to-one. but that is not required for dimension calculation. If the objective is to use F(.) to predict the future behavior of trajec- tories, then it is sufficient to have the map F be one-to-one. In which case n > 2 boxdimt.4) is needed. Knowing the current state in Ft -11 is sufficient to predict the iuture of the traiectorv tat least in the ,hort ,uni. In the I I I 3The I ,ituation 0I Fig. lb. on the other hana. prediction on the periodic ornit would ,tlt be possible. except when the trajectory was at the rmrdpoint ot the "figure eight.' If the objective ,s to compute the Lyapunov exponents of the system. it is necessary to ask which exponents are to be cor.. ;uted. For a simple example. .uppose the attractor .4 is a periodic orbit. Then the best p.issible result of the examination o F(.A) is o observe that t is a Lvapunov exponent. The other exponents. presumably all negative. cannot be observed without introducing perturbations. IMore generally. if an attractcr .4 lies on a manifold of dimension ti (as a 2.2-dimensional attractor might lie on a three-dimensional manifold), it will certainly he impossible to measure more than in true exponents from an embedding, cven tf thL reconstructed image F.) lies in R" ith n > m. There are no criteria for determining, the smallest manifold containina 1 Theorems 2.3 and 2.5 say that if t > 2. boxdimt.l ). then almost every F is an embedding of all smooth manifolds that lie in .1. The ,mooth manifolds we have in mnd are the surface corresponding to the unstable directions on the attractor .1. that is. the unstable manifolds. Under an embedding, the differential information is preserved along ,ilooth directions. ,uch as unstable manifolds. indicating that positive Lyapunov exponents should be computable from the image F(.I). The stable manifolds, on the other hand. will be likely to intersect in a Cantor set. The image of a Cantor set in F(A) ma,, he quite com-.1 pressed. For example. a set which is the product of five Cantor sets whose dimensions sum to 0.5 might be mapped to a one-dimensional line in F(A). It seems difficult to recover any exponents in these directions from knowledge of the reconstructed dynamics in FA). self-intersection results in Section 2.3 are aimed at another kind of 59 Sauer et al. question. A relevant experiment involving a vibrating ribbon is described in refs. 8 and 26. In this case. the Poincare map has an attractor whose dimension was experimentally calculated to be 1.2. The investigators were interested in determining the eigenvalues of the linearization of a period-3 point on the attractor. Using a delay-coordinate map of the attractor into R 2 did not result in a one-to-one map, which is consistent with our results in Section 2.2. Theorem 2.10 of Section 2.3. which deals with self-intersection. suggests that the subset . of A on which the map into R 2 fails to be one-to-one should have dimension at most 2,= 2 x 1.2- 2 =0.4. They found that the self-intersection set looked like a finite set. If .1 indeed has dimension 0.4 or less. as we w..i expect. then the set . would be unlikely to include the periodic point in question. and the delay-coordinate map would be expected to be one-to-on- in a neighborhood of that orbit., Numerical investigations of the dynamics near the periodic orbit revealed that the dynamics did uppear to be two-dimensional, and the researchers \%ere ,1le to estimate numerically the eizgenvalues of the orbit at these points. 3. THE DELAY COORDINATE MAP AN.) FILTERS 3.1. Main Results So far. we have defined the delay coordinate map .- F(Ih. ,,.v from the hidden phase space RA to the reconstruction space R". Under Ntiitable conditions on the diffcomorphism ,, the delay coordinate map Fit. ,,i is an embedding for almost all observation functions 11. In this formulation. information from the previous a time steps is used to identify a .tatc of the original dynamical system in RA. For purposes of measuring quantitative invariants of the dxnamical systems, noise reduction, or prediction. it may be advantageous to create an embedding that identifies a state with information from a larger number of previous time steps. However, working with embeddings in R" I, difficult for large i. A way around this problem is to incorporate large numbers of previous data readings by "averaging" their contributions in ,ome ense. This problem has also been treated in ref. 7 To this end. generalize the delay-coordinate map F(h. ,r R' -- R". F(h, g)x = th(x). h(g(xn..... hig" where the superscript T denotes transpose. by defining the fi!tered deluvcoordinate map F(B, h, g): R' - R" to be F(B. h, g)x = BF(h. g)x (3.1) Embgdotogy , 597 where B is an i x w constant matrix. Thus. each coordinate of F(B. h, g)x is a linear combination of the w coordinates of F(h. gjx. Here we are considerng the case where . is a diffeomorphism. for notational convenience. Everything we say applies to a flow P by setting g equal to the time - T map of the flow. We will call it- the iwindow length of the reconstruction, since there are w evenly-spaced observations used. We call tnthe reconstruction dimension. since R" is the range space of the map. We may as well assume that n <, w and that B has rank n: otherwise we could throw away some rows of B without losing information. Assuming that B is a fixed matrix restricts the filter to be a linear multidimensional moving average MA filter. Autoregressive tAR) filters in general can change the dimension of the attractor.' ,o, If B is the identity matrix (denoted 1). the map is the original Takens delay coordinate map. \s stated in the previous ,ection. in that :ase. t(1. ' t h,. 11)is almost always an embedding ab iong .:b n i, greater than twice the box-counting dimension of the attractor Ind the periodic points of period p less than n have distinct cigenvalues and make up a set of boxdim < p,'2. Under Fl: ring, some complications are caused by the existence of periodic cycles. On the other hand. the next theorem states that in the absence of cycles of length smaller than the window length it. every movine filter B gives a faithful representation of the attractor. 3 I1 3average Theornm 3.1 (Filtered Delay Embedding Prevalence Theorem . Let U be an open subset of R' , g be a smooth diffeomorphism on C. and lt .1 be a compact subset of U. boxdimA.)=d. For a positive integer n > 11. let B be an it x it matrix of rank I1.Assume ,, has no periodic points Of period less than or equal to t Then tor almost c%erv \mooth function hi.the delay coordinate map Ft B. i. L - R"i. ' I. One-to-one on .1. 2. An immersion on each closed ,ubset C of a .mooth manifold contained in .4. The probe space for perturbing I can be taken to be any ,pace of polynomials in k variables which includes all polynomials of total degree up to 2w. Furthermore. in case it,< 2d. the results of Theorem 3.1 hold outside exceptional subsets of .4 precisely as in Theorem 2.10. For example. consider the3 x 9 matrix I ii i 0 00 0 B=I0 0 0 1 1 1 0 0 0 0 0 00 (0 0 (3.2) 598 Sauer et al. Then F(B. h.g x.,= ( lx) + hi gt.x n + hgit.) x). ( viI" Y" 1)+ hI g4( .x" )I + hh4i .( , O(h( .l '1)+ h( t.v)) + hi g'(x)))) Although the map F(B. It. ,., uses information from 9 different lags. the moving averaae*" reconstruction space is only 3-dimensional. According to the theorem, if the dynamical system ,has no periodic points of period less than c = 9. then FiB. h. ,,is an embedding for almost all observation functions I. Remark 3.2. When the diffeomorphism , has periodic !,,oints. ,:ertain special choices of filters 3 will cause seif-intersection to occur i,ti periodic points. However. under the genericity hiypotheses on the oA'narn!. cal system of Theorem 2.5. for example. almost all choices of an Ii , matrix B imply the conclusions of Theorem 3.1, This follows from Remarks 3.4 and 3.,. A more detailed 'oew of the effect of periodic points otf the dynamical system is given in Sections 3.3 and 3.4. 3.2. Examples of Filters In this section we will list some examples of filters that may he useful in given situations. The easiest example is a simple averaging filter. For any integers in. n. let B be a n x in matrix of form (111a. ) In I'm ... I in where there are in nonzero entries in each row. In the presence of' noise. this filter :,hould perform well compared to the more standard delay-coordinate embedding which uses every rith reading and discards the rest. A more sophisticated noise filter was suggested in ref. 5 for a slightlk different purpose. and elaborated on in the %ery readable ref. 2. where it i*S used for dimension measurements. It is based on the singular value decomposition from matrix algebra. also known as principal component analysis. Let Yt, .... Y.L be the reconstructed vectors in R". where L is the length of the Embedology 599 data series. Following Broomhead and King. 5 1 define the L x tv trajectory matrix where the i are treated as row %ectors, The ot'artance m"ctrux of this multivariate distribution is .41.4. The off-diagonal entries of .4'.4 measure the statistical dependence of the variables. The singular 'alue decomposition''"' of the L x it- matrix .1. where 3 I = V'SLUr .4 where 1 .. an L L orthouonal matrix. C I,, a it x it orthogonal matrix Ithis means that 1' 1=/, U '= i). and S is an L t diagonal matrix t meaning that the entries a., of S are zero if i - ). By rearranging the rows and columns of V and I. ,e can arrange for the . mnqdur ralu,. of. I to s rt, a :,. .. 0. The bottom L - rows of S are zero. The ,ingular value decomposition suggests the uhe of the lilter 8= (. That I.N.instead of plotting the %ectors V ..... i, in reconstruction space R". plot the ectors U, ..... .'.. One immediate positive consequence of this change of variables is the ,tatistical linear independence of the new variables. The covariance matrix of the new trajectory matrix () is .A,) r. IU= SrS. a diagonal matrix. In practice. one can do better than 8 = U . This 14 because some of the nonzero ,ingular Nalues are dominated by noise. .\ rule of thumb ih to ignore (by ,citing to zero I all singular \alues below the noise floor of the experimental data. Ignoring all hut the largest A ,ingular %alues Is equivalent to letting the filter B in Eq. IS.1I be the top A rows of Lr. The rows of L r are orthogonal. so B is still full rank. Theorem 3,1 implies that F(B. h. g)j will typically be one-to-one and immersive. This program was followed in ref. 2. in the context of measuring the correlation dimension of chaotic attractors in a stable way. They used a filter B that consisted of the rows of Lr that corresponded to singular values above lo - '. 600 Sauer et al. 3.3. Conditions on Periodic Orbits Which Imply One-to-One For special filters B,conclusions I and 2 of Theorem 3.1 can fail. but only for periodic points. That is. some periodic points of period less than ,'may be mapped together under the map F(B. It. .). For example. assume B=( (0 i , ) 1 4 4 4 I- -0 4 1 and assume that g has a period.4 orbit, that is. g.(x) = v. Then for any h. F(B.h. g)maps all four points of the period-4 orbit to the same point in R' so F(B. h. g) fails to be one-to-one. There is no way for any observation function to distinguish the four points. ,ince their outputs are being ,iveraaed omer the cntire cyclc. Thus. the filtered delay coordinate map luk. tor ail observation functions i. to be one-to.one. A similar problem occurs with the filter 0,000 0 3.61 t Now F(B, h, . "= t!h(.\') + h(g-(.\'))), I((¢.") + ht( g:\I1v M. \ssume that ihe period.four orbit o, , constts of \ =,t\,, . v.. V1 = .t Vol. and x,= &.\,). Now \,, and ,are mapped to the hame point in thc reconstruction space R' by F(B. h. v i. and the same goes for .\ and v'.Again. the map cannot be one-to-one tor any h. A second obvious problem can be illustrated when the dynamical system has more than one fixed point. No matter how Itis chosen. the filter -- I- B= 0 t-0 3 37 0 t) t 3.7) maps all fixed points to the origin in R', violating the one-to-one condition. In each of these situations, the underlying dynamical system g may dictate that some periodic points will become iu,...:fled under a particular 1'Enibedology ft '601' 'filter B. no, matter how- aeneric-' thd obser~aionri function I:.,On, the -other hand.,these identificationis, occur. only at periodic ,points. Further., even, in, ihe, case of, periodic points., it turns out, that the restrictions onl B edxem-; plified- by 'the, three cases aboveare the only restrictions. That-.is. ifthds&,are, avoided. then' F(B. Iz.g), is one-to-one for a-prevalent set of observation functions h~. To be more precise about tihese restrictions'. we need to -make, some, definitions. For each positive integer p. denote. by ..4. the set of period-p' pointsof g lYind on 4. That is. A,P' .\ .4,: g (x) =,. . 'Let 1,, denote the a x-i identity matrix and (.jdenote greatest common, divisor. WVe wvill use !he convention that (p. 0)= 0. For integers p > q> . define the P x Ip -(p. q)) matrix I IA2. I Iminn. 3 50 3 Define C " to be the r. x (p- (p, q1))) matrix formed by repeating the block (',., vertically, and for a positive integer it. define' C";: to be the matrix formed by ?he.top, ' rcws of C"~ Theorem 3.3, Let C be an open subset of. R'. let q be' a-- smooth diffeomorphism on C. and let .jI be a compact subset of' U of box-countinig dimension d., Let it and it be integers satisfying ivw n> 1. Assume (hat B is an it x it- matrix of. rank /1'wvhich saUtisFc-s:, Al.. rank' BC;:: > 2, ho xdim(. I, for all, 1I p < v rank bC~ oxdimAjib'r a litI p~w Then for almost, everv' smooth function/h. ,F(B. It. -' is one-io-one (in Remark 3.4. Note that 'rank C,,, = p -T(p. qj). and so rank. C"IN mill'wi. p- (-p, q) I It follows that rank BiC" min~n.-p: and rank BC". 3 2 p1 1 for B= /,,-and also -for almostuevery it x it matrix B. To illustrate the restrictions that Theorem, 3.3 puts on moving average Filters, assume that B) is the S,x 6 matrix 0.5). In particular. the filter B must satisfy condition \2 for'p =4. q = 1. which means I 0 o) 1 0 rank,.B - 0-I0 010 > boxdim .4 602 Sauer et al. The rank on the ieft-hand side is Zero. however. and-if there exists any period4 orbit. the filter t3.3) fails this condition. This is consistent with .hat we have already noticed: in the-presence- of-a- period4 orbit, the-map F(B. h..gJ is ndt one-to-one for any h. The filter 13.6) satisfies the above conditio-dhas lone as there:ate finiteiv many period4 orbits. However. it fails condition A2 for P = 4. q 2. which .equires I) tankB -1 0 1 boxdim.4 1 0 This is aiain consistent with our earlier observation. Finaily. if there exist fixed points, the tilter i. l'aiis the condmon XI for ) I-if there exist fixed points. That is because condition AI requires rank B I > 2-boxdim A1, Since the rank on the left side is zero. the cohdiigdn fals unlegs the .et of fixed points is empty. 34. Conditions on Periodic Orbits Which .lmo!V. an Immersion Therc are also rather obvious situations When. cttaui -filters cause F(Bh Iig) 16 fail' anan immersion. Asstiethav- -is a circle that has a fixed point x. Assume-that the derivative of if-qt v-is -2. Consider the filter B . 3.9) In this case. the map F(B. h.g) cannot be an immersion at x forany observation function h. For a tangent vector c in T,:.= RI. the.derivative map is 603 EmbaologY Vh(xI V h( " " xj D F( . h ., " \ V h lg - =6 I. "r , i _gtx x" I D" X I iDx OlVh(xjic. =t) I ) 7 Vh1( x1"1. - .,r I .. so-the tangent map of F(B. h. -, at x is the zero map. In the case of an ie-dimensional- manifold A! with a iixed potnt X. .t can be checked that fo- a filter B of this. type. F(B. I. -,I wlii fail to he an Iimmersion f r all.h as ionri- as -the iinearization " . . as- on_: .env'aiue ot .. .is n the one-to-one case. the immer ion-u%%il fa.i oniv ,,r ncriouic points. To be precise. given numbers c,.c,. define the . <rp matrix :1D'~... cjf.- "' (' 3.10) (310 where I,, denotes the p x p identity matrix. For a- positive integer %i.let ..... cj:If the .... matrix formed by the top w rows of D c,) be the c, are distinct. then rank D,(c, ..... c% = min, w. rp, .. I.- . . . . . Let C he an open subset of R'. !ct , n'e .I ,mouth d iffe9morphism onan C. and let .1 be a compact ,ubset_7-n-*,i a.. , ,mooth • L. , .ssume-0t,'. Let w and n b iefiid-NftKsfvingivw-. '.. -manif~fold in the linearizations Dg" of periodic orbits of period p less than or equal to w have distinct eigenvalues. Assume that B is an n, x w matrix of rank ,, which satisfies: Theorem 3.5. A3. 5 . rank BD;*(l.,. . > oxdim In<. and for all subsets linearization at a point in A., P 4.' r ... - for all I< p < iv of. eigenvalues of the Then for almost every smooth function h. F(B. h. g is an immersion on .4. Remark 3.6. See Theorem 4.14 for a- proof. Note that since rank ( = min w. rpl for distinct eigenvalues ,;,. it follows that rank BD= min{n, rp,} for the original delay coordinate case of B = 1,, and also for almost every n x ivmatrix B. 6~. Sauer er ak. TO illustrtethe- condition X) is asafixed point with Un eiOgenfaiue of -rank ~D(-2 >Q. but :~. o i~ 39 hn~z~ 2. .at -c6nditi~ii reciuiret' that. OD 2r1j 4. PROOFS This section contains the proofs of the esults stated above. After some fundamental- -lemmas, w6 !ivye -thc -proofs of the WVhitney forms of the qmbedding theoremts. These follow 'Umma-4-I1. The proofs.-of the -delay-coo rdinate- forms involvinia 11lters. Theorems 3_3 and 5. :oilow immediateiy frm. Theorems -',L3 and 14-- _rSpeCtIVeiy. FhIS ,:CEuon concludes with the proof of -Theorems 2.- and .-. which ard sed:ai, casds of Theorems 3.3 and 3'.5. Lemmra 4.1. points in R'. and.it, Let i ar's /. be positivc i.atesvers. .... it,, in R. c,.. j%...v, distinct L-in- R'. 1. there exists a Oolynomial Ii in A variables of degree -at most 1 - I such that l'or = I... it. hIv, ,= it.. 2.- There exists a polynomnial-/i in-k variables of dearce-at mnost n such that- for i .t.V/~,~v) Proof. L. We may assumc. by linear chanize of coordinates, that the first- coordinates of ..... ..are distinct'. Then. ordinary onfe-\ .nable inter- polation- guarantees such a- polynomial. 2. First assume A = 1. There exists I poi,nomial of dciiree at most n - I in one variable that inter-polates, the data. The antidcrivative- Is the In the general case. by a linear changc of coordinates. may assume that -for. each j L..k. the ifh coordinates of I,, are distinct. The above paragraph shows that for j =Ik -there is a polynomnial of degre at most n in the jth coordinate . whose derivative h\%. interpolates the 1ath coordinate of it, for i ~i.n The sum of all k of the-se polynomials is a polynomial of degiree at most n which satisfies-the conclusion. Lemma 4.2. Let Ft'x) = x + h be a map from R' to R". where Nt is an~tn x t matrix and R". For a positive integer r. let - - 0.be the r th largest singular value, 01 M. De'note by- Bp, the ball centen.- a..the origin 'Embedalogy -605, .of -radius,p in RK. and' by. B the bal c~rtrtd ar theiorigin of radius,6 in, R.Then Vol ;B% ) , Proof. Note that- -decreasing aiV 'v singular, value of A- does-not dtcrease the Ieft-hiand,,side. Thus we ntay asgumne that the singular values 'O jlf..atis*ya. LetJM=_ VSUrbe, the sinIgular--value tuicOmnposition of M. Here S is a diagonal: matrix with 'entries a and all, other. entries zero. V is an inx n orthogonal matrix. and" U is~a ux torthogonial matrix. Sihke the columns of 'U and J each- form an orihonormal, set. we recop-hizc AID., =jVSL'r ,_as an r-dirnehsional, ball of radius a olvinL!T in R1. In fact. the first r columjns of V-mdhtniied- by th-e factbr r,,) d di which, span _A1B,,. The set, F n-~ -Bp,consists of the, vectors in B0 whose image~by A! lan 'ds in a ball, of radius 6iin R". This is a-cyliridfrial subset of B',. with 'base dimcnfsion r and -base radius,a. The subset thus has i-dimrensionial volume less than W6aj'C~ ',, where , r~~,) eoe the volumeo ihe r-dimc-nsional: unit ~ball. The volume-OTAB, is p'C,. so . Vol 11,F(B,s) (Iap q Lemmiia 4.1. Let S be a.bounded subset of R'. boxdimi) =-d. and let-~ ,~G be 'Lipschitz maps from- S to R". A\ssume that for each v InS. the--rithiarg"s-sin ulaf-valtie of-he ,rx trnatrixM~ :G H GCJ.v is at least T>O0, For each :ce R' define G,, Gj-i-+ x.G.. then for almost every in R', ihe-set. G--'(0) has lower box-countina dimension at most, d- 1%If tr>d.L then, G;(01 is empty for almost every X.. Proof. For a positiv'e number p. define the set B, to be. the ball of radius p centered at- the origin in R'. For the purposes of proving the theorem, we may replace R' by, B.. For the remainder of the proof. we will, say that G, has some property with probability p to mean that the Lebesgue- measure of the set 'of a e B0 for which G%has the property is p times the measure of B.. For example. if xc-S, then Lemma 4.2 shows 606 Surta ,that '+ 'J e.iG~ for cC-3B., wh -Probabiltatios Ldt_ D > d'. and'let,.., - 0 be- such that-for 01 < c~ , the foll'owing,two, facts hold. First, S can: be covered -bV i: "~k-diriihsionAi balls B(x. c)o radius-e. cenitered, at xqS. Second.,by the zLipschitz- codition there -exists a constanit .C such that-the irbagt under any G'.~'..o~.~~bl nR intetse~cting S is contained' in a, -C-baliL i R*7. For ,thd-remaindif Of the prdof., we assume c <-j) The probabilit -y that the set Gj((.,: ij ii o is ,at most -the Probability 'that ;G,(x )I < Cc. which is a constaint, times L'. s ince 1)and' r, are fixed. For' any,positiv& number It. the probdbilitv that at least V! of the c'n' images- -G (B(x. cy)- contain 0, is at most, -C V'"f' Therefore. G,'0.can be covered by fewe& than Mt'='i: " of, the i:-balls. except with probability at -most re 1.As lone as h > D-- r 'this probability canbe-made as small as desired' bv deceasing i:. Let p >.0. There-is a sequence ;;a pprouching o such-that (J'0 can-,be-covered.-bv fewerthan zi: "halls~except--for~ probability at most, 1 Thus, -the lower'boX-Co untinpg' dimension of G. "(0), is at 'mo'st h.except for a probability p, subset, of :(. Sne/>0wa' arbitrar., lower ,boxdi'r1 fG;'01v<I for almost- e"ery :c. Finally. :since -this 461&d for all h.> "d lower bdxdimt G 7(0)) I-r 3 -r. Remnark4. In case boxdimi Sydoes not, exist. the hypotheses Oif the lemma can b'e slightly weakened by allowinii d to, be the lower hox counting, dirridnsion of' S. A slight- adaptation of -the' -proof sho s that boxdim can, be replaced 'throughout Lemhma 4.3 by Ha4usdorff dimension. In,particular.i ir > HD-S'). -thenG;7'M, isempty for altnost every c in R'. If in 'Lemmba 4,3 we assume that rank(M,) ;4dfor each xye.S-instead: of the assumption on the siniiular values, then 6,10V-Is empty for almost every x. That is because one can apply Lemma 4J3 to the' set S, c- S E rth largest, singular value, of' Af,,>_aT to gev -I~l)~~.then SU>S, implies G;"(0) =,. We state- this fact in the next lemma. ' Lemmna 4.5. Ldt S be a bounded subset of R'. boxdimiS) = d1. and let Go, G .. . G, be Lipschitz maps from S to R". Assume that- for each x in S. the rank of -the n x t matrix GVb 1 G, (x-. is at least r. For each i c-R' define G, =G + 1 xG,. Then for almost every 2 in R'. the set G,-'(0) is the nested countable union of sets of lower box-couritihg- dimension at most d -r. If r >d.1 then G-'(0) is empty for almost every 7. 60 tmb eddg-V I Lemma,,4'.6. Let A "be a compact subset ,oflRk."beLet.F, ,,., F, Lipschitz maps ftom A.to R-". For each, integer r ,., let S7. be the -set of" pafrs "-Y in.A for .which the trx-r, matrix 'a j-(X' Flo").... . F( -F 'has ,rank r. and, let d, =lower. boxdim( S, Define F.- F)- ._,, .4,7: R?. Then for =. .. outside a measure zero subset of R'. the ,flowing hold': I. fd .<r for Aintegers r >,O., then-the map F, is one-to-one. 2. If d , r for some integerr >. 0. then for every (i > 0. the lower box- II Proo. V:or t= 0.... deline G, I ( i. On the Net, S.. -i the rank of the n x t matrix ;s r. Ii If r> d,, Lemma 4.5 show that for almost every x e R'. the origin is not in the image of S, under the map G, = ,.",, or equivalently. ; F,(y)1Tor v ,i' in S,. If r > I, for all r. then F, i., one-to-one. since pair x les- in' some S,. H If ,.. d,, let (A x A ,, =, (x, y Iq.4 x A: ,x - Yl ti, bebthe subset of(-distant pairs ,of points in . x A. Since (A x A),, is compact for any J > 0. the minimum of the nth singular value of V,, in A< .I , ,. greater than 0. Lemma 4.3 shows that for almost every i. the orin i i ..i I 1)J) 1< for a subset of (..I x 4 ),, vith lower box-counting dimension at most a, - r. Therefore the (-distant self-intersection subset (F,,)of .A.which is the image of this subset under the projection of (A x .A), to A. has dimension at most d,.-.r. 3 o,(x) 3 1each '1. , Theorem 4.7. Let I be a compact subset of R' , lower boxdim(.-l) = d. If n > 2,d. then almost every linear transformation of R' to R" is one-to-one on A. Proof. This follows immediately from Lemma 4 6 and the remark following it. Let 'F,l be a basis for the nk-dimensional space of linear transformations. For each pair x - y, the vector x - v can be moved to any direction in R" by a linear -transformation. In the terminology of Lemma4.6. S,, =Ax-Aand S, is empty for r-#n. Since lower boxdim,) - 2d< it. almost every F, = P,F, is one-to-one on A. I Io -608,Sure~i Remhark 4;.. It :is initerdsting. tha t, -no statement similar toTheoremA47 can be made if 'box-coUnting, dirnesiii : epacedx' by, Hausdorfft dimension. In. an- Appendix -to -this, work 0provided- by. Kanh. examples are tofistructdd of' comfpact subses .4)6f. a&y .Efclid an, spacO Rk that'haveHausdorff, dimension, d,= 0. and, such that -no pojecion to -A"f~r it < k, 'i .one-to-one onA. This strikinfg, difference between, tboX-c6untintg, dimension, and' Hausdorff dimension, is -related to the fact that Hausdorff dimensionv does not work well With products.. Extra 'h vpo theses are needed on, C. in par-7 ticular on 'the. Hdusdorff -dimension- of the product, A x A. to prove- an' analogue, to Theorem 4.7. For example., M46i has shown (see ref 17 and its correction in. ref. 9. p.,611), that if ii,> HD(A 5<A) +J-. thenr the conclusion (if theorem 4.7 ag-ain 'holds, Of' course. using. Lemma 4;3 n Remark 4-.. ,it turns out that only it.> I-DA x<.4) is required: Theorem 4.9. Let A be, a compact, subset oft i.nd ]Lt,it > H-D( .{. -.1 ). Then almost eyery linear transformation- of P% o R is one-to-one on A. It- was showvn in ref. 10 that under the hypothese of Theorem 4.7 almost every orthwgonul'projection is one-to-one -(and -in-flact ha~sa, Hbider continuous inverse). Definition 4.10. For a compact, differentiable mnanifold 11. 'let =(X.Tu): Xv e Al, v T%At I be the tatkgew bundle (if Al. and 'let =(x, ',A) .v) e T(MA); Irl = 1 'denote the unidt tangent bundlile o' Nt Lemma 4.11. Let A1be a, compact subset of'a smooth mnanifold' embedded in R'. Let F, 1 . P,: R R" be a set of smooth. maps fromn an open neighborhood 1U of A to R'11 rt- each positive' tnteer r-. let S, b-,, the subset of the unit tangent bundle St.]) such that the it, t matrix has rank and let i, =l1ower boxdimiS'). Define: F U-R". Then the' following 'hold: = F,,- +, . ,: I.- If df- < r for all integers r > 0. then for almost, ever;' : e R?'. (Ihe map F, is an imtmersion on.. 2. If dl, >, r for some r >, 0. then for almost every 7 e R'. F-, is an immersion outside a subset of .4 of lower boxdim, <, (1 - r. Proof. For i =0,..., 1. define G,: S(A) - R" by G,(x. vti= DF,(x) v. If r> di, for all r > 0, then Lemma 4.5 applies to show that for almost every G (0) r)S, is the empty set. Since S(A) is the union of all S, G'() G.- Em-dolgy ,, . , A' ,3 009 ' i I.", SF, I. is'empty. Thus. no ufit.tangent vector issmapped to, the origin -anirhimersion. ,and F 1 is Incase,.;ld -for some ,r. there -is a.,ositiVe !lower bound -on the singular vaes, o'the Q., on S(A); Lemma,4,.3 imp!ies thatl therel-ss .s.bset of unittangent vectors-.. of lower- boxdim 4,-r, that, can map to zero., The projection 0f this subset into .4 has lower boxdim,;!,- . 3 Proof of'TheoremS 2.2. 2.3, and2.; 10. Theorem 2.2 is a special case -of Theorem ...To. prove the 'latter. we need to show that a prevalent set ofmaps areoie-to-one and immersive. Let F,...,.., be a basis for the set of linear transformations from the notation, of Lemma 4.6. the set S,, = .4x.4 .1 and S,= ,0 Rk - R", In for r z . Sincc'boxdim,4 x A) F7 is one-to-one on A for almost n,t2d< F, are. added. the rank of .1,, every ze R'. If any other maps F. .... cannot drop I'or any pair .\v. so almost every linear combination of F.is one-to7one on .. .... The -proof of the immersion half uses Lemma4.1 1 instead of C is a subset of a smooth manifold of Lemma 4.6. Since boxdiml{) = d, dimension at most d.'and therefore boxdim S(C) < 2d- I. In the -notat .on of Lemma4.11. S,,=S(C) and S,= 0 for r On. Since i>2d>2d- I boxdim S,, the proof follows from Lemma 4.11. The proof of Theorem 2.10 is similar, except that the second part of the conclusions of Lemmas 4.6 and 4.11 are used. For example. in the use x .4- J and S, = 0 for r ti as before, but now of Lemma 4.6. S,, = A4 boxdimt.4A x A)= 2d> it. Thus for each 5> 0. for almost every F, the j) has lower box-countin dimension at 6(F, 6-distant self-intersection set most 2d - i. The immersion half is again analogous. I Let ' be an open subset of R'. let g: U'- L be a map. and let h: U-, Rbe a function. Let w <i - be integers and set w= Iv"-iv- +1. For I <i<i, set g,=g,,--. so that .,=g" and Definition 4.12. g,, =g ". Let B be an i x w matrix. Define the filtered delay.-coordinatemap F,,:(B h. .0: U-.R"1 by r F,,I(B. h. g)(x) = B(h(gl x) ). hM g2(x) .... h(g,,x))) = B(h(g" (x)),.... h(g" Ix))) r Theorems 2.7, 3.1, 3.3, and 3.5 are corollaries of the next two results. for which we will use the following notation. Let g denote a smooth diffeomorphism on an open neighborhood U in R '.Let h ,.... h, be a:-basis for 4th6, polytnomials in, k.arables, of .degr6 at,. most l.' f. a srmooth funicition iZ) on i~adrR. deffine- Ii. -r7 - ',hFor, each p9l4eitgrp enote, by':. tias t, period-p -p6intsof lynonA That x.1~ txA: g~x . ~the~rniatries C b-e as in iheorem, _31.3 . Theorem 4.1,.Lx be a, smnoth diffeohqrphism-on an, open. ,neiahborhood- U 6f,' R ' a~dltAb b~atsbe fU odm K d. Let it and- w it r bhe-integers. iv< it w-- Assumne'that, the It' it X w maffix-ff'satisfie's: A2. tlank BC"' > 2 b6xd.imni.- )for a1lf I <p rank PC> boxdimfi A~ for all I <, q~ <:P < Let ., It be a -basis for the polynomials in k variables of degree at most, 2W. Then for any smooth functionvi, on R', and for almost every x R'. 'the I'dilowinm: told: If it > U. -then FitB, It,. g j: U' R" isone-to-one_ of I-[ 2.If it -, 2. then for -every Ji > 0, the i-distant self-intersectiotv et 1;Fw h, ), 6) has lower box-counting dimension at most 2d- it 'Proof. For i I.. defin-. I. /hI, (g 1 By definition. F(B,., 2 )~ F, To use Lemma 4.6. we need to check for each x vthe rankoof the matrix Al"= I F,(.) - F,(y I.. .(xVI - .(yII which can be written as q<_ 2w. the :, are distinct, and J = iV is a ivx q matrix each of whose rows consists of.zeros except for one I and one 1. By part I of Lemma 4.1. the 3 EMbedblog'y61 rank of H is. q. We divide-the study oif"the rank of. Af ,, i-8JH into-three I cases. Case I: .vand.Yare notboth periodic with-period <-i, Invthis case. J,, is upper or lower triangular_and, ank,,,,) w.Since B.. I. and H are onto linear transformations, the product BJH is onto and has rank nt. The set of pairs x yv of case I 'has box-counting dimension at most 2d. and rank(M.,) = n.If g has no periodic points of period < w,we are done. and conclusion I (respectively.,'2) of Lemma 4;6 implies conclusion V: (resp., 2) of the theorem.. The remaining twocases are necessaryto deal with periodic points of period < ii, We show that conclusion I of Lemma 4.6 appliesin both cases. Case 2: v and y lie in distinct periodic orbits of period , Assume p and -q are minimal suchthat ,i;(.1 =x.. -,(yv= ; and that I < q < p , w. In,this case the matrix J,, contains a copy o .,Since H is onto, rank .1,,.=-rank BJ,,H= rank-.BJ,,. By hypothesis. rank BJ, . rank, BC' > 2. boxdim .. ,,which is the box-counting dimension of the set of pairs treated in case2. By Lemma4.6. for almost ,very xeR', '.(xJ-#F,(.-) for every such pair x # y. -Case3: Both v and v lie in the same periodic orbit of period < w. Assume P andq areminimal such that g(-x) = x.g(x)= y. and that q < p , . Since v.' nd *vlie, in the same periodic orbit, the column space of J,. contains the column space of C" . Thus. rank BJ,, I= rank BJ, - rank BC > boxdim .4,p, which is the dimension of the pairs x # iyof case 3. Now Lemma 4.6 applies-,to give the conclusion. | aI Theorem 4.14. Let g,be a smooth diffeomorphism tn an open neighborhood L' in R'. and let .. I be. a compact ,ubget of a smooth "i-manifold in C.Assume that the linearizations of periodic orbits of period less than w have -distinct eigenvalues. Let a < itbe positive integers as -in Theorem 4.13. and assume that the it x it- matrix B satisfies: A3. rank D (;,, ,>boxdimtA +r- 1) for all I p< i. I<r<, and for all subsets ,..,, of eigenvalues of the linearization DgP at a point in .-1 . Let h ...... h,be a basis for the polynomials in' k variables of degree at most 2w. Then fr any smooth function h,) on R .and for almost every ;c e R'. the following hold: . If n > 2m. then F(B, h., g): U-- R" is an immersion on .4, 2. If n < 2n, then F(B, h, g) is an immersion outside an exceptional subset of A of dimension at most 21n -t- 1. Proof, To, app, 'matrix Lemma,4.l ,D F,.( \) -we need,to check the rank of the,,ii=x v .. DF.I x(.v )), (4 .1) for each Lin vx. the unit tangent bundle S(A )LFor a given observatlon ThAction h. the derivative of7F(B. h g)his hFBl. l'=&I'a, U,. Vh~g'-()) g (v)/ If x is not a periodic point of period less than w, then g""- v)..... (.x are "l.vI distinct poinis. The facts that ,, is a diffeomorphism and r ?0 imn'l that D9"(.V)L'?=( for all i, Thercfore by Lemma 4.1. part 2. the set of %cctors 'DF(B. hi, :O.x,: ;c-=_ R'' ipans R". In the notation of, Lemma 4 II. the subset S, contains allpoints of, S(.ij:that are not periodic with period, less than ,w, and d,, = lower boxdimiT,,) < 2m - I. If g has no periodic points of peribd less than w, the proof is finishcd. by Lemma 4.11. If x is a periodic point of period, p <u. then HP It'l v B DF(B. It. g .)t= IfD Hl D where •,=g 9 i '{IX)= vp, bi, = VIZ~x, ) wt'-"Dgt. , )..- Dg{."x1}Dg" (.xkt Dj= Dg(x, I)"".DgxtI) Dglx,,)... Dg(x,) Each matrix D, has the same set of eigenvalues ,. ... and by hypothesis, they are distinct. If u t.... I is a spanning set of eigenvectors for u,,, DI, then it checks that uji= Dg(x,_-1 )...Dg(x)u, for I <i<p, 1 <j.n defines a spanning set (uj, ....u;,, of eigenvectors for D,. Thus. if !1! EribedologV 613 lot =7 , auis the eigenvector expansion of wi then the eigenvector expansion of w is Z,". 'a~uq, which has the same coefficients. i Ii Thus DF(B. h. q)(x)Iv can, be written as Bftimes the is-vector V. " • l0 "! .. . 0 0 0 ... 0 /.,o, I'Io 0 0 ,. 0 4M 0 ... o 0 ... 0 HI .+ + 1011)0 ,,; ..t.;,0 0H0 l r (42) (0 0 .. 0 ;" ', ;.i ...0 To find the rank of the matrix (4.1),for (x, v) where x is periodic, we need to find the span of B times the vectors (4.2) for I = h, = T x,,. i e R'. Assume that the eigenvector expansion of v has exactly r nonzero ....a,. By Lemma4.1, part 2, the set of vectors 'Vht,(.,): coefficients ot e R', spans RI. Then because the it,, I < i n.are linearly independent. the vectors of form (4.2) span a space of dimension min[w. rp as x spans R. I,,, Therefore, for this iv. ri. the span of the vectors 14.1) has dimension By hypothesis. the hoxdim of such equal to the rank of 1_D.,. . pairs (x. v) in S(4 ) is boxdimt..I) + r- 1. By hypothesis. the rank of the it x t matrix (4.1) is strictly larger. so that Lemma 4,11 applies to give the conclusion. Proof of Theorem 2.7. Apply Theorems 3.3 and 3.5 with B= 1,5. According to Remarks 3.4 and 3.6, the conditions AI-A3 translate to p>2.boxdim(A.), p/2>boxdimA,), and min{n. rp' >boxdim.4,)+r- I. respectively, for I < p < n and I < r m. Thus, the hypothesis boxdim(A,.) < pi 2 guarantees that AI-A3 hold. Proof of Theorem 3. 1. Since Ap is empty for I < p < w. the conditions AI-A3 of Theorems 3.3 and 3.5 are satisfied vacuously. I 614 Sauer-er al. APPENDIX. HAUS,ORFF DIMENSON-'ZERO SETS WITH NO ONE-TO-ONE PROJECTIONS Intai Kan The purpose of this Appendix is to construct a Cantor set C-R" whose Hausdorff dimension is zero and which has the property that everyprojection of rank less than ti is not one-to-one when restricted to C. Definition A.1. The iautisdvrt s-drnetnstota outer iieasurec'uI a set K is "'(K)=limn inf where the L.!' U |.ntimum is taken over ail covers C ' .I' K vith the diameters o the C. uniormity less than o. The -.auurlt ad.wension ol a nonempty ,et K is the unique valu o1's such that 1K)= f. if t<s and ty"(K)=0 if '. Example A.2. We construct tl-.e subset C of R' as the union of two sets .4 = iJ',., , and B =,', 8, each of Hausdorff dimension zero. with the property that for any projection P of rank less than ti theimages under P of A and B intersect. and thus P is not injective when restricted to C. The set .1, lies on a face of the unit m-cube and a = (a1 .,a,..,,) is in .Af, if it satisfies the following restrictions on the binary expansion a, = a!aa(I ... of its coordinates: I. If i=,#. then a = 0. 2. If i:n and k i.o. then either Ia)a.=1) for allI =_0.k, .,t]: or (1:) at'= I for all Ie(.Vt, f. .. I]. Here the sequence 0 =Mo0 < .t ... increases sufficently rapidly ,o that limtfM,. 11 ,) = -..If i =n. then the orthogonal projection of .. , on the ith coordinate axis is a Cantor set which can be covered by 2" intervals of length - . where r, = k + 7". 1 (V,,- _ . Thus. 1., can be covered by 21'- "' cubes with edges of length 2 , Since r, Af,. we see that lim. - (i- I ),,. - I =0 and both the lower box-counting and Hausdorff dimensions of A, are zero. Since .4is the union of rn copies of A,,, we see that both the lower box-counting and Hausdorff dimensions of A are zero. 'Department of Mathematicai Sciences. George Mason University. Fairfax. Virginia 22030. EmnbeddobgY :615 'The-'set ' lies on ,a face of 'the .unit ,n-cube opposite A ,4, andb is -in B, if it, satisfies the -following, restrictions on tuhe' biniary e.xpansion of 'its, -coordinates:' V.Ifi~.then b,'=: 1. 2. If i t n and ,k 0, then either ta) h'1-'0,foral1-' IG( or (b) h'=,,l 'f6r~aI I,''k IM J1.k +]. k MA,21; -V'k Here H,,L is as above. The lower box-countine and *'Hausdorffdimensions of B are zero. The Hausdorff dimension of C = AukyB, is zero. Let P denote a projection ofi ranik less than in, Let~vc; i U1 , v. ...,)i the null,".spaice of P be chosen so that for all i and r, 1,for some particular n. We now show -that P-r estrictdd to C is not injective by 'finding some 1)= B, and a e.., such that r = h --. Using t~he 'binarv* expansiofr notation. %v,def-ine it and 1)as f'ollows: IfIi =u. then d (Yand h'= 1,. 2~ Ifj'11 and k >0, then '1a). a~ 0 and b" r' for all ,C VU., MA.I ; and. b) a',> (v'i. I-) mod'2 arid h>, I for all IcoordinateI Clearlv we have, v = b a.and ,by the definition,,of A,, and' B., we also have aeC 4 and 1,e B,.3 *additionially * ACKNOWLEDGMENTS The rdsearch of T.S. and J.:\.. was supportc' d by the Applied and Computational, Mathematics Program of DARPA. that %)f J.A.Y. by, AFOSR and the U. S. Department o1f Energy Bgasic Energyv Sciences ), "and'that if M.C. by grants to the Santa Fe Institute. including core funding from the John D. and Catherine T. MacArthur Foundation. the National Science Foundation. and -the U. S. Department of Energy. REFERENCES L.H. Abarbanel. R. Brown. aznd J. Kadtke. Prediction in chaotic nonlincar systems: Methods for time series with broadband Fourier spectra. preprint. 2. A. N1. Albano. J. Muench. C. Schwartz. A. Mees. and P.Rapp. Singular value decomposition and thc Grassbereer-Procaccia uaorithm. Piws. Rev'. .1 38:30-17-3026 (1988). 3. V. 1.Arnold. Geometrical Mtetlhods tin the Theori. ol Ordinart' Differential Equationis (Springer-Verlag. New York. 1983). 4. R. Badji. G. Bropgt. B. Derighetti. M. Ravani. S. Ciliberto. A. Politi. and M. A. Rubio. Dimension increase in filtered chaotic signals. Phiy.. Rev. Lett. 60:979-982 (1988). 5. D. S. Broomhead and 0. P. King, Extracting qualitative dynamics from experimental data. Plwsicat 2OD:.217-236 (1986). 616 Sauereuet al. 6., M. Casda Ii. Nonlitiear prediction of chaoic time series Phymrca 3-46:335-3S6 11989,1. -CasdaalL -S. -Eubank& -D. -Farmcr, ad J. -Gib~on. Stae-sbace rccbnsrc=.:on-_in..the presence oi noie. preprint. 3. W. Ditto. S. Rauseo.; an 'd M. Spano. Experimental control- of chaos. P!:. Re. left. 65:32 11-z--14 11990). .M. 9. i.-P. Eckmann and D. Ruelile. E-codic thecorv of chaos and stranize 2ttractors-, Ra-. Meutd Phfs. 7:6174~56_(1985). 10. A. Eden. C. Foias. B. Nicolaenko. and Rt. Temnam. HIder-continuity for ie.in'. erse of -'Maiiis-projection.-Compies-Rendus.-to aippear; 11. K.' Fdlconer.-Fractul-GeoirneirrM ile'.. New York-. 1990). 12. 1.- D. -Farmer. aind, J. Sidorowich. Predicing chaotic time efies. -Phui. Rer. :Leir. 59:845-848 (1987). I3 1 . Farmer-and J. Sidorowich. Exploiting chaos to piedict-the., Wuture and reduce noise. Technical Report LA-UR-88-901 Los Alaros National Laboratory u1988). 14. G. Gjolub -and t. Van--Loan. Mfairixr Ciiinputatiums..,nd cd. (Johns -Hopkins Uhi'.crsity Press. Baltimore. Miaryland. l91. 15. E. Kostelach and- J. Yorke. Notse reduction: Findine the- mimolet d.1namicat s%,icn consistent ".nh the data. 1-Pua 41 Di_" ,- 146 11990). 16. E. Kostelich and 1. Yorke. Nise reduction in dynamical -t.ic .':. - 38:14%-I 652-I l9881.17. R. Mafia. On thc dimen'sion (if-the compact invariant,- -ets off certain nonlineair nips in Lecture Notes in Mu:thilinuuc. No.-IS99 (SprinL'er-VerlaL'. I98!). Ig; P. Martcau and H; Abarbanel. Noise reluctivin in ch- ic. time Nec.N u-,inel 'caled probabilistic methods. preprint. 19. P. Mattila. HausdoriT dimension.' orthogional projections -and -intrsections- u'ith planes. Ann. A-cad Sc. Fcnn. MthI. 1. 2 "4 (1975 1. 20. F-. Mitsckc. M. \18ller. .ind W Lance. Meawuuifi filtered chaotic -Ienais. Piwi. Rer. 1 37:4 S-452 1- (I (AX ). 21. N. Packard. J. Crutchfield. D: F-armcr. and R. Sha..-Gcometry from ti line -crics. Phi. Rev. Lcit. 45:7 12 (1980). 2.W_ Rudin. Real- and (Ctptw Attahis. 2nd ed. iMcGraw-Hill. New York. 110741, 23. J.-C. Roux and H. Switncv. Topolocv Of chaos in a chemical reaction. in Vwtnar Phenomei~na tit Chemical Dv-ratturs. C. Vidal and A~. Pacault-. eds. i Spriket. Bterlin. 111.1(1 '4. B. Hunt. r. Sauer and J. Yorke. Prevalence: N\translation-in'.ariant -.lIm(N~ :%crv'Il in. iite-dimcnsional spaces. prcprintt. 25. T. Sauer and 1. Yorke. Statistically N.elf-simiiar %ets. preprint. 26. J. Sommerer. W'; Ditto. C. Grebom. E. Ott. and M. Spano. Experimental confirmaion (if the theory for critical exponents %;fcrises. Plhs. -Leit. .1 153:105 -109 11991 ), 27. F. Takens. Detectine ttraniie attractors in; turbulence. in Le-wure Notes~ tit 1lii',iitiic. X98 (SprinL'er.Verlae. 1991). 28. 1. 1ownshend.* Nonlinear prediction of speech signals., preprint. 29. H. Whitney. Differentiable manifolds. . Ign. M~ath,. 37:645-680 (1936). 30. 1. Yorke. Periods of periodic %,olutionsand the Lipschitz constant. Prnt (Im.Wtful. So. 22:509-5 12 (19691. Printed by Catherine Press. Ltd.. Tempelhof 41. B-8000 Brugge. Belgium -Nonlinearity 4 (1991) 1183-4212. Printed in the UK A numerical. procedure for finding accessible trajectories on basin -boundaries* U: 'Helena E 'Nusset§, and James A Ybrket + tlnstitute for Physical Science and Technology, University of Maryland. College Park, MD 20742;-USA tDepaftment of Mathematics, University of Maryland. College Park, MD 20742,. -I USA Received 291ianuary 1991 I, Accepted by J-Sinai Abstract. In dynamical systems examples are common in which two or more attractors coexist, and in such cases the basin boundary is non.empty. Thc asiri borndary is either smooth or fractal (that is. it has a Cantor-like structure). When ihere are horseshoes in the basin boundary, the basin boundary is fractal. A relatively small subset of a fractal basin boundary is said to be 'accessible' from a basin.However, these accessible points play an important role in the dynamics and, especially, in showing how th~e dynamics change as parameters are varied. The purpose of this paper is to present a numerical procedure that enables us to produce irajectories lying in this accessible set on the basin boundary, and we prove that this procedure is valid in certain hyperbolic systems. AMS classification scheme numbers: 58F12, 58F13, 65005 1. Introduction Dynamical systems often have:quite different behaviour in different open sets, each open sethavingits own attractor. These open sets maybe the basins of attractors. We are interested in the boundary on the common boundary between such- open sets. The common behaviour may be either smooth or fractal. A point p on the boundary of an open set U is accessiblefrom U if there is a cui-ve'lying in U U {p) which ends on-p. The'basin boundaryis the set of all points on the boundary of a basin of attraction such that each open neighbourhood of p intersects at least two different basins of attraction [GOY1]. If the basin boundary is smooth, then each point on the basin boundary is accessible'from two basins. In particular, if the basin boundary is a curve, then all of its points are accessible. When the basin boundary is * Research in part supported by AFOSR, and by DARPA under the Applied & Computational Mathematics Program. Permanent address: Rijksuniversiteit Groningen, Fac. Economische Wetenschappen, WSN-gebouw. Postbus 800, NL-9700 AV Groningen, The Netherlands. i 0951-7715/91/041183 + 30$02.50 © 1991 IOP Publishing Ltd and LMS Publishing Ltd 1183 184' IHfENusie and , A Yorke fractal, ontlyia rel tively msmall subset0f the: basin §6undary consists- of adcessible points. -and generhlly no points that are accessible' from a ,basin will be accessible frf Ather basin; A collection. of papers have assumed"- that investigators can :pro ... accessibleitrajectories onbasin boundaries [AS ], [AY], [GOYl], [HJ], but ,no rigorous procedures have been presented; For more details, see the discussion in section 6. Studying dynamical: systems; one often observes transient chaotic behaviour , apparently due to the presence :of horseshoes. It is well kn6Wn [MGOY]' that transient chaos is present Whenev -there isa fractal basin boundary separating the basinsof.two or more attractors. Foi example, for-suitably chosen parameter values, the-Hn6n map 'has attractiig periodic orbits with. period 3 and 5, and also a non-attracting chaotic invariant set in the basin boundary, and one observes that the duration of the transient chaotic-behaviour-of many trajectories is rather short before they. settle dowr 'o. one of these two periodic attractors, Other famous examples with chaotic traasicnts, due-to a-bounded non-aitracting invariant chaotic set in-the basin boundary, are the-forced damped pendulum and-the forced Duffing equation. Transient chaos is also present if. there is a chaotic invariant set-in the interior of the ,closure of' the basin. In this case, the basin boundary -can be either fractal , or smooth 1[KG], [NY1], [N t21. Let M be a smooth d-dimensional manifold without boundary with d ; 2, and let F-be a C3-diffeomorphism from.M to itself. "For xi y in Mwe denote by p(x, y) the distance betWeen x and y. A set S.c M is positively invaiantif .F(S) =S, 'aindis invariant if-F(s)=s. ,For xEM- and a closed set ScM, we write P(x,S)= min{p(x. ,,: y e S). An attractor A is an invariait, compact set in M such that0(1) there exists an open -neighbpurhood"U of A such that, for each x E U-the distance p(F-(x), A)-- when n- 4 c; and (2) there is a point x eA such that the closure of the- trajectory (F"(x)},,. 0 equals A. A generalized. attractoris the union of finitely many attractors. We say a~region'isan openand boundedset in M; a transientregion is a region thatcontains no attractor. F6r an attractor (or. a generalized attractor) A we-.say, -the domain of attraction of A is-the set of all points -x in M.-for which p(F(x), A)- 0'as n--*'. The basin boundary is the: set of all. points, ie M :for which each open neighbourhood has, a non-emptyintersection with at least two different domains of attraction, see [GOY1]. In the-literature, for-an attractor A the notions 'domain of attraction, of A' and- 'basin of A' -are often equivalent. On the other hand, in other studies of dynamical systems, the notion 'basin of A' is defined as-the region in M that is the interior ofthe closure of the domain of attraction-of A. Thereforefor an attractor (or generalized attractor) A we define basin {A} to be the interior of the closure of the-domain-of attraction of A. We would like to emphasize thatba~in(A). is associated with attractor A and may include Cantor sets of curves ,that are not in-the domain ofattaction of A; that is, the trajectories of all the points on these curves will not converge to the attractor A. In the forced pendulum example in section 3- we show numerically that basin{A} does include such an invariant Cantor set of-curves. We -will. be studying transient, regions- in cases where the trajectory through almost every initial point eventually leaves the region. We investigate special trajectories that remain in such a transient region for all positive time. In [BGOYY]i [GNOY] a numerical method (involving the bisection procedure) for finding trajectories on the basin-boundary was presented. The papers [NY1], [NY2] introduced the PIM triple (refinement) procedure and the accessible PIM triple Accessibletrajectorieonbasin boundaries 1185 (refinement) procedure.and'Both these numfeficAl-trajectbries refinement procedureg respectively, enabi6 us tothat obtain numerical ,trajectories, accessible stay (for positive, time) in a soecified transient region in M. 'In [NY2]. these two iefinemenr procedures were shown to be valid for uniformly saddle-hyperbolic dynamical: systems, for which the dimension of the, unstable ,manifold of any nonwandering point in the transient region was assumedto be one dimensional. Let R be a transient region 'for F. The stable setS(R)'of Fis (xe R :F"(x)'eRfor n,= 0,1,2,. .. ; the unstable set U(R) of F is x eR :F-"(x).e R for n = 0, 1, 2,... }. The set of points x for which F'(x) is in R for all integers n is calledthe, iniariantset Inv(R) of F in R, that is, Inv(R) - S(R) n U(R). A component of S(R) (resp. U(R)), which contains a point of Inv(R) is called a stable (resp. unstable) segment. We call Inv(R) a chaotic Saddle when it includes a Cantor set. These notions are illustrated in the following example. Example 1. An S-shaped horseshoe map is an invertible map that squeezes, stretches and folds a rectangle into an S-shape area as illustrated in the figurebelow. We consider the S-shaped horseshoe map g, which is defined on a neighbourhood of connected set W, where W is the union of a rectangle E and the two half disks DA and Da as indicated in the figure. Assume (1) g maps W into its intcrior, (2) the intersection g(E) n E consists of three horizontal strips, say H1 , H-2 and H3 , and (3) the half disks DA and Da include fixed point attractors A and B respectively. Let V1, V2 and V3 be the vertical strips in E (stretching the full width of E) such that I'j) =,Hi, I ---i -< 3; see figure 1. "'It is well known, see e.g. Guckenheimer andHolmes [GH], that under ', reasonable assumptions, almost every point will be attracted to either A or B, the stable set S(E) of g with respect to E is a Cantor set of vertical curves, and the unstable set U(E) of g with respect to E is a Cantor set of horizontal curves. All components of S(E) are stable segments, and all comiponents of U(E) are unstable segments. The intersection C of the stable set S(E) with the unstable set U(E) in E is a chaotic saddle. Note that all the points on the chaotic saddle C stay in the box E for-all time under all forward and all backward iterates of the map g. The set of points in E that are on the basin boundary is the stable set S(E), and the basin boundary of g is fractal. One might choose the transient region R to be the interior of W minus two small closed balls that are centred at the attractors A and B. Ia-c6mpact, 1 U 12 H13 '9,,: Figure 1. S-shape horseshoe map: vertical strips in the rectangle E are mapped into horizontal strips in E, namely -:V) = H1, F(V2) = H2 , and F(V3) = H3 . The half disks DA and DB each contains a fixed point attractor, and each is mapped into its interior. 1186 H E Nusse and TA Yorke We assume throughout that (1) foir the transient region R the set Inv(R)_is non-empty,, and(2), there exist two generalized attractors A and B; afid'each point ,in R'that escapesroffm R, under iteration-of the map Fis either in basin{A). or in :basin{B}, and the basin boundary. is the common boundary of basin{A} and, basin{B}. We will refer to R\S(R), the complement ofthe stable set S(R) in'the transient region-R, as the transieit-set.Recall that a point p,in S(R) is accessible from an open set V if there is a continuous curve K ending at p such that K\(p} is-in V. We investigate the cases where V iseither basin{A} (or basin(B}) or isthe transient set R\S(R). In-this paper we emphasize points accessible from basin(A} rather than from basin{B}, just to simplify notation. Obviously, if a point p in S(R) is accessible from the transient set R\S(R) and p is on the bas boundary, then p is-accessible from either basin(A} or basin{B}. On the other hand, S(R) can contain points which are notin the basin boundary, and such points might beso numerous that they block the access to the basin, boundary, that is, every curve in basin{A} that goes to an accessible point wouldnecessary pass through points of S(R). Thus no points of the basin boundary would be accessible from R\S(R). Naturally S(R) would have its own accessible points, but these would lie in basin{A} (or basin{B}). This situation occurs in- the previously-pmentioned pendulum example. Hence, S(R) might contain points on the basin boundary that are accessible from basin{A} (or basin(B}) but not.accessible from the transient set R\S(R). In example 2 below, S(R) contains such points in the basin boundary, Therefore, the accessible PIM triple procedure [NY2J for finding accessible points on S(R) is, generally speaking, not a-procedurefor finding accessible points on the basin boundary. We would like to point out that there are cases where S(R) equals the set (basin boundary l R), (though this condition may be hard to verify). In such cases the ASST method (involving the accessible PIM triple procedure) might be used for finding accessible trajectories on the basin boundary. Example 2. In this example, we illustrate the fact that S(R) can contain points that are not in the basin boundary, and for simplicity we present one-dimensional maps. Consider-two one-dimensional maps with attractor A (which is -,o) and attractor B (which is +-). Let f and g be the piecewise linear maps of which the graph is given in figure 2(a) and -2(b) respectively, such that g(y) =f(y) for all y < 1. 3 Figr(.On-iesonlmp) y adg(h grah f" .1 n-yregvnin2atn I 21 13 pq p y - Fgue 2. One-dimensional maps f and g (the graphs of f and -g are given in 2(a) and 2(b) respectively). When we choose the transient region R to be the interval (-2,3), the stable set S(R;f) equals the basin boundary, and the stable set S(R;g) is strictly larger than the-basin boundary. AcCessible trajectorieson b'sin boundaries -1187 Let p'and q denote the two fixed points of g in (1, 0), and write m = -min(g(y):y,>>p}. Assume, 1 <m <2 <p <q <3< g(m), see-figure 2. The maps are constructed-in such a way that basin(A},and basin{B} of g andf-coincide. Hence, both-maps have the sam6 basin boundary andit is contained in the interval [-1, 1]. .Note that the basin boundary is the set of all pointsih [-, 1].that stay inside [-1, 1] under all positive iterates of the map f (or g), and the basin'boundary is fractal. Onthe other hand we have, all points in (1, oo) go to attractor B under forward iteration of the-mapf, whereas basin{B} for g includes a chaotic saddle in the open interval' (2,3). When wechoose the transient region R to be ,the open interval (-2,3), the stable sets S(R;f) and S(R;g) are the sets of points that stay in R under all' forward iterates of f and g ,respectively. We have the basin boundary equals the stable set S(R;f), but,the stable set S(R; g) -is, strictly la'rger than the basin boundary. It can be shown thatpoints of S(R;g)-S(R;f) can be found arbitrarily close to each point of the basin boundary. We-would like-to address the following problem. Accessible basin boundary static restraint problem. Given a segment J that has one end point in basin(A} and one end p6int in basititB), describe a procedure for finding a point on the basin boundary (in I n S(R)) which is accessible from basin{A). We will state a procedure (the accessible basin boundary refinement procedure) for finding accessible -points in M on the basin boundary. We will show it is valid (guaranteed to work) for the same class of hyperbolic dynamical systems as in [NY2I, namely hyperbolic systems in which the unstable manifolds are one dimensional. All the procedures are based on our presum ed ability to specify an initial point p and compute the time TR(p) its trajectory takes to escape from R. For applications, we need a 'dynamic' version of the 'static' problem above, since we want to produce numerical trajectories that are accessible from basin{A}. The 'dynamic' problem that is associated with the 'static' one is the following. Accessible basin boundary dynamic restraint problem. Given a line segment J that has one end point in basin{A} and the other end point in basin{B}, describe a procedure for finding a numerical trajectory on the basin boundary that starts on J and which is accessible from basin(A}. The ideas of the 'accessible basin boundary refinement procedure', which solved the 'static' problem, can be applied to solve the 'dynamic' problem, in such a way that implementation is possible on a computer. For more details, see the discussion in section 6. The organization of the paper is as follows. In section ,2we present the 'accessible basin boundary refinement procedure'. Then, n section 3, we discuss some examples in which the straddle method involving this refinement procedure has been used. The main result for the validity of the refinement orocedure for hyperbolic systems is stated precisely in section 4, and this-result is proved in section 5. Section 6 is devoted to the discussion of ths associated numerical method (the accessible basin boundary straddle trajectory method or ABST method) and related 1188 H E Nusse and J A Yorke ntimerical methods. Finally in section' 7i the case- of d-dimensionAl hyperbolicsystems, d -_3, and smoothness of Fare discussed, 2. The accessible basin boundary refinement procedure Let -the manifold M, the diffeomorphism F; the tr" ient region R, and generalized attractors A andB be as'before. Recall that We as:. .- that each point that leaves R under iteration of F is either in basin{A} or in basin(B}. The escape time TR(x) of a point xin R is defined by TR(x) = min(n -_1 :F"(x) 0 R}, and TR(x) = if F(x) ER for all n :, 1. We say, TR(x) = 0if x o R. Let J be an unstable segment in R. The notation {x, y} for a pair, means that x and y lie on J. Since J is homeomorphic to an interval, we may assume it has the ordering of an interval. For {x, y} we always assume for convenience that the ordering on J is such that we may write x <y, and denote [x, y]i for the segment on J joining x andy. Let L cJbe any-connected subset of J. Assume L intersects the stable sei S(R) transversaily, and let (a, b) be a pair on L. For each e >0, an r-refinement of {a, b} is a finite set of points a =go<g 1< ... <g = b in [a, b]j such that (E/2).- p([a, hi) < p([gk, gk,+]j e -.p([a, bi,) for all k, 0 - k N- 1. We say the pair (a, b} is a straddle pair if a Ebasin(A} and b e basin{B}. We call (a, b) a proper straddle pair if (a, b) is a straddle pair, and at least one of the points a andb is in the interior of L. If (a, b} is a (proper) straddle pair, then we call the interval [a, b]j a (proper) straddle segment. Our objective is to describe the 'accessible basin boundary refinement procedure' that selects in a unique way a proper straddle pair from any E-refinement of a givenstraddle pair (on J., When we rcveatedly apply the procedure to the. end points of the ever decreasing straddle segments (with lengths converging to zero), the resulting nested sequence converges to an accessible point p in the basin boundary; of course, this point p is in i n S(R). The point p that we find is accessible using the curve [r, p]j for some r in I nbasin{A}, so we say p is 'accessible from the left' ('accessible from basin{A}'), that is, from the side containing r (in basin{A}). We could alternatively have chosen to approach from the right and we would expect to find a different point on the basin boundary. Since almost-every point on J has finite escape time (see section 4), we can assume that all points of all refinements are chosen with finite escape time. We now describe the accessible basin boundary refinement procedure which is the refinement procedure that generates a uniquely defined proper straddle pair from a given straddle pair. This procedure plays a dominant role in the method that generates a numerical trajectory on the basin boundary that is accessible from basin{A}. A slightly improved version is stated in section 4. Let (a, b} be a straddle pair on a curve segment J such that a is contained in basinfA), and b is contained in basin{B}. Let P=Cxi:0--i<--N(e)} be any E/3-refinement of (a, b), we of course have P c J and a = xo < xI <... < XN(e) = b. We choose the proper straddle pair {a*, b*} from P in the following way: (1) select b* to be the leftmost point of P that is in basin{B}; (2) define m to be the minimum of the escape time of the points in P to the left of b*, and write a° to denote the rightmost point to the left of b* that has the minimum escape time m. I I 1189 Accessible tiajectories on basin-boundaries (2a) If'm <TR(a) then choose a* =an; otherwise, (2b) if m =-TR(a) tihen the choice of a5 depends on the grid P*consisting of b* andallthe points in P to the-left of b* (that.is,P* = (X : P:x E (a, b*])). (i) If the grid P * is not an r-refinement of (a, b*}, then, choose a*=a; otherwise, (ii) if the grid-P* is an e-refineineht of (a; b*} then choose a*to be the-idjacent point in. P* to the rightof a', unless b* is thattadjacent point, in which case choose ,a* = do Remark. Assume that e >0is suitably chosen. In case, of step (2b) the equality a*= a0 does not occur and'one has a*> a0 . (1) As the accessible basin boundary refinement procedure is, -applied, re5 peatedly ,step (2a) only Occursat most finitely,,inany times, andthe segment-[a, a* ] in (2a) may include points that are in basin{B}, However, once step (2b)-occurs, step (2a) will, never-occur again. When step (2b) is applied, the entire segment [a, a*] (n0,'1,st the grid points) is in basin(A) .but,[a, a*]may include points that have escape, timeinfinity. We would, like to emphasize, that all'the points between a and,a* in step (-'b) whose~escape time is finite, go to attractor A. This-is why the method produces an accessible point as.the refinement is repeated. The problem ofcourse isto find E small-enough. (2) When aI and b* have been chosen, iftthe grid consisting of a*, b* and all the points in P between a* and b* is still an e-refinement of the pair then set a*= a and b*-b and apply step-(2b). Repeat this until the grid {x e P:xe [a*, b*]} fails to be an i-refinement of (a*, b*}. Notice that in cases when only step (2b), is repeated, the point b does not move. (3) Under hypotheses in section, 4, it is possible to repeatedly apply the accessible basin boundary refinement procedure obtaining a sequence of straddle pairs that convergesto an accessible point on~the basin boundary. I Example 3. The purpose of this example is to illustrate 'the accessible basin boundary refinementprocedure in a graphical way. We choose E = 0.1. Let (a, b) be a straddle pair, and let P be an E/3-refinemerit of (a, b}. We assume that P is on 3 TR TR , * ag, ;* --h*s | . , . . ab 3. The accessible basin boundary refinement procedure. In figure 3(a) the grid igrid on (a. b#] is not an e-refinement of {ab*) and so a does not move; in figure 3(b) the on (a, b*] is an,6-.finement of (a, b) and so a moves to the right. 1i90 HE Nusse and A Yorke the.str ght line segmeni, that joins, a with b and that the grid points are equalIly s-paced,.so P consists of 31 gridjpoints. In figure 3 the escape time of a grid point x ih Pis represented by,a stir, if x is'in basin{A), and it-is represented by-a dot if-x is in basin{B}. In, figure 3(a) We have b*=xq. The grid P* = {x e P:x e [a, b*]}. is not an 6-refinement of (a, b*}, since-the'distance between two adjacent points equals IIb'-:ai/8 "Which is greater. than t.Jib* - all. Hence, we choose a* = a. In figure 3(b) we ha'e b* =x2o. Thegrid P* = (xe P:x c (a, b*J} is an t-refinement of {a, b), since the distance between two adjacent points equals 1ib* - a 11/20 which is smaller than e lib* - a1. Since TR(xo) = TR(x8) = TR(x o) = m, we choose a*= xi as indicated in the figure. 3. Applications The objective of the paptr is to present the, accessible basin boundary refinement -procedure which enables us to obtain accessible-numerical trajectories on the basin boundary. We also prove that this numerical pre-idure works in-ideal cases. While we believe that the hyperb'oiicity hypotheses (stated in section 4) are often satisfied, they are nonetheless, in practice difficult or impossible-to verify. While chaotic attractors are usually not hyperbolic, the sets we look at are fo0attractors. We do observe that frequently we can successfully use the procedure to'obtain pictures of the accessible- points on the basin boundary. In all the examples below, the pictures were-obtained by using the Dynamics Program [Y]. In these pictures, basin(X) is obtained as follows: for a 960 x 544 grid, use-each grid point as initial value and assign to each grid point a colour (respectively, no colour) if its trajectory converges to X (respectively, stays away from X). The set of coloured grid points is in basin(X}, and the non-coloured grid 1{X}. In all. the pictures for which one of the numerical points are outsi,' .plied in order to produce a single numerical trajectory, have procedures has be been obtained by selecting e = 1/30 as default value (see also section 6). 3.1. Hnon map Let the diffeomorphism F acting on the plane be given by F(x, y) = (p - x2 + M. y, x). The map F is equivalent under a change of variables to the Hdnon map -(1-p'X2 +Y, 14'X). For a first example, we choose the parameters p= i.81257970 and g =0.02286430; these parameters are due'to Grassberger and 'Cvitanovid (personal communication). For these parameters attracting cycles-with period 3 and period 5 coexist. Let D, and D2 be closed balls of radius 0.01 centred at one of the points of the attracting period 3 cycle and 5 cycle respectively. We choose the transient region R to be the open set {(x, y): -2 < x < 2, -4 < y < 4)minus the closed balls D1 and D2. Let A and B be the attractors with period 3 and period 5 respectively. The white area in figure 4(a) is basin{A}; the black area is basin(B}. By using the bisection procedure (see-also section 6), we obtain a straddle trajectory (that is, a numerical -Accessible trajectories on 'basin boundaries (a) 1191, ' (b) Rpgme 4. (a) The white area is basin(A) and includes the period 3 attractor. the black area isbasin(B) and includcs the period 5 attractor in thc region -2 <x <2. - 4 < y< 4 of the Hdn6on map with parameter values p =1,812 579 70, ju 0M022 864 30. (b) Straddlc trajcctory using thc bisection procedure for the Hdnon map (p =1. 812579 70. ju-0,0228643) in the transient region ((x~y): -2<x<2, -4<y<4) minus two cloed balls of radius 0,01 centred at a point of each attractor. The three saddle periodic points on the basin boundary that are accessible from basin(A) and the five saddle periodic points on the basin boundary that are accessible from basin(B) are indicated by st~raight and curved arrows respec'tively, trajectory) on the basin boundary consisting of more than, 100 000 points (actually' tiny intervals); the result is presented inmfigure 4(b). By using the accessible basin boundary refinement, procedure we obtain a period 3 saddle when the left point a is chosen in'basin(A), and a period 5 saddle when the left point a~is chosen in basin{B). The accessible period 3',ahd period'S saddles on the chaotic~lsaddle are indicated by arrows in figure 4(b). Therefore, the set of all points 4ccessible-from basin(A} are the stable manifolds ~of the points ofthe period 3 saddle, and all points accessible from'basifi(B}, are -the stable manifolds of the points of the 'period s saddle.- p I For a second example, we select.the values p = 2.66, u = 0.3. The map F has two attractors A and B, Where A .and B denote the attractors infinity and a cycle with period 3 respectively.- The' box ((, .y) :-3 <x <3I -3 <y < 3) contains a chaoticsaddle, and We select 'the transient, region R 0o b,-the open set {(x, y): -3 <x < 3, -3 <y,<3) minus tie-ball of -radius 0.005 centre.. at a point of attractor B. Usiiig ~the bisection procedure results in one numerical tra tcctory, that hasbieen'p'?-sented 'in figure 5. By using the accessible- bisi boundary refinement procedure-we obtamin a period 1 saddle when the left pointa is chosen in basin{A), and a period 3 saddle when the left point az is,chosen in basin{B}.. Thea,accessible period 1 and period 3 saddles-on the chaotic-saddle are indicated by arrows in figure 5. So, the set of all- points accessible from-basin({A} is the stable manifold of the-period 1 saddle, and the set- of 1192 H E Nusse and J A Yorke * 9 Straddle trajectory usingu the /," 'forFigtire the5.Hdnon map (p = 2.66. -0.3)bisection in 'he procedure transient i region ((x, y): -3 < x < 3. -3 < y < 3) minus a closed ball * I of radius 0.005 centred at -a point of. attractor B (the, ,,.,< "'.,.period 3 attractor), The fixed point on the basin boundary ., ... ,,:..-."that-is accessible from basin{A} (where A = o), and the three saddle priodic:points onthe basin boundary that are accessible from basin{B) are indicated by curved and straight arrows respectively. all points accessible from basin(B) are the stable, manifolds of the points of the period 3 saddle. For a third example of this map, we select the parameter values p =1.405, -0.3. The map F has two coexisting attractors, namely, a period 2 cycle (attractor A) and the attractor infinity (attractor B). The box ((x, y): -3 <x <3, -3<y<11) contains a chaotic saddle. Basin(A) is the white area in figure 6(a). (the two points of attractor A are marked by a dot in the figure), and basin{B) is black in figure 6(a). = (a) (b) Flgure 6. (4) The white area is basin{A) and includes the period 2 attractor, the black area is basin{B} (where B oo)in the region ((x, y):-3 <x <3, - 3 <y <11) of the Hinon map with parameter values p = 1.405, pu= -0.3. Attractor A is marked by two dots, and a saddle fixed point in basin{A) is marked by a cross. (b)Straddle trajectory using the bisection procedure for the Hdnon map (p = 1.405, pu= -0.3) in the transient region &(, y):-3 <x <3, - 3 <y <11) minus a closed ball of radius 0.2 centred at a point of attractor A. The three saddle periodic points on the basin boundary that are accessible from basin{A} and the saddle fixed point on the basin boundary that is accessible from basin(B} are indicated by straight and curved arrows respectively. 2 I 1193 Accessible trajectories on basin boundaries We select the transient region R to be the open set {(x, y): -3 <x <3, -3 <y < 11} minus the ball of radius 0.2 cenitred at Apoint of attracto" A. Using the bisection procedure results in one numerical trajectory, that has been presented in figure 6(b). The PIMtrip!e proceduremay resiult in a saddle fixed point that-is in basin{A); this saddle point is marked. by across in figure 6(a). If we select the transient region to be the region R minus a ball of radius 0.2 centred at this saddle fixed point, then applying the PIM triple procedure results a similar numerical trajectory as in figure 6(b). Notice that the ball including the saddle fixed point is in 3 basin(A}. By using the accessible basin boundary refinement- procedure we obtain a period 3 saddle, when the left point:a is chosen in basin(A}, and a period Fsaddle. when the left point a is chosen in basin(B}, The points of the accessible period 3 saddle on the chaotic saddle are indicated-by arrows in figure 6(a). So, the set of all points accessible from -basin(A) are the st'able manifolds of the'points of the period 3 saddle, and the set of allpoints accessible from basin{B} is the stable manifold of the period;1 saddle. Note that the invariant set of points in the transient region consists of at least three basic sets, namely, (1) the period 2-attractor, (2) the saddle fixed point !n basin(A} and (3) the chaotic saddle on the basin boundary. 3.2. Pendulum We consider the differential equation x"(t) + vx'(t) + sin x(t) =f cos(t). UJ I Ipendulum i (a) (b) Figure 7. (a) The white area is basin{A} and the black area is basin(B} (where A =-(-.472 615, 2.037084) and 8 = (-0.478014, -0.608233) are fixed point attractots) in the region {(x, y):-r <x <,r, --<y < 4)of the time-2:r map of the forced x'(t) + 0.2x'(t) + sin x(t) = 2 ces(t). The three saddle periodic points on the basin boundary that are accessible from basin{A) -are indicated by arrows. (b) Two stfaddle trajectories using the PIM triple refinement procedure for the time-2;r map of x'(t) + 0.2x'(t) + sin x(t) = 2 cos(t) in the transient region ((x, y): -ir< x <,, -3 <y < 4) minus two closed-balls of radius 0.05 centred 3t the fixed point attractors A and B, one trajectory in both basin{A} and basin(B}. The two saddle periodic 2 orbits on the stable set that are accessible from the transient set R\S(R) are indicated by arrows. 1194, H E Nusse aid J A Yorke We choose the parameter values v = 0.2 and f =2. For these parameters, the time-2 rmap has two stable fixed points A and B. In; figure 7(a), !basin(A} is coloured white and basin{B} iscoloured black, It was already observed [GOY2] thatthere was -transient behaviour in the basin(A} and basin{B}. We choose the transient region to be the rectangle {(x,y):-jr<x <#, -3<y<4) minus two balls (of radius 0.05) centred at the attractors A and B. By using the PIM triple procedure for two different transient regions, we obtain twonumerical trajectories. The result for the choice of the interval with end points (-3, -3) and (-3,4) isa trajectory lying in basin(A); and the segment from (-3, 4) to (3, -3) results in a numerical trajectory lying in basin{B}. Both trajectories are presented in figure 7(b). By using the accessible PIM triple procedure we obtain period 2 saddles, see also the discussion in section 6. The result for the segment from (-3, -3) to (3, 4) is-a Lperiod 2 saddle on the chaotic saddle in basin(A), and the segment from (-3, 4) to (3, -3) results~in a period 2 saddleon-the chaotic saddle in basin(B). The points of these accessible period 2 saddles on the chaotic saddle are indicated by arrows in -figure 7(b), The set.of all accessible points on the two chaotic saddles are the stable manifolds of the points of these period 2 saddles. By using the accessible basin boundary refinement procedure we obtain two period 3 saddles: one is accessible from-basin{A}, and the other one is accessible from basin{B). The points of the period 3 saddle that is accessible from basin{A} and is on the basin boundary, are indicated by arrows in figure 7(a), The set of all points on the basin boundary that are accessible from basin{A), are the stable manifolds of the points of this period saddle. A similar result as above holds for the points on the basin boundary that are accessible from basin{B}. 3.3. Complex quadratic map We consider the quadratic map in the complex plane given by z,+= z,+ 0:32 + 0.043i. For this system two attractors coexist, namely, a period 11 attractor (attractor A) and the attractor infinity (attractor B). Let D be a clcsed ball of radius 0.05 centred at a point of attractor A. We choose the transient region R to be the open set {(x,y):-l.35<x< 1.35, -1.35<y < 1.35) minus the ball D. The basin boundary straddle trajectory resulting from the bisection procedure is presented in figure 8(a). The accessible basin boundary straddle trajectory resulting from the accessible basin boundary refinement procedure, a trajectory of which all the points are accessible from basin(A) is presented in figure 8(b), and the accessible. basin boundary straddle trajectory of which all the points are accessible from basin{B} is presented in figure 8(c). The choice-of this equation was motivated by the picture of the Julia set in [PRI. The reader should compare our figure 8(a) with figure 25 in [PRI. We would like to point out that the basin boundary of thissystem (the Julia set) is two dimensionally unstable; thus our results are not valid for this example. Accessible tiajectories on'basin boundaries -1195 LI Figure 8. (a) Straddlc trajectory using the bisection procedure for the complex quadratic map + 0.32 + 0.043i in, the transient region ((x, y): -1.35 <x. y < 1.35)minus a closed ball of C/ "fl J J SA (period 11attractor). (b) Straddle trajectory the accessible basin boundary refinement procedure for the complex quadratic map that is from basinA). (c) Straddle trajectory using the accessible basin boundary refinement Cusing 'accessible . , procedure fom-the" complex-quadratic map that is accessible from oasin(B). 4. Results In section 2 we presented the accessible basin boundary refinement procedure for finding a point on the basin boundary in the transient region, which is accessible from basin{A). First, -we formulate a refinement procedure which is a slightly improved veision of the accessible basin boundary refinement procedure. We will describe inductively how to refine our proper straddle pairs. Given a straddle pair {a,,, b,,}, we have a,, is contained in basn{A), and b,, is contained in basin(B}. Given any e/3-refinement P,={x:O< i N(a)} of {a,,,b,}, we of coursehave a,, =Xo<x, <X...<xN(,) = b,..-We choose the next proper straddle pair {a.+t, b.+t} from P. in the following way. (1) Select b,,+t to be the leftmost point of P,, n basin{B}. J.)6 H ENusse aidJ A York (2) Definei,, = min{TR(x):x EP, and x < b,,..}; a% =inax(xi Q.:x <b.+, and TR(x) = m}. (2a)Ifmn < TR(aq) then choose an~1 ao+1 ;-otherwise, (2b) If mn,, TR(ai) then in order to choose a, we write -Q. = {x .[a, faP,b,,),+I= minimum of the set (x Q, :a%,<x < unless this set is empty in which case a,, = a I = max{x e Q :x < bn+ and TR(x) = TR(a +t)}. "ase (i) If Q,, is -not -an &-refinement of (an, bi,+,), then choose a,,+,=an; -wise, Case (ii) -fQnis an s-refinement of {a,, b,, ) then choose a,,.=-. = Remark (1) For the convenience of the reader, if e > 0 is chosen suitably, then )TR(4,)=R(a+). N( that a'+|< b,+ andmrn- TR(a°+,) Qn might-fail to be an s-refinement of (a,,,b,,+,} in that the distance between some r..ir of consecutive points in Qn might be bigger than E -"p([a., b,+,]. (2) Under the hypotheses below it is possible to repeatedly apply the improved !-.:iement procedure above-obtaining a sequence (-(a,, b} },,, that settles down to an accessible point on the-basin boundary. ,an <-an+ <a1' In the descrir-ion of-the refinement procedure above, we assumed that there exists an E> 0 f,,. which every E-refinement -of a straddle pair (an, b,) includes a, proper straddle pair {a,+,, b.+},) such that_[an, an+dj is-in basin{A}, and the letgth of the. straddle segment, [a.+, bn + s at most (I - e/2) times the length of the previous straddle segment [a,, bnk. We will justify the seconcepts. Let the n !old M and the diffeomorphism F be as in -the -introduction. We assume that A and B are two generalized attractors -such that each attractor is contained either in A-or in B. Recall that a subset A of M is hyperbolic ifit is closed and F-invarin' nd the tangent bundle TAM splits into dF-invariant sub-bundles E dFis uniformly contracting and uniformly expanding respectively. -And E" on v, A hyperbolic set A is called saddle-hyperbolic if dim E' - I and dim E" >- 1. In [NY2] we defined a region R to be a-saddle-hyperbolictransient region if R satisfies all the following conditions: (Al) R is a transient region: (A2). hyperbolicity properqynv(R) is a non-empty saddle-hyperbolic set; (A3) boundary property: U(R) n aR is mapped outside the closure f of R; (A4) intersection property: each non-trivial component y of U(R) is an unstable segment, that iF,y intersects Inv(R); note that such a segment y must intersect S(R) transversally. In this paper, we say a transient region R satisfies the basin boundary property if (1) each-point in R\S(R) is contained in eil'er basin{A) or basin{B}, (2) the sets R nbasin{A} and R fl basin{B) are nc -npty, and (3) the R flbasin boundary is oasin boundary into itself). We define a positively invariant (that is, F maps region R to be a basin boundary transient region if R is a saddle-hyperbolic transient region and R satisfies- the basin boundary property. Accesible trajeciorieson basin bodnd4ries : 1197 For a basin boundary transient region R, and- £ >0, the properties _(Al) and -(A2) imply that the escape'.time of almost every pointpoint on an unstable segment .3 is finite. (A result due to Bowen and Ruelle [BRI shows-that -S(R) has-Lebesgue measure zero.) Hence, one may-assume that such azrefinementdoes not intersect -the stable se- S(R). The basin boundary property implies thateach point that Ilim escapes from R under-iteration of~the hap F is either in basin{A) or in basin{B). If R is a basin boundary transient region; then the escape time map T restricted to an unstable segment JT U(R) has the following two properties (see [NY2). (i)-All the points in a chosen segment [a, bkjon J will escape from R if and only if no t-refinement of(a, b }:includes-a PIM triple (that is,:a triple (p, r, q):onJ such that TR(r) > TR(P), TR(r)> Tk(q), and p([p, q]j) < p([a, bk)). (ii) TR is locally constant on an open subset of full measure of J and'if-TR(x) < and x is a point of discontinuity of' TR then lim inf,_, TR(y) = TR(x) and sup,,., Tq (y = TR(x) + 1. We- assume throughout that dim E" = h For, the,sake Of simplicity, we -assume that-d- 2; the moredifficultcase d 3-will-be-discussedin-section 7. From now on, We-will assume that R is a basin boundary transient.regionfor F, and that J cU(R) .denotes an unstable segment. The proof of the proposition below, Will follow immediatelyan the propositions 5.1 and 52. I I:1 Proposition There exists a -finite set-of periodic points P" in-Inv(R) such that (1) each pointin P" is accessible from R\S(R), an&(2) for x E S(R), the point x is accessible from R \S(R) if and only if x E Ws(p) for some p EP". Corollary. Each accessible point on the basin boundary is-in the stable-manifold of some periodic point. Since J is an unstable segment, recall that this implies that both ends of J are in the-boundary of the transient -region-,R. We know by the intersection-assumptionthat J intersects the stable set S(R). Obviously, if (a, b} is a- straddle pair, -then there exist proper straddle pairs in every E-refinement of- (a, b}, for each- e, 0<e <0.5. The next result deals with the convergence of the sequence of nested proper straddle segments*[a,+,, b+,I] c [a,, bn]j on J. A sequence of straddle segments {fan, b.l}j}. 0 on J is called a straddle segment sequence if {a.,,, b,+I} is in -an E-refinement of the straddle pair {an,-b.} for all n. We say {[an, b.]j}n ., is the accessible straddle segment sequence if (an, b,,) is selected using- the accessible basin boundary refinement procedure for all, n. For every E, 0 < e 0.5, each straddle segment sequence {[an, b.n}jlh;,() converges to a point on- the basin boundary. In section 5 we will, show that there exists e > 0 (depending on F and R) such that for every accessible straddle segment sequence {[an, bn,] }, there is an integer N - 0 such that ,f6r every integer n > N the straddle segment [a,, a.+Il, is contained in basin{A). This number e also appears in the result stated below. The main result stated below implies that the accessible basin boundary refinement procedure is valid. Theorem. There exists E>0 (depending on F and R) such that every accessible straddle segment sequence converges to an accessible point on the basin boundary. 1198 -F ENusse andl'A Yorke' 5. Proofs 5.1. Preliminaries, Let the. manifold ,M; the distance p oh M, and'diffeomorphism F be, s before. We assume, that R; is a, basinboundary transient region for the diffeomorphism -F; and that there.are generalized ,attractorsA -and B such :that each ,point that',eventually leaves R is eitherinbasin{A} or in bisin{B}, Recall thatthe non-wandering set-9 (that is,. the set of-all points x in M such thatfor every open neighbourh6od V of x 'there exists n- 1 for which F"(V) n Vis non;empty) can uniquely be decomposed, into a 'finite-collection of disjoint closed invariant subsets and on each of these subsets-F has a dense orbit; these maximaIinvariant subsets of 0? appearing in the decom-osition are, called thebasic sets (see'e.g. [GH -for the definitionsand several prop&, . s of uniformly hyperbolic systems). From now.on,,iet r denote a basic set of'F From the definitin Of Inv(R) it. follows immediately that either r a Inv(R), or r n'inv(R) is empty. Thus, we can decompose Inv(R):-'into finitely many basic.sets. Note that, 'rl Inv(R) is empty' does not imply 'r R is empty', and 'r R is non-empty"does not imply 'i n Inv(R).is non-empty'. Recall that 'for z E9, thestable manifold WS(z) of z is the-set of points'x for which p(F(z), F (x))-- 0 asn--co';-and the unstable manifold Wu(z) of zis theset of points x for which' p(F-(z), F-"(x))--0 as n The local, stable manifold Wfo (z) of z (of ) is the setof points x in W(z)'such that -p(F'(z),'F"(x))<I3 for;all integers.n -0, and the local utnstable manif0ldW W"(Z) of, z is the set of points x in W"(z)such that p(F'(z), F-"(x)) I3 for all n _ 0, where P > 0. When the -stable or, unstable manifold is a curve, we write Wlvc(z) 'and W'-(z), for the .two components of W'(oj)\{z}, where a is either s or u. We call F a trivial basic set if r consists of one periodic orbit, and we call' r a non-trivial basic set if r includes more than one-periodic orbit. Assume that r is non-trivial; we callxF periodic if there exists m EN. such that F' has no dense orbit on F. and'we~callF non-periodic if it is not periodic. We will:,see below that the structure of Inv(R) is essentially controlled by-finite sets of.pedidic points. Recall -that x in Inv(R) is accessible from an open set V if there is a curve y such that y\{x} lies in V. If wechoose V to be the transient set R\S(R),.and if x in Inv(R) is accessible from R\S(R),it isalways possible to choose 'this curve y to be a piece of the unstable manifold WU(x), that is, -, can be chosen to be either Wr(x) or W-(x). Notice if x is accessible from R\S(R) and y W (x), ,then x is not a limit point of W"o(x) nl 9. Similarly, if we choose V to be the open set'R\U(R), and if x in Inv(R)'is-accessible from"R\U(R) it is always possible to choose this curve y to be a piece of the stable manifold W'(x), that is, 7 can be chosen to be either WfZ(x) or'W (x). Applying a result due to Newhouse and Palis [NP], we obtain the following. Proposition5.1. There exists a finite set P of periodic points in Inv(R), P = P" U P', such that each point-in Inv(R) that is accessible from R\S(R) is in W'(p) for some p in P", and each point in Inv(R) that is accessible from R \ U(R) is in W"(p) for some p in P'. Proof. For a proof, see Newhouse and Palis [NP]. 0 Accessible trajectorieson-basin boundaries 1i99' Palis, and, Takens. [PT] have shown that there exist regions in -M,whose boundaries are segments in the stable and unstable manifolds 6f,these finite sets of periodic points P' and P", suchthat the intersection of-the unionof these regions with the saddle basic set r is a Markov partition forT, see Bowen [B] for the notion of Markov partitibn, Proposition,. 2. Assume f.is a non-trivial non-periodic basic set in Inv(R), and let, z E r be fixed,.Let P' and, Pu.be asabove. There exist finitely many disjoint regions Ribeing diffeomorphic images of the square B =[-1, ] x [-I, 1], say R =gI(B), I< i -<N for some N EN, and a connectedsubset fu of W"(z) such that: (I) rn R, is.1 non-empty for all i; (2),raUO R,; (3)F(aR)cU. 1IRi and F-'(a,,Ri)c ..1 9,,R, where aRi= g,(((x, y): xI l J, lYj < 1)) and cj,,R =j(((x,y):IxI --, y =,I)) are connected subsets in the stable set WS(PU n F) and the unstable set Wu(P fl r)respectively; and (4)for every i, :-fnRi -consists of exactly one -component -and a(U n Rt) a ula I 1I i<N. Prbof. For a proof, see Palis and'Takens [PT]. 0 Recall that R is a basin boundary transient region, and F a basic set in Inv(R). From now on, let, the point z e r, the regions Ri, 1 i <N, and the segment Ju c Wu(z) be as in proposition 5.2. There exist a C', stable foliation 5P on a neighbourhood V. of r and a C'" unstable foliation 9'u ownA neighbourhood Vu of r, for some a > 0. Since it is no restriction to assume that every region Ri is contained in V.frl Vur, 1 < i _ N, see [PT], we will do so. Let r: R - WU(z) be- a C3 parametrization, and define a projection ;r:rUi.I R, nlIu by taking in each region Ri the projection along the local stable manifolds into the intersection 1u with that region, I -_i < N. This projection can be extended from r to the union of the-regions R by projecting along the leaves of the foliation 5P. This extension will also be denoted byr. The following result says that for some iterate K, the map F can be viewed as expansive along unstable segments. Proposition 5.3. There exist a positive integer K and a C ' map q' :U., r-I(lr fn Ri)-- R defined by IV(x) = T - o ;r oFK 0 T(x) such that Ip'(x)l > 1, for some a>0. Proof. For a proof, see Palis and Takens [PT]. 0 5.2. Proof of the theorem Let J c U(R) denote an unstable segment. Recall that both end points of J are on the boundary of the basin boundary transient region R, and that J intersects the, stable set S(R). Recall also that if a point x in-R eventually leaves R, then x is either in basin{A} or in basin{B). We define for every integer k > 1: Ck(J) = {x eJ:T(x)>k} Dk(J)= {xe J: TR(x) =k}. 1200, H E Nusse and J A Yorke In particular, Ci(V) =J.,Hence, foi eachinteger k -I we have Ck+tI):is the set of pointsin, Ck() Whose escape time from Ris,at~least k + I hence, :Ck+ 1(J)is the set of points in I that stayjnR under P-k. The p6ints in-, which stayin R under all iterates~willbe denoted by C(J). For ever' k _>'I, we write Dk(J; A)= (x r: P(J):x-e basin{A}} /Dk(3;'B)- (x e Dk(J):x Ebasin{B}}. The 'basin boundary propetty':nowimplies that for every k > 1: Dk(.) = Dk(; A) U Dk(J; B). Notice that basin{A} and~basin{,B} are disjoint open sets, so, that if there are nbasin(A) and b eJ n basin{B}, then there is apoint x in [a, b~J with points a Jfi TR(x) = -. Observe that Q is a component of DA(J) if and only if Q is a component of either Dk(J;A) or Dk(J; B). Forleach k i 1 we-have: Ck(J) = Ck+i() U'Dk() = J=Ck.I1 )U k Ck+,V) U D(J; A)-U O(J; B) k k D(J) = CkI(J) JUD,(J;A)U U D(J; B) that is, J is the union of the set of points CA+I) whose escape time from Ris at least k -L 1 and, the set of points D(J) whose escape time from R isj, and each of those p, ',s is either in basin(A) or in basin{B}, where 1 --j --k. We write D.J) = U DkJ)= U DkV;A) U U DkV; B). k-I k-I "k-I = fnl. 0 C(), and J = C.(J) UD (). Let I- c a component of Ck(J) that includes a point of basin{A) (or basin{B}). The following result then says that for some fixed positive integers (depending only on F and R), C contains a component of UI, D+,(J; A) (or U,- D+,(J; B)). In particular, s does not depend on k. The following lemma (basin boundary combinatorial lemma) is used to prove the 'basin boundary geometric lemma' which follows. Note &- (() Basin boundary combinatorial lemma. Let X denote either A or B. There exists an integer s -_1 such that for every unstable segment J and for each- integer k > 1 and every component C of.Ck(J), the following holds. If C includes a point of basin{X}, then there is an integeri, k:, i< k +s and a component D of D,(; X) such that D a C. Proof. Let U be a neighbcurhood of Inv(R) on which a C"+U stable foliation F exists, for some er > 0. Select the minimal integer v _-1 such that for each basic set F of F" the following holds, either F is a fixed point or r is a non-trivial non-periodic basic set. For each non-periodic basic set 1"of F", let 1"r and the regions Ri(J), 1 -<i - N(F, be as in proposition 5.2, and let Ur be an open neighbourhood of F Ur) consists of N() open such that (1) Uffr)Ri(F) c Ura U, (2) the set xTj(.ln intervals and its closure consists of N(F) disjoint intervals, and (3) the map Pr in proposition 5.3 may be extended to rTj'(lj. l Ur). For each trivial basic set F, let Ur be an open neighbourhood of F in U such that Ur does not intersect UA, for each basic set A in Inv(R)\F. I, Accessible trajectories on basin. boundaries i: 1201 Let, Li, ... , L (r)be the components of rw(I -lUr); these finitely many components are Open intervals in R. Select the minimal integer K(F) - I such that the map q'r: tr((Iu fl Ur)'-* O defined by Tr(X) - o "ro FKr.o tr(X); satisfies 19(x)I> 1, Define the map lr:rrr(X) (I l Ur)- 4 -by -1 o r oF o 'Now we define the-N(F) x N(F) matrix Ar by Ar(i, i) = 10 if Vpr(Lf) =L otherwise for all 1 < i, j - N(F). Since F is a non-trivial non-periodic basic set of.F", the matrix Ar is primitive. Choose the minimal -integer m(r) 1 such that all the entries of the matrix A ' r)are positive. We define the intbger s(F) as follows. If r is a non-trivial non-periodic basic set, then define-s() = m(r). v, and if Fis a-fixed pointvof F" define s(r) = v. Now, let s be the smallest common multiple of {s(F):Fis a basic set of F'). Let re(R) be the number of -basic sets of -F" in lnv(R), and write Inv(R)= 1J1±) Gk We associate -with Inv(R) a directed-,graph G as follows: G, consists of the points FA, 1 < k <m(R), and there exist a path-from Fr, to Fi if there exists a point z eFi such that W"(z) fl W(F) is non-empty. Notice that for each k, 1 k m (R),there exists a path in G, from Fk to itself. Let J be an arbitrarily chosen unstable segment. Select an integer > I such that Cg(J) is-contained in U. Let,& denote the number of-components of Cg(J), that is, Cg(J) = U I C(J). From the definition of the matrices associated with the non-trivial basic-sets, the-directed graph G, associated-with lnv(R), and the choice of the integer s, and'using the techniques in [Null and [Nu2I, we can associate a (0, 1)-matrix Mj with-C,(J), which is defined by * I M I I to if ;,, o Fv(C,(J)) otherwise C,(J) for all 1 - i, j,</R, where ;rj is the projection on J along the stable leaves. We will assume that the C :,'s are-numbered in suchaway that the matrix M. is written in-the normal form, that is, _'M 11 0 ............ 0 ................ M.'1M., where each Mkk is an Nk x Nk matrix which is either irreducible (that is, for each pair (i, j) there exists t EN such that the (i, j)th entry of-the matrix (Mkk)' is positive, l-i, j<Nk) or a 1xI null matrix, l<k -rn and E'..INk=& for some m, <N. This assumption on the C4;i's is no restriction, since for every non-negative square matrix B there is a permutation matrix P such that PBPT has the normal-form (see Berman and Plemmons [BPI). In-particular, each irreducible Mkk is primitive,-and if Nk --2 then Mkk equals Ar for-some non-trivial nonperiodic basic set F in Inv(R), and from the choice of the integer s it follows that all the entries of (Mkk) ' are positive. Let X denote either A or.B. Let integer k --1 be given. Let C be any component CQ(J), and assume that C includes a point of basin{X}. We first assume-that k . The definition of Mj, the choice of s, and the results in [Nul] and [Nu2] yields that S<m 1202 R E-Nise and J.A Yorke there 'exists aninAteger i, -k i < k + s' and a component D of D1(J;X) such that D c:C. This ,result together With the .Lfinitions of Ak(J) and D,(J; X) imply immediately that also for I k - I one has -that C includes, a component of Df(J X),for. some i, k i k + s, Since J -was arbitrarily given; we have shown the following. There exists an integer s 11'such that. for every unstable segment Jand, for, each integer k _ I-and eyery component C of Ck(J), the following holds. If C includes a point of basin(X}, then there is an integer i, -k - i <k +s ,ahd a -component D'of D(J; X)-such that D c C, where X denotes- either A or B. This completes the proof of the -basin boundary combinatorial lemma. 1 From now on, let's be-as in the 'basin boundary combinatorial lemma', and let G = F. ,We now consider the escape time of points under G. For every-point x in R, the escape time TR(x) of x under G is defined by T"(x) = min(n ->-N:G"(x) I R and T,(x) = 6 if G"(x) E R for all n - 1. We say that T'(x) = 0 if x q R. We define for every integer k > 1; c (V)= (xc-J: T'(x) ->k) Da(J; A) = {x e J: Ta(xt) =k and x c basin{A)) D(J; B) ={x EJ.: Ts?(x).= k and x E basin(B}}. Hence, for each integer k --'I we, have C7+,(J)is the set of points in Ck?(J)whose escape time under-G from R is at least k + 1; hence, C+ 1 (J) is~the set-of points in j that stay in R under G' . The points in J which will stay in'4h under all iterates will be denoted by CQ(i). For each k > I we have: Ck(J) = Ck+ 1(J) U DkJ; A) U D (J; B) = c'+k 1 (I) U I'D(J;A) U U'D (J9B) that is, J is the union of the set of points Ck'+1(I) whose escape time under G from R is at least k + 1 and, the set of points D5(JA)-in basin{A}'(espectively, D(J; B) in basin(B}) whose escape time under G from R is j,. where 1 k.'We write DM(J) = U D/(J; A) U U D(J; B) k-1 k-l Note that 'C$(J)'= nk=o Ck (J), and J =-C(J)U D.(I). Lemma 5.4. For every integer k > 1, we have: (1) Dk(J;A) = -s D,(J: A); Dk(J; B) = % D,(J; B); '(2) C(J)= C(J) and DG(J) = D (); (3) each component of D(J) belongs to either basin{A} or basin{B}. Proof. The proof is left to the reader. 0 Note that'the set DWkJ) is the set of points x rJ with finite escape time (that is, T1(x) < w). The following result- says that, if the value of the escape time map TG cha ,7es then it changes in steps of 1. Denote the length of a connected subset L CJ by p(L).- 1203 AcCessiUle traectories:onbasin boundiries T-JumPh properiy. ',F&,-evcf, y. J Mihp([x,,yJ) < s 4 J With T Gx) <* there exists s >0 uch-that each - TR(y)li. Proof. Apply lemma,,5.4 andthe Tjtimp lemma in [NY2]: b0 The following, lemma for, G implies 'that if an unstable segment y has a n C(.y), that intersects basin{X}, ,then there is a point p of CG (r) h basin{X} with escape,time k, and- the- length of the component D of DG(y; X)' including p is at' least 6.p(C). Bain 'boundary'geometric'lemma. Let X-denote either A or B. There exists 6 >0, such, that for every unstable- segment J, and for each integer k ;>1 and every component C of Ck(J), we have: If;C includes 'a point of basin(X},>then there isa component D of D((J; X),such that D c' and p(D)Ip(C) : 6. Proof. From the geometric lemma'II in [NY2] applied to G, there exists 6 > 0 such that for every J in U(R),.andfor every integer k._1l,the following holds: (1), each component of Ckr(J) contains components of Ck(J) anc, C' + (J);:and (2) if C is any component of Cf(J), then' every component D of DG(J)flC satisfies p(D)lp(C) > 6, and every component 'U of C+l(J)n C satisfies P(U)IP(C):, 6. Let X denote either A or B, and let J be any unstable segment. Let integer k > .1 and component C of C(J)'be given. Assume ihatvC includesa point of basin(X}. Applying the basin boundary combinatorial lemma yields that there exists a component D of Dk(J; X) such that D : C. From the geometric lemma II in [NY2], since D is a component of Dk(J), and the definition of 6, we obtain p(D)Ip(C) 6. Since J,k and C are assumed to- be given arbitrarily, we conclude for each unstable segment J,for each integer k - I and every component C of CG(J), if C includes a point of basin{X), then there is a component D of D'(J;X) such that D c C and p(D)Ip(C) : 6. This completes 'the proof of the basin boundary geometric lemma. 0 From now on, we fix 6 as in the basin boundary geometric lemma. Before we prove the theorem, we present a non-intertwining.property for the escape time map as well as an auxiliary observability result for accessible straddle pair sequences. We call a'pair {p, q} a balanced pair if T'(p) = T'(q). Non-intertwining lemma. Let {p, q} be a balanced pair, let P be a 62 -refinement of (p, q}, and assume that TR(xi) _TR(p) for every x in P. If each point of P is in basin{A) then [p, q]j is-contained in basin{A}. Proof. Let {p, q} and P be as in the lemma. Assume that~each point of P is in basin(A}. Write in = min(TR(x) :x c [p, q.,}. The assumptions 'T(x ) >_T"(p) for all xi e P', 'P is a 6-refinement of {p, q)', together with basin boundary geometric lemma yields that'm = T'(p). Hence, [p, q]j is contained in a component of Cg(J). If there exists a component D of D (J) including [p, q],, then D is a component of Dmr(J;A), and we are done. Therefore, from now on, we assume that [p, q], is not contained in a component of Dr(J). This implies that there are at least one 1204: ilk Nuse and J.A kokke comnponent of CI~(J) in the interior of Jp, qj, and-atAkast'Wo comiponents o6f 'D(J) which havea nn-e.mpty ,irtersection'.,withi[p, qjj. Let D' be a component of,.DgJ uh htD0 p J, snnemt.B the basin, boundary geometric lemma,, We 'have p(D)/p((p,,qj,)> &. Since P is, an 8refinemenit Iof {p', q, it follows that P flD is noni-empty. This fact and, the assumption "that, eachlpqiht of P is.in basin{(A} imply thit D is 'a,component- of D1(J; A). This implies that [p, qh h D~J)is-contained in basi{} Let C be any component of C-,G,,J) in'the interior or ['P, qhj. Applying the basin boundary, geometric lemma we get thit'p(C)/p((p,,qj) >6. If C includes a point of basin{B}, then- C includes a component D of- L),' B), and by 'the basin boundary geomeic6lemnma, we have p(D)/p([p )=(P(D)/p(C)) -(p(C)/p([p, qkj))* 62. Hence, if Cincludesa point of' bsin{B} then every.0 2-refin enent of (p, q}',inciudes a point of basif{'B}. Sijnce Pis an "62-refiniement of (p, q}) andY does not coniainva -point of basin(-B}, it follows that C includes no point of basinfB). Since C is arbitrry- we get. that each component, of Cl-jJ that'is in [p, q~J contains no ,point of basin(B); Therdforejp, qj, 0 Cli/, 1 j) is contained .in-bAsin{A}. &ecause of '[p, q] ",((p, q], n D'(J) U'((p, q], 0 ))-the coni..~ision is that [p, qj, is, contained iin basin{A)-- This completes the proof -of the ,nonintertwining lemma. 0 Basin boundary observability lemma. L 63/3-refinemdent of~a straddle pair (ao, bo}, and assume T(x,) > Tr'R(aO) for (;eryxi in P. Let (a(,, bI) be, the straddle pair in P, in which b1 is,selected- as -in-the accessibl e -basin boundary refinement procedure. Let, al be defined as in the improved version of the accessible basin boundary refinement procedure. f'P'isa'-refi ,nement of (a(O, b1}, then r a(), aI') is in basin{A}, and T"(al) = T"(a)) + 1. Proof. Let P, (aO,bi), and all be as in the lemifia, and assume that Pfnl[au,,bJ, is an e-refinement of (a0 , b,}, where E= 6'. Let rn =-ffin{T (x):x iE[a0, b'lj. Let a" and at be defined as in the improved version of the 'accessible basin boundary refinement procedure. The assumptions 'TR(Xj): : TR(aO) for all x, e PF, 'Pn [a1O,bljis an 6-refinement of -(ao, 'b1}', together with the basin boundary geometric lemma yields mn = t(a) Hence, [aO, bill is contained in a component of Cr(J). By definition, we have a0 - ao. We 'show first that [at, a~,iscnandn basin{A). App lying the* T-jump property and the basin boundary-geometric -lemma we~ obtain that there exists a component D of Gf,'+ 1(J; A) such that D is in the ineror'o,( bil,, and p(D)/p([ao, bl,) > 6. Therefore, at exists and T'(at) m + 1. The definition of a' and lemma 5.4 imply that [a(, a+], is contained in aI =T(a)R~a If basin (A). Recall that (a+,, a',) is a balanced pair, that is, and a, are in the same-component of D.G, 1 (J) then [a+, al], is in basin(A}, and we get that [a', af], is in basin{A}. Now assume that a+ and a, are in different components of DMG+ 1 (J). Then, '[at, all], includes at least one component C of lemma we have C%.72(J) in its interior, and by the basin boundary geometric +()p(a, a11j)> 6. This implies that P n [a', a 'j is a 62-refinement of (a+, a"). Applying the non-intertwining lemma yields [a;, a,isnbaiA)adweotn also this case that [a', all, is contained 'in basin{A}. We conclude: basin{A} +1 fa': a'l, and TG(a',) = Tg'O inch. Accessible tfaje~ories onwbasin boundaries I 1205 'Ifa ao, , tewit,.follows, ifiiiediAtely, from,,the concisiioai, aboveihAat [o, a I'D is contained in basin{A}. From now ,on, we asswiie a6ol Recall tha a, 9?}is a balaniced pAir. f no and a? -are in,the same comont ofDG)te~[, ?,i n baisin{A}. f -a6 and a? -ared in different components ofD V~(), then [do a?), includesat least one co~mponent C -of, C.,,(J). Since p((ao,, aJ,)/p([ao, bill)---: p(C)Ip([ao, a?1j,> 6,,and P fl [a0 , AIvis: a 62*refinemnent of (aO, a,)', applying :the non-intertwining, lemmia -we obtain-JaO,-,aj, is in basin(A},, Since, [a?1 al'kj is in basin{A}, the~conclusion-is that [a0, aili is contaidd in basin{(A}. This completes, the proof'of'the, basin boundary observability lemma. 0, Proof-of the th'eorem. Let 6-ba as in the basin boundary geometric lemma, and ,choose t= 63. Let {[aij, b.], ,,.,:be,an -accessible straddle 'segment sequence, that is, (ao, --b,) is a straddle pair an&, a,, bJ)is obtained by the improved version of the -basin ~boiindary refinement procedure forall n :-::,. For-n >_, let P',be an sf3-refinementv of (a,, b,,}), -and let m, be as -in-the improved version of the accessible basin bouindary refinedment, proicedure. By the basin boundary geometric wve obtain m, =,,min{T"(x):xO(a,, b.+1 ,}. We will -show, that there, exists4 an; integer N >-0 such that for every integer n :-:N tfolowingroerties hold. ()Ta)=m,(P)I(a+)-TG(ai)I < 1, and (P3) [a,,, '4+1 is contained- in basin{A}. Notice that we do not claim that ITRG(x) -- TG(a )I - 1 orall x-C- (an a---,whr f N. From the T-jump property and the basin boundary-,geometrid lemma, together with- the -assumption that ([a,, is obtained- using the accessible basin procedure, we have ,that if T~r(a,,) > in,, then T'(q,+ f)= in,,, for each a -:0. This property implies that there exists a minimal integer Ni - 0 such that Tg(x,)_: m = T'(ajy) for each xi E 1%,. Hence, (P1) holds for N, We now show that (P2) and (P3) hold for this integer N. Iaccessible I-lemma I:boundary Case 1. Pv is not an E-refinement of (UN, bN+I1). Then av+I = UN, -so [ay, aN+ I j is contained in basin (A) and T'(xi) --mv+ I = TR'(aN+ 1)for each xi EPV+ 1. Therefore, (P3) holds, while (P2) is obvious since aN = aN'+I. I Case 2 PNv is- an E-refinement of The basin boundary observability lemma imolies (P3) since [aN, aN+ '), is contained in basin (A). It also implies (P2) since T"(xi) -_inf.+1 (UN, bN+,}. T"(av+I) = T (aN) + I for each xi E PNv+I. By induction, one obtains the desired result. This completes the proof of the 1 1~ 6. The numerical procedure and related numerical methods 61. The dynamic problem Now we return to the 'dynamic' problem stated in the introduction, namely, to describe a procedure for finding a numerical trajectory on the basin boundary which is accessible from basin{A}. (Recall that the basin- boundary of basin{A} is the boundary of the closure of basin{A}.) We assume we are given a straight line segment that intersects- the basin boundary transversally and has one end point in 1206 H ENusse and J A Yorke basin (A) and the other end point in basin (B). In the statement of the results, we assume that a straddle pair and! its e-refinement lie in a connected subset of an unstable segment, and that ail unstable segments intersect the basin boundary transversally. However, from our proof of the theorem iifollows that a similar result holds if we replace the unstable segment by a straight line segment so that we assume that. every e-refinement of a straddle pair fa, b} is in the straight line segment [a, b] from a to b, and that *[a, b] intersects the basin boundary transversally. A straight line segment [a, b] straddles the stable manifold of a point P if (a, b] intersects this manifold transversally. In the caseswe study, that is, a e basin{A} and b Ebasin{B), the stable manifold of P will be replaced by a (fractal) basin boundary and (a, b] will straddle a subset of the basin boundary. 'Furthermore, in practice [a, b] will be very short and will be extremely close to the invariant set Inv(R). The numerical procedure goes as follows. (1) Choose (with some experimenting) a straddle pair (a, b} and let I denote the line segment from a to b. (2) Apply the accessible basin boundary refine-ent procedure (that is, refine and choose a new straddle pair (x, y} in / and then replace I by the straight line segment from x toy. Repeat this process until the length of Iris less than some distance a (for example, a = 10-). If the initial a and b are less than a apart, then the pair is not changed, Given any initial straddle pair (a, b), we will write (ao, bo) = ABS 0 ((a, b}), for the straddle pair resulting from step 2. Note that Ilao - boll < a. 'ABS' is an abbreviation of 'accessible basin boundary straddle refinement'. (3) For each integer n ::0, and straddle pair {an, b,) such that Ila,,," b,11 <a, compute the refinement for the image pair {F(a), F(b)), and write {a, 4t, b.+1) = ABSo(F(a,,), F(b.)}. Thus we obtain a sequence ((a,, b,},)).o of straddle pairs. Note that only F(a) and F(b ) and a are relevant to the computation of {a,+,, b,+,) = ABSo((F(an), F(bn)}), since ABS((F(a,,), F(b,))) is a straddie pair in the line segment from F(a) to F(bn). Write I.for the line segment from a,to b,. Since the system is hyperbolic and the matrix of the second parti" 'erivatives D2 F will be bounded on the closure of the region R, there will be a bk -:id on the curvature of the curve F(In), and F() will deviate from the straight line segment L, from F(A) to F(b) by an amount proportional to IL I2 , where ILI denotes the length of L,. We thus obtain a trajectory of tiny straight line segments 1,, and to the precision of the computer (about 10-14) we usually have 1,,+ c= F(I,,), and selecting any point x, from I, perhaps the midpoint, we have that Ix,,+. - F(x,,)l is small, typically of the order of a. Since computers can never be expected to produce true trajectories (except in trivial cases such as fixed points), we may say {x,,.o is a numerical trajectory with precision a. Despite the complexity of the construction, we will refer to x,+, as the 'iterate' of x,,. We call the sequence of intervals (I},,),o an accessible basin boundary straddle trajectory or ABST trajectory, and we call the numerical procedure above that generates the sequence {,,no, the accessible basin boundary straddle method or ABST method. Notice that each interval straddles a piece of the I Accessible trajectories on basin boundaries I i207 basin boundary. After-a few iterates, the sequence {x,},, 0 resembles a subset of the non-wandering points in'R which are accessible from'basin(A). 1n this paper we have shown that our procedure, (the accessible basin boundary refinement procedure) is valid in ideal situations. We find that theaccessible basin boundary straddle. method works well in practice even in less than ideal cases, in particular, cases where hyperbolicity seems to fail. If e is chosen too large, then (a., b.)},;.() would still be a sequence of straddle pairs with a, e basin(A}' and b, ebasin{B), but the sequence would not be accessible and probably would not settle down tea periodic orbit. In practice we find that, in most cases we study, the method appears to work well for E= 1/30. In computing the sequence of straddle pairs (a,, b,) defined by the accessible basin boundary refinement procedure, once case (2c),holds, then it can be shown that every e-refinement of the proper straddle pair (a,b) includes a proper straddle pair. For the examples in this paper we find that the accessible basin boundary straddlc method leads (in all-cases- but- one) to accessible fixed-points or periodic points, in agreement with the fact that all the accessible points for two-dimensional saddle-hyperbolic systems are on the stable manifolds of finitely ma., periodic points. The e'.ceptional case is the example of the complex quadratic map of which the basin boundary is two-dimensionally unstable, and the result due to Newhouse and Palis does not apply in this particular case. 6.2. The accessible set on the basin boundary We have seen above that in many interesting cases our numerical method (accessible basin boundary straddle method) produces a periodic trajectory on the basin boundary that is in Inv(R). If P is a periodic trajectory in Inv(R) that is accessible from basin(A), then all the points op the stable manifold of P are accessible from basin(A). Therefore, we need a numerical method that produces the stable manifold of a periodic point. In [YKYJ a procedure has been presented that can be used for the calculation of stable manifolds of saddle periodic points of the diffeomorphism F. The calculation can be made with a guaranteed accuracy, in particular, it can be used to calculate the pieces of the stable manifolds of the periodic points that we find. As illustration, we present in figure 9 the stable 1Figure 9. The stable manifold of the fixed point of the Hdnon map (with p = 1.405, is = -0.3) that is accessible from basin{B}. .18 FlW ENusse and JA Yorke rmanifoldkof the period- 1 saddle in the'example of the Hdnon map for which the attractor infinity (attractoi A) and a period 2 attractor coexist. This stable manifold of the saddle fixed point constitute the accessible set (accessible from basin{A)' on the basin boundary. 6.3. Related straddle trajectories In this subsection we review briefly 'straddle trajectories' that are obtained by methods which are based on refinement procedures such as the'bisection procedure [BGOYYJ, [GNOY], the PIM triple refinement ptocedure[NY1u, [NY21 and the accessible PIM triple refinement piocedure [NY2]. The methods were used in the applications presented in section 3 and the refinement procedures above are related to the accessible basin boundary refinement procedure. Thesedstraddle methods are numerical methods for obtaifiing&trajectories-on the basin boundary and on chaotic saddles. For clarity of the exposition and in order that this paperis self-contained, we describe these methods; see the references above for details. Straddle methods involve ,s refinement procedure in which 2 points on a curve segment are replaced by two new points. In some cases the points have different roles. Usually each of the refinement procedures takes a pair of points and returns a pair of points; such a returned pair is on the line segment joining the two points ofthe original pair. The dhstance between the two points in the returned pair is smaller than the distance between the points of the original pair. Straddle methods consist of applying the refinement procedure repeatedly until the points in the resulting pair are less than some specified distance a apart, say o = 10- '. If the points in the original pair are already less than a apart, then no refinement is carried out. Next apply the dynamics; that is, apply the map F to each of the two points of the w pair. resulting pair, giving The basic numerica, ,nethod takes a pair (as, b,} which is separatedby at most a distance a, and applies the map F to each of the points of this pair. If the new pair (F(a ), F(b.)} is separated by less than a, then it is denoted (a..,, b,+), and -ocedure is applied repeatedly until a pnir with separation otherwise the refineme at most a is obtained, awu it is called {a,+,, b,,.t}. However, ii. ter to produce the first pair {ao, bo), the method starts by applying the refinement procedure on the given pair (a, b}, whose points are presumably more than a apart. Writing I,,or '[a, b,,] for the line segment from a, to b, and to the precision of the computer we usually have 1,+ a F(,). We call the sequence of tiny straight line segments { a straddle trajectory. 'e a numerical BST method. the 'basin boundary dynamic problem' is to dmethod for finding a trajectory on the basin boundary. The refinement procedure for straddle pairs is particularly simple. Let {a, P} be a straddle pair such at a e basin{A} and P3e basin{B}. We define y to be the midpoint of the straight line segment [ar, /3], that is, y = (a + P)/2. If y e basin{A} then we choose cr* = y, * = P; otherwise, if YE basin(B) then we choose cr' = (r, /3*= y. This refinement procedure is also called the bivcction procedure. The solution to the 'basin boundary dynamic problem' is the straddle trajectory using the bisection procedure. We call the sequence of tiny straight line segments j.I}, o a basin boundary straddle trajectory or BST trajectory, and we call the I! Accessible trajectories on basin boundaries 1209 straddle method above that generates the BST trajectory {i),,, the basin boundary straddle trajectory method or BST meihod. Notice thit each tiny, line segment in a BST tjiiectory straddles the basin boundary. A BST trajectory typically resembles (after a few iterates) a basicset.in the basin boundary. SST method. the 'saddle 'dynamic restraint problem' is to describe a numerical method for finding a trajectory that remains-in a specified transient region for an arbitrarily long period of time. First, we describe the refinement procedure that is involved in the current, straddle methliod, Let fa, b} be a pair such that [a, b] intersects S(R) transversaiiy. The notation (x, y, z) for a triple means that x, y, and z lid, on [a, b] auid y is between x and z, and we assume for convenience that the ordering on [a, b] is such that x <y < z. For each e > 0, an e-refinement ofa triple (x, z, y) is an E-refinement of {x, y) such that it includes z. Let (a, y, P) be a triple on [a, b. We call (a, , f3) an Intet 'orMaximum triple if both TR(y) > TR(a) and'TR(y) > TR(1)); and we call (a', V, ) a PIM triple if (a, ,,fy) is an Interior Maximum triple and lit - all < Ilb -all. Let (a, V, fP) be an Interior Maximum triple, and let P be an, E-refinement of .). The procedure that selects in 'the refinement P any PIM tripld (a, ,*, f) is called a PIM triple (refinement) procedure. The solution to the 'saddle dynamic restraint problem" is the straddle trajectory using the PIM:triple procedure. We call the sequence of tiny straight line segments Y.),..() a saddle straddle trajectory or SST trajectory, and we call the straddle method that generates the SST trajectory 1 the saddle straddle trajectory method or SST method. Notice that each tiny line segment in an SST trajectory straddles a piece ofa (chaotic) saddle. An SST trajectory typically resembles (after a few iterates) a basic set in the chaotic saddle. ASST method. The 'accessible saddle dynamic restraint problem' is to describe anumerical method for finding a trajectoryon the stable set S(R) that is accessible from the transient set R\S(R). The refinement procedure that is involved in the current straddle method is a PIM triple (refinement) procedure in which a PIM triple (a*, y*, P*) is selected from the E-refinement P of.the interior maximum triple (a, c, b) such that [a, a*] is in-the transient set R\S(R) (hence, [a, a*] does not intersect the stable set S(R)). This refinement procedure is called the accessible PIM triple (refinement) procedure. The solution to the, 'accessible saddle dynamic restraint problem' is the straddle trajectory using the accessible PIM triple procedure. We call the straddle trajectory {(I}),,o an accessible saddle straddle trajectory or ASST trajectory, and we call the straddle method that generates the ASST trajectory {If4,.o, the, accessible saddle straddle trajectory method or ASST method. An ASST trajectory typically resembles (after a few iterates) a-subset of the non-wandering points in R which are accessible from the transient set R\S(R). In most cases that we have investigated we find that every 6-refinement of two points {a, b}, when e is chosen to be 1/30, includes several PIM triples. In [NY1], [NY2] we find that the ASST method leads to accessible fixed points or periodic points, Which is in agreement with the fact that all the accessible points for two dimensional hyperbolic systems are on the stable manifolds of finitely many periodic points. In [NY2] we have shown that the two PIM triple procedures are valid in ideal situations (hyperbolic systems). We find SST and ASST methods work well in (a', y, 1210 H E Nusse and JA Yorke practice-even in less than ideal cases. From the examples in [NY1]; we have seen that-the SST method-works quite well for, a-variety of dynamical systems. Most pictures in section 3 for which one of the numerical straddle procedures has been applied in order to obtain a single numerical trajectory, have been obtained by selecting e = 1/30 as default value, and neglecting the first 10 iterates. We chose e to1 be-somewhat smaller (0.01) in the ABST method for the Hdnon map (parameter values p = 2;66, -- 0.3). 1 6.4. Shadowing It is important to- ask if such straddle trajectories obtained by one of the straddle methods (BST method. SST method. ASST method. or ABST method) represent true-trajectories of the system. In other words. does-there exist a true trajectory of the system that shadows (i.e. stays close to)-the numerical trajectory obtained by a straddle method? When a map is sufficiently hyperbolic on the invariant set in question. Bowen [BI obtained a result saying that each noisy trajectory in the an-wandering set can be shadowed by a true trajectory if the perturbation is small: see [BI for the precise statement. Recall that Inv(R)-satisfies the 'no cycle condition' rk(.% if whenever basic sets 17k(1) .... 1) is a sequence of basic sets in Inv(R) for which the stable set of rk(i) has a non-empty intersection with the unstable set of I', +lIfor all I <_i < k(M), then the stable set of rk(1f) does not intersect the unstable set of rk(i). Assuming Inv(R) satisfies the *no cycle condition' and 6 is sufficiently small. we can show that every BST or SST trajectory of a two dimensional uniformly hyperbolic system with a fractal basin boundary or a chaotic-saddle, obtained by the BST method and SST method respectively, can be shadowed by a true trajectory (for as long as the saddle straddle trajectory can be computed). I 7. Concluding remarks 7.1. Higher-dimensionalsystems One of the ingredients in the analysis of the validity of the accessible basin boundary procedure in dimension two, is the existence of a C +" foliation -1y on a neighbourhood of a basic set. The proofs of the basin boundary geometric lemma and the basin boundary combinatorial lemma require the existence of such a stable foliation (see also the proof of geometric lemma II in [NY2], on which the proof of the basin boundary geometric lemma is heavily based). For d = 2, the existence of such a foliation is guaranteed by a result due to De Melo [MI. Unfortunately, the existence of a foliation a' on a neighbourhood of a basic set in higher dimensions is not known, see e.g. [PT]. Let from now on, the dimension d >_3. Let Fbe an Axiom A diffeomorphism, let R be a basin boundary region such tho. dim E" = 1, and assume that for each basic set r in Inv(R) there exists a C' staule foliation T on a neighbourhood of 17, for some a > 0. Then the conclusion of the theorem is again valid. The proof is almost the same; instead of proposition 5.2 one should use the properties of Markov partitions of basic sets; see Bowen [B]. I : Accessible trajectorieson basin boundaries 1211- Z2 Orderof diffrpntiabilityof the diffeothorphism + I We-assumed that the diffemorphism-F is C3. This assumption implied the existence of a C' expanding map, for some cr>0, in proposition 5.3; If F is of class C2 , then it is known that such an expandngmap is C'. We would like to point out, that the H61der exponent cr is only used to obtain (2) in the proof of, the Geometric lemma I in [NY2I; the proof of the basin boundary Geometric lemma depends indirectly on this result. Fortunately, -we can prove Geometric lemma I in [NY2] (in particular the property (2) mentioned above)for the C'-map .9of proposition 5.3 by combining the techniques of the proof of proposition 6 in [Ne] and lemma 5.5 in 2 [Nul]. Thus in fact, it is sufficient to assume F is C to guarantee the main result of the paper. 7.3. An ad hoc numericaltechnique I 3 [GOY1] describes an ad hoc straddle technique for determining accessible periodic saddle points on the basin boundary. In [GOY1] it is.issumed that there are two attractors A and B. The objective in [GOY1] is tofind a saddle periodic point-on the basin boundary that is accessible from, basin(B}. This-method worked-on several test problems but had no rigorous foundation. The objective of this paper is to attack the problem raised in [GOY1] and we find a straddle method (ABST method) which has a rigorous foundation. i 7.4. Examples 3 By using the SST method, in the example of the H6non map with parameter values p = 1.812579 70, It = 0.022 864 30 the resulting SST trajectory gives virtually the same picture as figure 4 (which was generated using the BST method). Also in this case, the ASST trajectory is similar to the ABST trajectory. In the second Hdnon example (p = 2.66, It= 0.3) we choose in the ABST method e = 0.01; the ASST method gives a similar result when E= 1/30 is chosen. 7.5. Smooth or fractal basin boundaries I 3 I IThe accessible basin boundary procedure is valid for smooth as well as fractal basin boundaries. References [ASI Alligood K T and Sauer T 1988 Rotation numbers of periodic orbits in the Hnon map Commun, Math. Phys. 120 105-19 MAYI Alligood K T and Yorke J A 1989 Accessible saddles on fractal basin boundaries Preprint [BGOYYI Battelino P M, Grebogi C, Ott E, Yorke J A and Yorke E D 1988 Multiple coexisting attractors, basin boundaries and basic sets Physica 32D 296-305 (BPI Berman A and Plemmons R J 1979 Nonnegative Matrices in the Mathematical Sciences (New York: Academic) 1~1& I- Nusse andfA Yorke [B] Bowen R 19-75 Equilibrium States and the Ergodic Theory of Anosov biffe6morn';sms 4.ecture Notes-in~Mahematis 470 (Berl in: Springer) [B] -Bowen R and Ruelle D 1975 the irgodk theory of Axiom A flows'lnvent. -atdh. 29 181-202 [GH) Guckenheimer I and H6lmei P 1983 Nonli -nearOscillatiotis, D%1,aiical Systems. and Bifurcations of Vetor'Fields'App'lied Mathematical Sciences 42 (Berlin: Springer) (GNOYJ -Grebogi C, Nusse H E; Ott E and YarkelJ A 1988 Basic sets: sets deteiine the dinriension of basin undaries Dynamical Systems: Proc0. University of Maryland 1986-87 (Lecture Notes in Mathematics 1342) ed J C Alexander (Berlin: Springer) pp 220-50 [GOYi Grebogi C. Ott E and Yorke J 1987 Basin boundary metamorphoses: changes in accessible boundary orbits Physica 24D 243-i52 (GOY2J Grebogi C, Ott E and Yorke J A 1987 Chaos. strange attractors, and fractal basin boundaries in nonlinear dynamics Science 238 632-8 [HI~l Hamimel SM and Jones C K R T 1989 Jumping stable manifolds for, dissipative maps of the plane Physica 35D 87-406 [KGj Kantz H and Grassbergcr P 1985 Repellers. semi-attractors, and long-lived chaotic transients Physica 17D 75-86 ][MI de Melo W 1973 Structural stabilityof diffeomorphisms on two-manifolds Invent. Math. 21 233-46 IMGOYI McDonald SW, Grebogi C, Ott E and Yorke J A 1985 Fractal basin boundaries Phvsica 17D 125-53 [NPJ Newhouse S and Palis J 1973 Hyperbolic nonwandering sets on two-dimensional manifolds Dynamical Systems M M Peixoto (NewYork: Academic) pp 293-301 [Ne] Ncwhouse SE 1979 The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms Pub!. Math. IIIESSOIOI0-51 e [Null Nusse HE 1987 Asymptotically periodic behaviour in the dynamics of chaotic mappings SIAM J. App!. Math. 47 498-515 [Nu21 Nusse H E 1988 Qualitative analysis of the dynamics and stability properties for Axiom A maps 1. Math. Anal. App!. 136 74-106 [NY1 I Nusse H E and Yorkc J A 1989 A procedure for finding numerical trajectories on chaotic saddles Physica 36D 137-56 [NY21 Nusse H E and Yorke J A 1991 Analysis of a procedure for finding numerical trajectories close to chaotic saddle hyperbolic sets Ergod. Theor, Dynam. Syst. 11189-208, [PR] Peitgen H10 and Richter P H 1986 The Beauty of Fractals (Berlin: Springer) (PTJ Palis J and Takens F 1987 Homoclinic bifurcations and hyperbolic dynamics. 161h C'oldquio Brasileiro Matemdtica.. IMPA 1987 IYI Yorke J A 1989 DYNAMICS. A Program for IBM PC Clones 1989 [YKYJ You Z, Kostelich E J and Yorke J A 1991 Calculating stable and unstable manifolds /ft, J, Bifurcation and C/haos I in press. Volume 156. number 1.2 I I PHYSICS LETTERS A, 3June 1991 Calculating topological entropies of chaotic dynamical systems Qi Chen. Edward Ott'I and Lyman P. Hurd ~ ~Lahoratri for I'Iasmna Re'~earch. L'nm'rviv if*3larvland, ('ol'eve I'aik..111 20 42. Received 13 Fetbruar 19I. ie%ised manuscript rcceiscd 26 March 1I") 1.atccepted lr publiwtion %\prii 1991 Communicated b AXP. Fordy I We present an ciielent kofritihm Ior wik ulai ng iopologit.i~ miopi eN W Ihd% klOOiAli.a "'OUS11 fi.1 ow hd .ipic% it) chaoiic altictors as \Nell a% ciiaoti saodkes, The quantitative characteriz.ation of chaotic processes has proven to be an important issue in nonlinear dynamics. Calculations of' Lyapuno\ exponents and ('ractal dimensions hiase been \er\ useful in this regard. \nother fundamental qujantity is thle topological entropy [ 1,2.1,w\hich characteriz.es the comnplexit\ of'the orbit structure o1'.4 gis en ch namical Stcei. Thle topological entropy i%imsariant untopological conjugac\ of'the dynamical systemIs (iLe. it is preser' ed by Lontinuous and not necessarily differentiable changes of' ariables), Ider The general definition of' topological entropy is computationall\ unwield\, so .alculations inaria- IFor Inential general definitions. an axiomi Ndif'feomnorphism I' (see ref. 131 f'or a definition of the conditions satisfied b\ an axioin \ systemn thle topologikal cntrop\ is thle gross th rate of the numbei ul periodic. points5 13 1. Let P),, be thle number of fi\cd points of' ihle it times iterated map P'". thIus. P,~ LOUfS thle niumbei of points of period it plus thle number of'points " hose period di'ides it. 'flhe topological entropy. W.1 . satisfies IgII lgP It (2) ('haoit *L sstems encountered inapphLations are often not axiom \. Nes ertheless. for non-axiom A stiuations. it is often assuimed that eq, (2 1continues to hold. ELen so. chait sstvims tend to he numeri, all\ un'stable. and this c in make it dif'ficult to obtami I suiffi-ientl\ large number ol periodic orbits to uise in eq. ( 2). Calculations based onl this method iequtire Ingenuit\ and Iiai'e been carried out in a few cases 14.i1. AnAIM* 11proac2h ducto Ne%lmhe and Yonmdi ponentmal gross t rate ol'a A-dimiensional volumle in thle plhine space j 11. F-or tso-(imenslonal mnaps. Nevwbouse uises these testilt to obtain numerical bounds (inthle eiitiopy bk .oiptmin the exponential !;.-o\\ III rat of thle length of a ty pical line segment. Recenils. a more sophismiated technique based onl !;enviatintg paritliuns ol haooic attractors has been proposd. Tis mnichud seems to yield precise estiatc'. on topological ci1uiopies. I luses er. genlerating partitions are usually difficuit to construct 16]. tills note. we introduce a new algoriihmn for cal- I) culating the topological entropy which is particularly Thus for N sufficiently large. %%e hiase thle approximation some aksaitages oser pre\sious methods. It applies Ii= lim '~'~ Also at. Department of Elcttriwai Engineering and Depai,meni of Physics. 48 simple :and efficient. and may in some cases have to chaotic attractors as \\ell as chaotic saddles. Consider anl insertible map of thle plane (. I-') j,'( v. ' ) Choose a compact \olume V. Normally we choose V to contain thle chaotic Invariant set of 11.17-9601/9 'S 0)3.50t 149 1 - Ulse%ivr Science Publishers B. t North-HollandI Volume 156, number I;2 PHYSCS LETTERS A. ,themap. Howe'r,.since:the topolosical entropy of F is bounded below by the entrpy -restricted to any subregion. our'algorithm obta...j lower bounds evenwhen this is not the case. This fact is useful if one does not know a priori bounds on the dynamics. We assume that under the action of the inverse map F-', all points in Vexcept for aset of Lebesgue measure zero (the invariant set and, its unstable rhanifold) eventually escape V.This is true, for example, for area -contracting maps such as the-djissipative H~non map. Consider the intersection of V with its preimages, V.=VnF-'(V)nF- 2(V)c ...n 3 June 1991 behavior of s,, for largqen. Alteriatively, wecan plot log N, Versus n and estimate in(F) as the slope of the fitted curve, (discarding a suitable number of small n values); To obtainr ai estimate of the number of disjoint strips in V,,, let T(x) denote the smallest value of n such that F-n(x) is not in V. We call T(x).the inverse escape time from V. Now consider a line cutting transversely across the stable manifold. Then this line also cuts through all strips in V, for large n, since each strip of V, lies basically along the direction of the stable manifold. Hence, N, is given by the num- F-"(V). For large-n. V, generally consists of disjoint elongated strips i. ing in the direction of the stable manifold of F for the invariant setcontained in V. In the limit n-,oo, V,, is the intersection of the stable manifold with'V. Let us denote the total number of disjoint components in-V, by N, (in the-case of the standard hnrseshoe map, this number is 2"). The theoretical oasis for our algorithm lies in the following model situation (see ref. [ 7 ] ). Let V be a rectangle whose sides are roughly parallel to the stable and unstable directions of the invariant set. If F(V) o)V consists of in horizontal strips and F-I(V)cnV consists of in vertical strips, and Funiormly contracts horizontal strips, and F - 1 uniformly contracts vertical strips, then F restricted to the non-empty invariant set A=no.=.,,, F'(V) is conjugate to the full shift on in symbols which has entropy log in and therefore the map F has entropy at least log m (see re,. 17] for details). Given the region V and the map F,often the above hypotheses are not satisfied, but are satisfied by an iterate, F" and a possibly smaller region V' :V ". Recalling that Nn is the number of disjoint strips in F-"(V) rnV the above argument implies that the en- ber of intervals where T(x)>n in a typical onedimensional line cut. In practice, we count the number of such intervalswhere T(x) >,n for successively larger values of it and calculate the quantity s,, up to a certain level, or until it converges within a given tolerance. Although h obtained in this fashion only gives a lower bound for the topological entropy, for all the systems where comparisons with previous calculations are available, this algorithm appears to yield very sharp lower bounds, We remark that in studying chaotic scattering in two-dimensional Hamiltonian flows, Kovdcs and TO have obtained a similar quantity, Ko, for the Poincar6 map on a surface ofsection, They call Ko the topological entropy of the scattering process [8]. Their method is similar to ours except that we use F - I while they use F (the topological entropy of a map and its inverse are the same). Using F -', however, allows us to obtain the entropy of chaotic attractors (this is not possible using the method of ref. [8], which was designed for chaotic saddles). We first illustrate our algorithm for the Hdnon map, tropy ofF" is at least logNn. Since h(F")=nh(F), X+, we define If the above hypotheses are satisfied by the region V and iterate i, the above estimate forms a rigorous lower bound. In cas ere explicit checking ofthese hypotheses is impr..cal, we examine convergence (4) =x. Set b=0.3, in the parameter range !.4,a<4.0, the invariant set of the Hnon map changes from a strange attractor to a strange saddle, and finally to a full 2-shift (horseshoe). For a sufficiently large, the topological entropy saturates at log 2. It can be shown that the invariant set of the Hdnon map is included in the square max( lxi, liy )<R, where [9] =a-x +by., '+ R= J(l + Ibl + [( I + Ibl ) 2+4a ")2 }. " Recall that the topological entropy of F restricted to V gives a lower bound for the topological entropy ofF restincted to V. This is the region V which we use for calculating the inverse escape time function. For simplicity, we take 49 Volume 156. number 1.2 a vertical one-dimensional line through the- origin .k =0. y=0 and calculate T(.x) at regularly spaced intervals. This is shown in fig. I for a=3.0, where the invariant set is topologically a full 2-shift (horseshoe). There is a natural Cantor set level structure in the inverse escape time function. At level 1.there ae two intervals from which it requires at least two backward iterations to escape the square V; at-level 2. there are four intervals from which it requires at least three backward iterations to escape V. etc. The intersection of these intervals is the intersection of the stable manifold of the invariant set with the vertical axis. I'sing a double-precision algorithm, weare able to calculate the inverse escape time function up to level 20. The algorithm is implemented as follows. Starting from the initial interval Q0given by the intersection of the vertical axis with V, we interpolate q, with auniform grid of N=50 points %ridcalculate the inverse escape time for each point with cutoff time n= 2, We find all the intervals 21's in the grid where the inverse escape time function is greater than I. We then interpolate again each interval Q,with 50 points, calculate the inverse escape time for each point with cutoff time n=3, and find all the subintervals 22's where the inverse escape time function is greater than 2, etc. Assuming each iteration of the Hnon map costs about 10 machine instructions and the topological entropy to be calculated is log 2.the whole calculation up to level 15 then costs approx- 21 I ,HYSICS LETTERS A U6 T~ • ,i 1,K11 I -3 03 -2 71 K0.7 0 1 2 i 3 Fig. 1.The inverse escape time function for the Henon map at a= 3.0. b=0.3 for a verical cut through the ongm =0. j =0. 50 0.91. !0.8? I 3 I imately 32 million machine instructions. On a 10 MIPS workstation. the whole computation takes approximately 3 s.We can achieve better precision by going to higher levels or interpolating more points in the grid. The calculation time typically increases with the level at an exponential rate given by the topological entropy. Usually, level 10calculation ( I million machine instructions, or 0.1 s on a 10 MIPS workstation) yields good estimates on the entropy for chaotic systems. (For instance, for the Htnon attractor ata= 1A b=0.3, level 10 calculation gives .s=0.660. while level 15 givess=0.670. a relative error of less than 2%. Note this value isconsistent with the one obtained in ref. [6].) In all our numerical examples, the logarithms are taken to be base 2. Fig. 2 shows the topological entropy for the H6non map at b=0.3 in the parameter range l.4.<a<3.0. It is calculated with 100 interpolation points at level 15. This figure seem,. to be identical (with better precision) with the one obtained by Biham and Wenzel [41. Note that there are plateau regions where the entropy is constant. This is because for any parameter value where the invariant set is hyperbolic, the topological entropy must be locally constant due to the structural stability of hyperbolic sets. The whole calculation with 260 parameter values takes about 50 min on a 10 MIPS workstation. We also apply our method to open Hamiltonian systems. Generically, the phase space of Hamilto- ----.r -3June1991 0.61.0 1.5 2.0 a 2.5 3.0 3.5 Fig. 2.The topological enirepy for the HWnon map as a function ofaatb=0.3.Thisgraphisobtained usingthe methoddescnbed in the text at level i5with 100 interpolation points. Volume 156. number 1.2 PHYSICS LETTERS A nian systems has mixed, components (101:' regular rotational motions of KAM type, andirregular motions with positive Lyapunov exponents. If the irregular component is noncompact, its only bounded invariant subsets are strange saddles. The topological entropy is related to the escape dynamics from the saddle [ 5]. We wish~to calculate the topological entropies of such systems. One example-is given by the area-preserving sawtooth map on the plane [ 5], Heref./(x) is a sawtooth function, x(6) where (.vI denotes the greatest integer in x. Note that f(x) is discontinuous on the line x=0, therefore the sawtooth map is piecewise linear with constant Jacobian matrix except on the line x=0. The nonlinearity of the map comes from this line of disconti. nuity. For K>0, the map is uniformly hyperbolic except on the discontinuity line, hence, there are no KAM curves in the phase space. The Lyapunov number A is related to the parameter K by J = I+ 0.5 [K+ (K2+ 4K)1121]. L ..er the action of the sawtooth map, almost all initial conditions inside the fundamental region V= (x: xIx0.5) escape to infinity. It can be shown that there is an unstable invariant set r in V [51. Fig, 3 shows this invariant set at A = 2.4. We will calculate the topological entropy for this invariant set as a function of.A. (We.iote that the t6pologicaientropy of the sawtooth map defined on the-plane is different from the topological entropyof.the same map defined ofi the torus. In,the-latter case. the, space is, compact, the chaotic invariant setis'the whole torus. which -contains r as a subset. The- topological en!ropy of the latter is g:. en by the Lyapunov exponent log A, the uniform expansion rate of a line segment.) When A> 3,we can show that the invariant set is a full 2-shift [5], therefore, the topological entropy saturates at log 2 when / > 3. For convenience, we choose the cut at x= -0.5. The topological entropy is shown in fig. 4 for 2< ,10.The solid curve is the entropy obtained by counting the number of periodic points of the map by using the coding scheme of ref. [ 5], the dots are entropies calculated with our algorithm at level 18. The agreement is excellent. When i <2. the convergence for both methods becomes slow, and we find it prohibitive to obtain the entropy value wthout going to a higher precision algorithm. We note that there is no apparent plateau structure in fig, 4,This is because the invariant set is not everywhere hyperbolic in this parameter range. We have also calculated the topological entropy for the corresponding invariant set of the standard map on the plane. The standard map is given by replacing the impulse function in (5) with a sinusoidal function [10]. t.o U, 0.8 0.4t O.6 0.2[ y 3June 1991 0.0 L0.4 .4 - ..... 0 1& I )1 1 -- -0.2 0.02 2.0 .0. 2.4 2.6 2.8 3.0 1 -0.41 -0.6 .0.5 .2 2.2 0 3 0.5 Fig. 3.The unstable invanant set for the sawtooth map at A = 2.4. Fig. 4. The topological entropy as a function of A for the sawtooth map. The solid curve is the entropy obtained by counting the number of n-cycle fixed points, the dots are the level 18 calculations with 100 interpolation points. 51 I Volume-]56. number 1 2 3 I.)- ,PHYSICS LETTERS A (7) -sin(2nt.)/2r. For moderately large values of K, (of -rder I), the I motion ,in the phase space has'both regu lar and irregular components.. However, When the parameiet K is large, the map is almost hyperbolic [ 10]. Therefore. the invariant set contaifned'in the fundamental region V= Kx: Ixi <0.51 is a strange saddle for large K. In fig- 5, we show the topological entropy in the parameter range 5.0 < K< 9.0 calculated using our algorithm atlevel 10 (again logarithms are calculated in base 2), We see that at Kz,8.4, the topological entropy saturates at log 3, indicating the invariant set is topologically a 3-shift. Indeed. this is the typical dynamics of the standard map for large parameter K. 3 June 1991 In conclusion, we have presented.aft efficient algofithm for calculating:the topological' entropy of chaotic dynamical systems. QC wants to thank the Aspen Center for Physics for its hospitality and Rex Skodje for discussions. This work was supported by the Office of Naval Research (Physics), by the Department of Energy (Scientific Computing Staff Office of Energy Research) and by the Defense Advanced Research Projects Agency. References We again note that in the entropy function there are obvious plateau regions where -the topological entropy remains constant, similar to the case of the ( I S,Newhouse. Entropy and volume as measures of orbit complexity, in: Lecture notes in physics, Vol. 278, The physics of phase space (Springer, Berlin, 1986) P.2 . H46non map. (21 P. Walters, An introduction to ergodic theory (Springer Berlin. 1982): N.F.G, Martin and J.W. England. Mathematical theory of entropy (Addison-Wesley, Reading, 1981 ). 2,0 .-4--1.5 I(31 R,Bowen, Am. Math, Soc. 154 (1971 ) 377: A.B. Katok. Publ. Math, IHES 51 (1980) 137. (410. Biham and W.Wenzel, Phys. Rev, Lett. 63 (1989) 819; P. Grassberger, H, Kantz and U. Moenig, J. Phys. A 22 I (1989) 5217. 151 Q. Chen, 1.Dana, J,D. Meiss, N. Murray and I,C. Percival, Physica D 67 (1990) 217: N. Bird and F. Vivaldi, Physica D 30 (1988) 164: 11 1.0' I.C. Percival and F.Vivaldi, Physica D 25 (1987) 105. 0,5 : •161G. 0,0_, 4 (71 6 8 10 K Fig. 5. The topological entropy for the standard map as a func. tion of K. 100 interpolation points at level 10. I U I I3 52 D'Alessandro, P. Grassberger, S. Isola and A, Politi. J. Phys. A 23 (1990) 5285. J.Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems and bifurcations of vector fields (Springer, Berlin, 1983), (8 Z. Kovdcs and T. TOI, Phys. Rev. Lett, 64 (1990) 1617. (91 R. Devaney and Z. Nitecki. Commun. Math, Phys. 67 (1979) 137. 110) B.V. Chirikov. Phys. Rep, 52 (1979) 262. I On the Tendency Toward Ergodicity with Increasing Number of Degrees of Freedom in Hamiltonian I| Systems Lyman Hurd Iterated Systems, Corp. 5550A Peach Tree Parkway, Suite 545 I 1 Norcross, GA 30092 and Celso Grebogia and Edward Ott b U ,Laboratory for Plasma Research University of Maryland College Park, Maryland 20742-3511 U ABSTRACT INumerical experiments on a symplectic coupled map system are performed to inves- tigate the tendency for global ergodic behavior of typical Hamiltonian systems as the I number of degrees of freedom N is increased. As N increases, we find that the fraction of phase space volume occupied by invariant tori decreases strongly. Nevertheless, due to observed very long time correlated behavior, a conclusion of effective gross ergodicity cannot be confirmed, even though extremely long numerical runs were employed. 3 1! a. and Department of Mathematics, and Institute for Physical Science and Technology. b. and Department of Physics and Astronomy, and Department of Electrical Engineering. 'The bAsic assumption in' statistical mechanics is thatf of ergodicity over the phase -space hypersurface determined by- the global constants, of the motion (e.g., total-energy,. total angular momentum, etc.). On the other hand, studie, of 'Hamilt0nian systems with few degrees of freedom (e.g., two) typically reveal the presence ofinvriant KAM tori in addition to chaotic orbits; and -the -existence of KAM tori yields motion that is grossly different :from that assumed in statistical' mechanics. A natural supposition reconciling the above contradictory views might be that, As the number of- degrees of freedomis increased, the tendency for global ergodicity increases. By "tendency for global ergo lcity" we meaii that, for systems with many degrees of freedom (the situation of interest in statistical mechanics), the, overwhelming majority of' initial conditions would be ergodic over effectively all of the area of the phase space hypersurface determined by the global constants of the motion. The purpose of this paper is to present numerical experiments which attempt to test this supposition in a specific case. In particular, wt study a symplectic map system (the symplectic condition insures that the dynamics is Hamiltonian). A closely related wcrl is that of Falcioni et al.1 For other previous relevant works on Hamiltonian dynamics in higher number of degree of freedom systems see Kaneko and Bagley, 2 Gyorgyi et al., 3 and the discussion and references in the book by Lichtenberg and Lieberman. 4 The main result of the present paper is that, for the system we study, the fraction of orbits on tori decreases very strongly as the number of degrees of freedom is increased, but there is still no conclusive evidence for effectively complete global ergodicity even over the very long times investigated in our numerical experiments. The latter is due to the extremely long time-scales, insensitive to machine precision, observed in the numerical experiments. The system we studied -derives from the standard map, x/ = x+Y, (1) y = y+ksinx'. In these coordinates the map can be considered as a map of the two-torus T2 , 0 < x < 27r and 0 < y < 27r. Given a positive integer N, consider the space (T2)N thought of as 2n-tuples (xo,y0,xi,.y, ... ,Xv-1,yv-). We define a coupled standard map allowing symmetric 2 bidirectional nearest neighbor interacti6ns , t x, SY _. Xi + yi, (2) = y j+ K sinxx + CK sin(x x.-.) + CKsin(x - where the indices are taken modulo N and xi, yj are taken modulo 2-r. Here C is the I coupling parameter to nearest neighbors. Letting K = k/(2C + 1), Eqs. (2) reduce to Eqs. (1) for N = 1. We call k the nonlinearity parameter. This map is symplectic since it can be obtained from the generating function, n F(xx') = 3 (x 2 =1 -xi) 2 + K cos x + CK cos(x - x+). (3) One checks readily that yj = OF/Oxi, y = -DF/Oz,. The original aim of our numerical experiments was the exploration of the relative Smeasure 3 3 of KAM tori as a function of the number of coupled maps. To this end, we first note that motion on KAM surfaces is quasiperiodic with all Lyapunov exponents zero, while motion not on KAM surfaces typically is chaotic and has at least one positive Lyapunov exponent. Thus we proceed as follows (see also Ref. 1). A cutoff value Cfor an orbit to be considered quasiperiodic was set and the number of initial conditions with largest Lyapunov exponent (LE) less than e counted. The run consisted of taking m initial I conditions uniformly distributed in the 2N-torus and iterating them approximately 106 times along with their tangent vectors to compute their LE's. A cutoff value e = 0.005 for the largest LE was set below which an orbit was considered quasiperiodic, and the ratio of the number of quasiperiodic initial conditions to the total number of initial conditions 3 3 3 3 was returned. When the coupling coefficient is zero, the volume of the KIAM tori decays exponentially with N. In particular, if f denotes the fraction of phase space occupied by KIAM tori for a single standard map, Eq. (1), then the fraction of the phase space (T2 )N for N uncoupled maps for which motion in the 2N variables (Xo, Yo,..., XN.-1, YN.1) is quasiperiodic is fN. When C > 0, the rate of decay was observed to increase dramatically. Results for the parameter values C = 0.5, k = 0.3 are displayed in Table 1. In this table the estimated measure of quasiperiodic (QP) initial conditions (sLcond column) is the fraction of 8192 1 3 ,randomly chosen initial conditions yielding LEs Aless than E. Maps J Estimated Measure of QP Initial Conditions J 106 1.000 0.403 0.048 0.002 0.000 0.000 0.000 1 2 3 4 5 6 7 Iterations 3.25 3.25 3.25 1.25 x 106 x10 6 x10 6 x 106 106 106 Table 1: Fraction of Initial Conditions Yielding Quasiperiodic (QP) Orbits Figures 1 show histograms of the observed distribution of maximum LE's for the 8192 randomly chosen initial conditions for N = ., 3,.... ,7 coupled maps. The case of three maps is presented twice with different numbers of iterations for the same set of data. The observed peaks get sharper but the effect is very slow. In most cases the following phenomena were noted: 1. The number of initial conditions following within the e bound for quasiperiodicity decreases rapidly as the number of maps increases. 2. The ohk peaks grew sharper with repeated iteration-but very slowly. 3. The histograms with more than one peak preservrd those peaks and they individually got sharper. These observati :us might lead one to conjecture that each peak represents a distinct ergodic component with its own maximum LE. We now discuss the behavior of six individual orbits for the N = 3 case, where the orbits are chosen so that their maximum LE's cal:ulated after 3.25 x 106 lay in distinct regions of interest of the histogram in Fig. 1(c). The calculated LE values for these six orbits are indicated by the arrows labeled with the letters (a)-(f) along the axis of Fig. 4 1(c). The pruojection of these orbits onto. the first two components (xo,'yo)'areplotted for 101 iterations in Figs. 2(a)-(f). Distinct orbits appeared to stay constrained in afixed regionof phase space, azpd this was also true when time series of 10' iterations -were plotted. Lyapunov exponents were then computed for each of these orbits-for a much greater period of time (3 x 108 iterations). The results are shown in Fig. 3 where the letters (b)l(f) labeling the curves correspond to the orbits shown in Figs. 2(b)-(f) and the arrows shown along the horizontad axis of Fig. 1(c). The first initial condition, which was presumed quasiperiodic, remained stable during the whole process, and in fact its computed LE reached zero to machine precision. Initial condition (f) also remained at a highly stable value. The remaining four, however, appear to have started to converge. slowly to a new common value. Figures 4 break down the curves in Fig. 3 (plus orbit (a)] giving the cumulative LE and a "local" LE which is calculated in 500,000 iterate bursts. Observe the gi'cat stability of initial conditions (a) and (f). Further studies were conducted for a variety of initial conditions and various behaviors were observed. 1 1.Some initial conditions "tppeared to lead to orbits whose LE's showed a great deal of stability (they remained essentially unchanged over the observed time scale). 2. Some initial conditions showed a high degree of stability at one value of the maximum LE but then "leaked" into a regime with a different LE. 3. Some initial conditions alternated between chaotic behavior and behavior very close to quasiperiodic. Il One effect of these observations was to call into question the reliability of the LE calculations in general. Many of these calculations seemed to be stable for greater than 106 iterates before changing value. Given the relative rarity of these "leaks," it was impractical to assign any numerical value to this diffusion. * 5 The histogram calculationswere performed on a"Connection, Machine, using (0f necessity) single-precision, arithietic,. The orbit calculations were performed, on a-DecStation I 3i00 usifig double precision. To examine ,the effect of machineprecision several ofthe -long-term LE calculations were done at both single and double precision. The observed behavior was qualitatively the same; the observed leakage between regions of different LE occurred in each case (at slightly different iterates). This work was supported by the Office of Naval Research (Physics Branch), by the Department of Energy (Scientific Comp-4ting Staff, Office of Energy Research), and by 3 the Defense Advanced Research Projects Agency. Ii U 6! I I I I I 1 I I REFERENCES ILJM. Fai-cioni, U._ M-arini Bettolo M-arconi, A. Vkiilpiani, Phyvs. Rev- A 44, 2263 1 (1991). We had completed our research at the time of the publication of the paper ofFalcioni et al. Because of the similarity of that work and ours, in this Letter vde shall be somewhat briefer than we otherwise might have been, and-will also emphasize that part 6f our work which is different fromthat of I alcioni et al. 1 12. ]K. Kaneko and R. J. Bagley, Phys. Lett. A 110, 435-(1985). [3. ]G. Gyorgyi, F. A. Ling and G. Schmidt, Phys. Rev. A 40. 5311 (1989). [4. ]A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springeri I I I I 1t Verlag,Berlin, 1983). 17 FIGURE CAPTIONS 1. Histograms of maximum -LyapUnov. exponents for 8192 initial conditions and N 2,3, ... , 7. - For (a) the -alue in the first histogram bin is about 2700, far off the scale shown. In general, the value in the first -bin is an estimate of the number of quasiperiodic orbits. 2. Projection of six individual orbits for N - 3 onto the plane corresponding to the first two components. 10 iterations are plotted. The calculated inaximum LE's for these orbits are (a) 8 x 10-5 (quasiperiod"c), (b) 0.0166, (c)-0.0676 [corresponding to the lower LE peak in Fig. 1(c)], (d) 0.1170 [corresponding to the higher LE peak in Fig. 1(c)], (e) 0.1191, and (f) 01300. These LE values are indicated along the horizontal axis of Fig. 1(c). 3. Maximum calculated LE as a function of the number of map iterations for the five orbits corresponding to Figs. 2(b)-2(f). 4. Cumulative and "local" maximum LE for the orbits corresponding to Fig. 3. 8 ii I I I I I I I 1 I I I -- I I I I~ ~ I -' I-~~ I~~I I 6 C Q bn Cl z liii I I U I I ~I I I 00 06 C1C C) C) C C)~~~ )4 co ) C ) C0 lx -q - c1 z CD00 4cq 00 A I I I I i I 4 -D 6o 0L I I I 1. .1 I . 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Cl I i Cl Cl -~ -~ 0 .5 I V 0 0 0 0o a I squouodxa~~0 Aon-eSj I II [ 3 uuax 3 ~x ;- c-, (1) 00 I0 CI- xx 3!a oxr- AIn r un-i-l 0 I- 0 C C 0 -4 0 0 sqU~UOdxaf AoundugrJ tuntuix-e- ~--00 0 1 4 1 -pI x C.c) 5 _ AI!d~r _C_ _ ___uox~ _ _ rrax r)iC) 00 00 i-t-~ £ xd I' _ _ _ _ _ _ __ Iq _ _ _ __ _ _ _ _ _C _ _C s~ueauodxq, Aound-egrj tunuiuxr 1 2 i- Ai r /d * ____ Ao _________ n -1r xe 2-m -00 ."~ r~I co 001 Cd~ LO Lo squaodxaAoud-e~j uintux-I I MtETAMORPHOSES: SUDDEN- JUMPS IN 'BASIN BOUNDARIES 1* by Kathleen T-. Alligood Department of Mathematics George Mason University Fa'irfax, VA 22030 and Laura Tedeschini-Lalli Department of Mathematics and Institute for Physical Science and Technology University of Me-yland, Colleae Park, MD 20742 On leave from- Dipartimento di Matematica Universita di Roma "La Sapienza" Rome, Italy 1-00185 * I I and James A. Yorke Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742 December 1985 I 1I 1) This research was supported in part by grants and contracts from the Defense Advanced Research Projects Agency, The Consiglio Nazionale delle Ricerche (Comitato per le Matematiche), and the Air Force Office of Scientific Research. METAMORPHOSES: SUDDEN JUMPS' IN BASIN BO UNDARIES Dynamical systems in-the plane can have -many,coexisting,,attractors. In order to -be able to predict long-term or asyrhptotic bahavior in such systems, it is important to be able to recognize to which attractor (final state) a given trajectory will tend. The set'of initial condi- tions whose trajectories are asymptotic to a particular attractor is called the basin of attraction of that attractor. In some systems that depend on a parameter, it has been observed that the boundaries of these basins are extremely sensitive to small changes in the parameter. Not only can a boundary jump suddenly but it can also change-from being smooth to being fractal. These changes, called boundary metamorphoses, are studied at length in [GOY). originally stated in IGOv, In this paper, we prove a theorem, which characterized the jumps in basin boundaries. The H6non map phenomenon. We fix f(x,y) = (A-x--Jy,x) J = 0.3 and vary A, provides an example of this resulting in a one-parameter, invertible map of the plane. The Jacobian of area contracting for all We will be looking specifically at the A. f is boundary of the basin of attraction of infinity. is the set of all points (x,y) such that J; hence, respectively. Ifn(x,y)l 00 as A = 1.314 A = 1.314) well This new set of black points has not gradually moved in from the boundary of the white region. certain critical value n + o.) In Fig. lb we see that the basin of infinity contains points which were previously (at within the white region. is (The basin of infinity Figures la and lb show the basin of infinihy in black for and A = 1.320, f A = A* % 1.3145, Rather, beyond a black points suddenly begin appearing deep in the interior of the white region. 2 As A increases, 7£jr Fiue11ato FiueIsostebsnoItrato fifiiyi lc o th S~) eo a =(-2j.) Wefx J=03InFg aAis134 n nFg bAi inraeIo13. Tecag nth ai fifnt lutae baiIondr up the thin bands thicken. This is a discontinuous change in the basin of infinity. In order to understand this phenomena, we must e*amine the dynamical behavior on'the basin boundary. At A - 1.314 (Fig. la)-the boundary is observed numerically to consist cf a saddle fixed point pl, and its stable manifold WS(pl)., (The stable manifold WS(p) fixed point as p is the set of points (x,y) such that n * ®. More generallyi, the stable manifold point Pk of period fnk(xy)' + Pk as k is the set of points n *. fn(x,y) + p WS(pk) (x,y) of a periodic such that Analogously, the unstable manifold of p,, n % . Such sets can be proved to be smooth curves.) is the set of points (x,y) such that unstable manifold of p at as shown irt Fig. 2a. At the critical value A - 1.314 f-nk(xy) + Pk wu(pl) are tangent (Fig. 2b). WU(pk) as One branch of the extends into the white region, A* % 1.3145, after which the basin boundary jumps Into the white region, we find that and of a WS(pl) S. Hammel and C. Jones [HJ] were the first to prove a theorem relating the tangency of Ws(pl) WU(pO) (called a homoclinic tangency) to basin metamorphoses. methods are different from ours, however. and, Their We want to relate these metamorphoses to the saddle periodic orbits which are found near the points of tangency and which we describe below. The complicated dynamical behavior which occurs at homoclinic tangencies has been studied at length in recent years, especially in the papers of Gavrilov and Silnikov CGS], Newhouse [N], and Robinson [R]. Under certain non-degeneracy assumptions, there are horseshoe maps defined on subsets of the plane near a point and WU(pl). Figure 3 shows a rectangle 3 qo of tangency of WS(pl ) B4 and some of its iterates IN 5,§ I IMEIO I'MMI _ _ _ _ _ _ _ _ _ _ (b ) _ I0 M5NIE. Fiur 2 Captio FiueIaad2 hwtesal n ntbemnflso fieUon eo ead attn eay ep ci ey 4,! under f. Notice that in two components. rectangle to Bn Bn f4(B 4 ) isLorseshoe shaped and intersects In fact,. for sufficiently large, there is a near the point of tangency is a horseshoe map. period h B4 .10 such that fn There is necessarily a saddle orbit of Bn fn(B ) (see, for example, CR)). On' -f these saddles will have a n "flipped" unstable ranifold (i.e., Dxfn at this saddle has an eigen- and sadd),e Pn. -1), 5 restricted n in each of the two components of the intersection of value less than I and the other will not. I We label the unflipped This orbit is called a "simple Newhouse periodic orbit" in [TY). The larger n is, the closer Bn will be to qo and Ws(pl). This corresponds to the fact that the length of time (i.e., the number of iterates of f) it takes for a point . move around the fixe 3 point p1 is determined by how close the point is to the stable manifold3 WS(p,). What we see (Fig. 4) is an infinite family of horseshoes, and a (pn} of simple Newhouse saddles (where Pn sequence and is in B-) such that (Pnl- qo. orbit Pn' for some n In the following theorem, as stated in [GOY], the saddle fixed point S corresponds to sion above, and the saddle orbit has period p, in the discus- T corresponds to a simple Newhouse n. The term "first non-degenerate tangency" refers to the following set (H) of hypotheses, (i) Wu(p]) does not intersect WS(p 1 ) for (ii) There exist points such that (iii) near ght), qo qo = fk(po) p0 for some in WU(pl) and 1, at A = A*. k There is a parametrization such that where ho - qo g'(h o ) = 0 and and A < A*. ht, -1 Wu(pl ) near g"(h o ) 10. q. in WS(p1 ) t 1 1, of qo WS(pl) is given by 3 4 V IA I.84-S Iq IP 12s I4 Ul Fiue3 ato Iiue ilsrtsahrehemp hivratsto h Iosso isi_34n f( ) h+I 8n nI nI f* 00e0 n) B qOI Figure 4 Caption3 Figure 4 shows the relative positions of two simple N{ewhouse saddles pnand pnIof periods n and n+1. respectively. i1l 3.Ii Theorem. Consider an invertible map f of the plane depending on a parameter A with a saddle fixed point or periodic orbit S.. We assume that the absolute value of the determinant of the Jacobian of f (or of fn ii the case of a periodic orbit of period than one at every-point of the plane. I :3S 3!I A* as value Assume that f n) -itless hLs a transition A increases where the-stable and -unstable manifolds of have a non-degenerate tahigency and then dross for the first time. Then there will be a periodic saddle stable manifold of in it at S for all T that is in the-closure of the 'A slightly greater than S* but is not A*. We prove the theorem with the aid of the following lemma. 3i Lemma. Let Pk be a simple Newhouse saddle of period described above) near the point unstable manifolds of :3 pl. unstable manifold of stable manifolu of Pn qo k (as of 'tangency of the stable and Then, for n sufficiently large, the crosses -(i.e., intersects transversally) the Pn+1" We postpone the proof of- this lemma due to its technical nature and' '5 proceed to show how the theorem follows. I crosses at a point x, then the forward iterates of any segment of containing x of ii If WU(pn) WU(pn+)) Su(Pn+) _ _ will eventually contain all of _ _ P(p ) in its set of limit points' (see Fig. 5a). WU(p) _ W WS(Pn+1) WU(pn) (the closure Hence Proceeding inductively, we have that _ _ iThis follows from the [GH]. A-lemma. See, for example, the exposition in 5 u n+II qOI Figure 5 Caption Figure 5a indicates that-the closure of in the closure of tangency qo0 is W Wu(Pnl .O Figure 5b indicates that in the-Closure of many simple Newhouse saddles. the unstable is contained the point of' -manifolds of infinity IN 5as I;thinner, 5 (see Fig. 5b), for- every m- W(p) e)wp mn Pt f-act, I thie proof of the lemma will show, thd horseshoe fn(Bn) contains a segment of Wu(Pn) around And- Approach p--. ri.-l* As Wu(pl1 ). tor the horseshoes become ' A slightly -larger than A for in :4tficiently large, the horseshoe together (1)And (ii),.we have (iii) W~p)C W (p) altouh bth riand after tangency-. 1(1 50 )- At to pi saddles A - -A*, and pn 1 Again N~otice that occurs for values of W(m n 'crosses W5 (p1 ) Expression (iii) is eiquivAlent to Wu(pl) Hence Onis in the closure of the portion of the plane bounded by from p1 to 9p Ws(pl) ,is invariant under -from f. The are in the interior of this region, and hence each one is a positive distance from the boundary For every I and Ws(phn4i) much smaller than the values 'ot m for which (see, for example, EGOY]). I3. for4 n sutficienly large. m are taken "sufficiently large" for this argu- ment, the crossing of WUp) I Wd(Bm), 5 W l(pl).., thus- will cut 'across IPutting 3 Fm(Bm). and hence .and A slightly larger than by the Ws(pl) A*, A-lemma. 6 of the basin of infinity. the theorem says that there is a n- such, that P -is in the closr e-of Pn -jump -inthe -boiindary at the -alue of n WS(pn+i) begins. -tee-s ; A - A*. The condition that .(2) Thus: there is.a Ws(p). - n, is suffienetly;,arge here: rbfers to .for which. the sequence of crossings -of WU(p) For the Henonmap with appears to be, 4. (see CGOY]). J =0.3 and A* -and 1.314,, This is supported hy computer evi- dence that for -A slightly greater than- 1.3145, the saddle ;p4 is on the. boundary of th 'basin of infinity. Non-degen~racy has not been proved for the tangency of the (3) map at 'At Henon .-1.314. However, theoretically, almost every such tangency will be6non-degenerat&. (4), The proof of tthe theorem characterizes the boundary after tangency by showing- that ther, are iWfinitely many saddles and their stable manifolds contained in V (p.) The fact that there is a: jump in the boundary is,, of course,, implied by this characterizat'ion. The existence Of such a jump can be demonstrated by A simpler., t6pologi.cal argument. Any path I connecting the left and right sides of Fig. 4) extends through the horse shoe image crosses Bn+j fn(Bn). B (cf,. f fn(B) (as shown in Fig. 4), a portion -of 'f(i) connects the left and right side of BAJ., If, at tangency crosses Br+1 r,. A > A*, some forward iterate of for all Proof of Lemma. r 2 n, (A =A*), fr(B then I U-I) contains r~n will then cross W5 p). qo. -so For Following the construction of ER), [TY] (see also [GH; Sec. 6.6]) we assume the following: (i) DF(p1 ) has eigenvalues 0 < , < I, X > 1, and v and X which satisfy vX <1. (ii) There exists a neighborhood 7 U of p1 in which the map f(x,y) is linear up to smooth changes of coordinates; i.e., f (Here we need an additional non-reconance U. in (x,y) for (Ax, vy) are not integer multiples of each X and v assumption--namely, that = other.) WS(p1 ) such that (O,qo) satisfy V : [p fk(poO) (Yy + x,y) f-n+k(V) For n, let the sense of Smale [S]. near fn (See Fig. 6.) (Notice that fn+kv) n W. Bn = fn*k(v) which is nearest [GH]), that know (see in ) 6 > 0, such that (x,y) E V. all , fk(po±c,O) n stretches Actually, since we Under hyposthesis (H), WS(pl). Bn restricted to is a I.,seshoe map, in Specifically, we use the following facts about such maps: (i) W2,n* Bn The saddle point of in fn which stays in (ii) fn(Bn) and Pn Wl,n* = be the connected Zomponent B may wind around a lot, we let of f-n+k(V) n W O - sufficiently large, n For such W. q W - :O,ae2] x [qo-BE, qo+8E]. (cc2 qo ± Be).) across (x-pc) o, Y, for some positive constants Now let (Oq )2 fk f c > 0 and for some p +E3 x [0,6] 0 -E, 0- WU(pl). there is a rectangular neighbor- Furthermore, (H). and Wu(pl) ard WS(pl) and (O,qO ) = and WU(pl) in (po,O) there exist points Specifically, hood WS(p I ) There is a non-degenerate tangency of (iii) intersect in two components, is contained in Furthermore, W1, Pn n Wl,n and and is the only fixed is the only point in Wl, W1 ,n under all forward and backward iterdtes of The only points which stay in (respectively, backward) iterates of WU(Pn)). 8 fn WI,n are in under all forward WS(Pn) (resp., n fn. (O,q0 ) Figure 6 Captionf Figure 6 illustrates definitions used in the proof of the Lemma. We arg~ue_ that 'the s table-,manifold of thriough Pn, (see' Fig,. !;Yt It, is edasily, seen-that-f( f ~~) ecursively,j le t Then L 'C L1 -and 1 Let Lb ph extends .(verttcaly,), O~e any horftontal segmaeftt in SB . is a parabol-a- which extenlds, through 0 n- w in 1= fn(~l for ,23,.. i a: -seq'uence of nested,,intervals with Lis n,~ I length(L1 Y( 14len~gth(L.) Hene 'Since f m(z0 ) is 16,nW f6r all Is one- point, call, 'it M > 1, t z. must* be, inW(P) This irgUmenit s-hows that Ws(p ) intersects the top and-bottom of and first leaves iterates of fn(Bn) I Bn f-1) thi'ough these sides. extends through the horseshoe first leaving the horseshoe through the "Ifeet"., (See Fig. 7). In order to prove ~that show that the horseshoe B Bn+1 Wu(pn) shows that A similar argument (using Spaaboa fn(Bn) since Xp < 1. (O,%,) we need to crosses through (Ovq 0 ) to n1 to the vertex of the right f Fn(B ), as shown in Fig. 8. It is easily seen by our assumptions dn YvVk(q,+Bc) Ws(Pn+l)e containing WUp) P be the distance from bundry intersects Let Q be the distance from (see Fig. 8). and let Wu (p) f that Q We conclude that - (n+l)+k(p0 .c) and P QC a 9. Y 00 n P - 0 as n n Figure 7 Caption Figure 7 shows parts of the stable and unstable manifolds of the simple Nfewhouse saddle pn. Pni I.I I.~ (On) I I sn+i ~Figure 8 CaPtiofl used in the proof of the Lemma. Figure 8 illustrates definitionls ------ REFERENCES [GH] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, anc Bifurcation of Vector Fields, Berlin, Heidelberg, New York: Springer, 1983. [GOY] C. Grebogi, E. Ott, and J. Yorke, Phys. Rev. Lett. 56 (1986), 1011. CGS] N. Gavrilov and L. Silnikov, "On three-dimensional dynamical systems close to systems with structurally unstable homoclinic curve I," Math. USSR Sbornik 17 (1972),.467-485; and part II, Math. USSR Sbornik 19 (1973), 139-156. [HJI S. Hammel and C. Jones, personal communication; and "A dissipative map of the plane--a model for optical bistability," Doctoral E ssertation by S. Hammel, University of Arizona, 1986. EN) S. Newhouse, "Diffeomorphisms with infinitely many sinks," Topology 13 (1974), 9-18. ER) C. Robinson, "Bifurcation to infinitely many sinks," Commun. Math. Phys. 90 (1983), 433-459. CS] S. Smale, "Differentiable dynamical systems," Bull, Am. Math. Soc. 73 (1967), 747-817. [TY) L. Tedeschini-Lalli and J. Yorke, "How often do simple dynamical processes have infinitely many sinks?" 10 preprint The Analysis of Experimental Data Using Time-Delay Embedding Methods Eric J. Kostelich* James A. Yorke Institute for Physical Science and Technology University of Maryland College Park, Maryland 20742 January 30, 1989 Abstract The time delay embedding method provides a powerful tool for the analysis of experimental data, including a novei method for noise reduction. In addition, we describe how the method allows experimentalists to use many of the same techniques that have been essential for the analysis of nonlinear systems of ordinary differential equations and difference equations. 1 Introduction Numerical computation and computer graphics have been essential tools for investigating the behavior of nonlinear maps and differential equations. The pioneering work of Lorenz [24] was made possible by numerical integration on a computer, allowing him to take nearby pairs of initial conditions and compare the trajectories. H~non [23] discovered the complex dynamics of his celebrated quadratic map with the aid of a programmable calculator. A 'Mailing address: Center for Nonlinear Dynamics, Department of Physics, University of Texas, Austin, Texas 78712 "l1 variety of classical and modem techniques has been exploited to find periodic orbits; their stable and unstable manifolds [20]; basins of attraction [251; fractal dimension [261; and Lyapunov exponents [17, 29, 351. In some cases, numerical methods can establish rigorously the existence of initial conditions whose trajectories have essentially the same intricate structure that one sees on a computer screen [11]. Unfortunately, until now experimentalists have not been able to apply most of these methods to the analysis of experimental data, since they do not in general have explicit equations to model the behavior of their apparatus. In cases where it is possible to find accurate models of the physical system, quantitative predictions about the behavior of actual experiments are possible [22]. However, all that is available in a typical experiment is the time dependent output (e.g. voltage) from one or more probes, which is a function of the dynamics. Until recently, power spectra have been the principal method for analyzing such data. For instance, Fenstermacher et a]. [19] relied heavily on power spectra to detect transitions from periodic to weakly turbulent flow between concentric rotating cylinders. However, Fourier analysis alone is inadequate for describing the dynamics. Other methods have been used to analyze time series output from dynamical systems. For instance, Lorenz [24] used next amplitude maps to describe some features of the dynamics; that is, he plotted z,+, against z, where z,, is the nth relative maximum of the third coordinate of the numerically calculated solution. Such maps are often useful, not only for investigating features of the Lorenz attractor [30], but also for instance in experiments on intermittency in oscillating chemical reactions [28]. In the past several years, the so-called embedding method has come into common use as a way of reconstructing an attractor from a time series of experimental data. In this approach, one supposes that the dynamical behavior is governed by a solution traveling along an attractor' (which is not observable directly). However, one assumes that there is a smooth function which maps points on the attractor to real numbers (which are the experimental measurements). In the embedding method, one generates a set of m-dimensional points whose coordinates are values in the time series separated by a constant delay [9]. For example, when m = 3, the reconstructed attractor is the set of points {X = (Si, si+,, -+2r)} where -ris the time delay. 'Existing numerical methods require the attractor to be low dimensional. 2 Takens [32] has shown that under suitable hypotheses, this procedure yields a set of points which is equivalent to points on the original attractor. The earliest applications of the embedding method may be called static in that the analysis focuses on the geometric properties of the set of points on the reconstructed attractor. For example, phase portraits and Poincar6 sections are used in [41 to help determine the transition between quasiperiodic and chaotic flow in a Couette-Taylor experiment. Another important static method is the estimation of attractor dimension from experimental data. for which there is a large literature [26]. In addition, various information theoretic notions can be used to find good choices of embedding dimension and time delay [21]. Certain recent applications of the embedding method are quite different in nature and can be called dyzamic in that information about the dynamics is stored in the computer for analysis. With each data vector xi, one stores the "next" vector, for example, xi+s for some S > 0. This makes it possible to compute a linear approximation of the dynamics in a neighborhood of x,, assuming that there is a low dimensional dynamical system underlying the data.2 In particular, a linear approximation provides an estimate of the Jacobian of the map at xi [9]. Eckmann and Ruelle [17] use linear maps computed in this way to integrate a set of variational equations and find the 3 positive Lyapunov exponents. In fact, the embedding method provides a powerful set of tools for analyzing the dynamics, the breadth of which may not have been realized by Eckmann and Ruelle. In this paper, we discuss two novel applications that are possible, specifically: * Noise reduction. Since one can approximate the dynamics at each point, it becomes possible to identify and correct inaccuracies in trajectories arising from errors in the original time series. Numerical evidence suggests that the noise reduction procedure described below improves the accuracy of other analyses, such as Lyapunov exponents and dimension calculations. I Simplicial approximations. Linear approximations can be computed at each point on a grid in a neighborhood of the attractor to 2This 3 Wolf material was first presented by D. Ruelle at a Nobel symposium in June 1984? et a]. [35] have proposed a different method in which nearby pairs of points are followed to estimate the largest Lyapunov exponent. 3 form a simplicial approximation of the dynamical system. This can be used to locate unstable periodic orbits near the attractor. We begin with a description of noise red iction in the next section. 2 Noise Reduction The ability to extract information from time varying signals is limited by the presence of noise. Recent experiments to study the transition to turbulence in systems far from equilibrium, like those by Fenstermacher et al. [19], Behringer and Ahlers [1], and Libchaber et al. [161, succeeded largely because of instrumentation that enabled them to quantify and reduce the noise. However, it is often expensive and time consuming to redesign experimental apparatus to improve the signal to noise ratio. In cases where the time series can be viewed as a dynamical system with a low dimensional attractor, the time delay embedding method can be exploited to correct errors in trajectories that result from noise. This is done in two steps once an embedding dimension m and a time delay r have been fixed. In the first step, we consider the motion of an ensemble of points in a small neighborhood of each point on the attractor in order to compute a linear approximation of the dynamics there. In the second step, we use these approximations to consider how well an individual trajectory obeys them. That is, we ask how the observed trajectory can be perturbed slightly to yield a new trajectory that satisfies the linear maps better. The trajectory adjustment is done in such a way that a new time series is output whose dynamics are more consistent with those on the phase space attractor. This approach is fundamentally different from traditional noise reduction methods. Because we consider the motion of points on a phase space attractor, we are using information in the original signal that is not localized in a time or frequency domain. Points which are close in phase space correspond to data which in general are widely and irregularly spaced in time, due to the sensitive dependence on initial conditions on chaotic attractors. In contrast, Kalman [3) and similar filters examine data which are closely spaced in time; Wiener [27] filters operate in the irequency domain. 4 Eckmann-Ruelle linearization 3 I. The discrete sampling of the original signal means that the points on the reconstructed attractor can be treated as iterates of a nonlinear map f whose exact form is unknown. We assume that f is nearly linear in a small neighborhood of each attractor point x and write f(x) Ax + b L(x) for some m x m matrix A and m-vector b. (The matrix A is the Jacobian of f at x.) This approximation, which we call the Eckmann-Ruelle linearization at x, can be computed with least squares methods similar to those described in [9, 17]. Given a reference point xrf, let {xi}!' 1 be a collection of the n points which are closest to Xref. With each point xi we store the next point (i.e., the image of xi), denoted y,. 4 The kth row ak of A and the kth component bk of 6 dre given by the least squares solution of the equation yk = bk + ak. x, (1) where yk is the kth component of y and the dot denotes the dot product. Figure 1 illustrates the idea.5 We mention three difficulties in computing the local linear approximations in the subsections below. 3.1 Il conditioned least squares There is a particular problem when one tries to compute solutions to Eq. 1 with a finite data set of limited accuracy that has not been addressed in previous papers [17, 291. Suppose for example that all the points in a neighborhood of x~f lie nearly along a single line, i.e., the attractor appears one dimensional within the available resolution. Although it is possible to measure the expansion along the unstable manifold at Xref, there are not enough 4The Icompute points xi are points on the attractor which are not consecutive in time. The subscript i merely enumerates all the points on the attractor contained within a small distance e of xr. In this notation, xi and yi are consecutive in time. 5 Farmer and Sidorowich [18] observe that the Eckmann-Ruelle linearization can be used for prediction. Given a reference point xi, find the Eckmann-Ruelle linearization Aix+ bi, xi+= Aix, + bi, and repeat the process to get the predicted trajectory f(x)=Ax+b Figure 1: Schematic diagram for the first stage of the noise reduction method. A collection of points in an e.ball about the reference point Xref is used to find a linear approximation of the dynamics there. 6 I points in other directions to measure the contraction. Hence it is not possible to compute a 2 x 2 Jacobian matrix accurately. Any attempt to do so will result in an estimate of the Jacobian whose elements have large relative errors. This kind of least squares problem is ill conditioned. The ill conditioning can be avoided by changing coordinatei so that the first vector in the new basis points in the unstable direction. 6 A one dimensional approximation of the dynamics is computed using the new coordinates; that is, we approximate the dynamics only along the unstable manifold. We recover the matrix A by changing coordinates back to the original basis. For example, if we are working in the plane and the unstable direction is the line y = x, then we rotate the coordinate axes by 45 degrees. The dynamics are approximated by a one-dimensional linear map computed along the line y = x. Then we rotate back to the original coordinates. (The resulting matrix A has rank 1 in this example.) This approach substantially enhances the robustness of the numerical procedure. 3.2 3 Sbox Finding nearest neighbors A second problem is finding an efficient way to locate all of the points closest to a given reference point. The dynamical embedding method imposes stringent requirements on any nearest-neighbor algorithm. The storage overhead for the corresponding data structures must be small, because there are tens of thousands of attractor points. The algorithm must be fast, since there is one nearest-neighbor problem for each linear map to be computed. We solve this problem by partitioning the phase space into a grid of boxes that is parallel to the coordinate axes. Each coordinate axis is divided into B intervals. (Figure 2 illustrates the grid in two dimensions.) Each point on the attractor is assigned a box number according to its coordinates. For example, a point on the plane whose first coordinate falls in the jth interval (counting from 0) along the x axis and whose second coordinate falls in the kth interval along the y axis is assigned to box number kB + j. The list of numbers is sorted, carrying along a pointer to the original data point. Given a reference point x, its box number is found using the above formula. A binary search in the list of box numbers then locates the address Uf X ef 6This is done by computing the right singular vectors [81 of the n x m matrix whose jth row is xi. 7 I . ..I• m• • • B'-:B B2-B+1 B 2 -B+2 B B+1 B+2 0 1 2 .... I 2B-1 2. ... B-i Figure 2: Box numbering scheme in 2 dimensions. The attractor is normalized to fit in the unit square. The bottom row of boxes rests against the x axis and the leftmost row of boxes against the y axis. and all the other points in the same box number. The search is extended if necessary to adjacent boxes. Only a crude partition is needed for this algorithm to work efficiently (typically we choose B = 40), and the grid is extended only to the first three coordinate axes. When the embeddi. g dimension is larger than three, a preliminary list of nearest neighbors is obtained using only the first three coordinates of each attractor point. The final list is extracted by computing the distances from x,,f to each point in the preliminary list. Although there are circumstances where this algorithm can perform poorly (e.g., when most of the attractor points are concentrated in a handful of boxes), the distribution of points on typical attractors is sufficiently uniform that the running time is very fast. Memory use is also efficient: a set of N attractor points requires 3N storage locations. In contrast, the tree-search algorithm advocated in [181 requires several times more storage (although the lookup time is probably slightly less). Because N 1i05 in typical applications, we believe that the box-grid approach (or some variant) is itae most practical. A survey of other nearest-neighbor algorithms is given in [2]. 3.3 Errors in variables There is a potential difficulty in the use of ordinary least squares to compute the linear maps. In the usual statistical problem of fitting a straight line, one has observations (xi, yi) where xi is known exactly and yi is measured. One assumes that yi = ao + alix + ei, where the ei are independent errors drawn from the same normal distribution. (Analogous assumptions hold 8 3 in the multivariate case.) In the present situation, however, both xi and yi are measured with error. It caii be shown that ordinary least squares produces biased estimates of the parameters ao and a, in this case [15, 10]. In practice this does not seem to be a serious problem, but statistical procedures to handle this situation (the so-c.dlled "errors in variables" methods) may provide an alternative approach to noise reduction. We consider this question in the appendix. 4 Trajectory Adjustment by Minimizing Self Inconsistency The Eckmann-Ruelle linearization procedure described above is computed and the resulting maps are stored for a sequence of reference points along a given trajectory (for the results quoted here, the sequence usually contains 24 Points). We now consider how to perturb this trajectory so that it is more consistent with the dynamics. The objective is to choose a new sequence of points *i to minimize the sum of squares wl - xill + 11*i - L-1(*i-1)l + ilI*+i - Li(*ki)I 2 (2) where L(x,) = Aixi + bi, w is a weighting factor, and the sum runs over all the points along the trajectory. Equation 2 can be solved using least squares. Heuristically, Eq. 2 measures the self-inconsistency of the data, assuming that the linear approximations of the dynamics are accurate. See Fig. 3. We say the new sequence {A} is more self consistent. The trajectory adjustment can be iterated. That is, once a new trajectory ki has been found, one can replace each xi in Eq. 2 by ki and compute a new sequerce {x*}. We place an upper limit on the distance a point can move. Points which seem to require especially large adjustments can be flagged and output unchanged. (This may be necessary if the input time series contains large 7 ,n the results described in this paper, the Eckmann-Ruelle linearization procedure is done using a collection of points within a radius of 1-6% of the each reference point, depending on the embedding dimension, the dimension of the attractor, and the number of attractor points. This results in collections of 50-200 points per ball, which gives reasonably accurate map approximations without making the computer program too slow. The weighting factor w is set to 1. 9 Figure 3: Schematic diagram of the trajectory adjustment procedure. The trajectory defined by the sequence {x,} is perturbed to a new trajectory given by {*i} which is more consistent with the dynamics. In this example the dashed line shows what the perturbed trajectory might look like if the dynamics were approximately horizontal translation to the right. "glitches" or if nonlinearities are significant over small distances in certain regions of the attractor.) When the input is a time series, we modify the above procedure slightly since we require a time series as output. The trajectory adjustment is done so that changes to the coordinates of xi (corresponding to particular time series values) are made consistently for all subsequent points whose coordinates are the same time series values. For example, suppose the time delay is 1 and the embedding dimension is 2. Then trajectories are perturbed so that the second coordinate of the ith point is the same as the first coordinate of the (i + 1)st point. That is, when xi = (.s,si+1) is moved to the point =( we require that the first coordinate of *i+l be ii+1. 10 5 Results using experimental data aof ! 3 U 3 3 '3 We note that the attractor need not be chaotic for this noise reduction procedure to be effective. Fig. 4(a) shows a phase portrait of noisy measurements wavy vortex flow in a Couette-Taylor experiment [12]. This flow is periodic, so the attractor is a limit cycle (widened into a band because of the noise) and the power spectrum consists of one fundam"rental frequency and its harmonics above a noise floor. See Fig. 4(b). Figures 4(c)-(d) show the same data after noise reduction. The noise reduction procedure makes the limit cycle much narrower, and the noise floor in the power spectrum is reduced by almost two orders of magnitude. However, no power is subtracted from any of the fundamental frequencies, and in fact some harmonics are revealed which previously were obscured by the noise. These results are significantly different from those obtained by low pass filteriag. Figure 4(e)-(f) shows the phase portrait and power spectrum when the original data are passed through a 12th-order Butterworth filter with a cutoff frequency of 0.35. Th,. dynamical noise reduction procedure is more effective than low pass filtering since the noise appears to hwave a broad spectrum. However, the method appeaito subtract power from a mode whose fundamental frequency is approximately 0.3 times the Nyquist frequency. We do not know exactly why this occurs. However, this peak corresponds to the rotation frequency of the inner cylinder and may result from a defect in the Couette-Taylor apparatus [31]. We do not consider this to be a serious problem, because the power associated with this mode is several orders of magnitude smaller than that of the wavy vortex flow. We emphasize that our objective is to find a simple dynamical system that is consistent with the data. It is possible for this method to eliminate certain dynamical behavior from an attractor if those dynamics have small amplitude. This situation is most likely to arise when there are not enough data to distinguish such dynamics from random noise. In the present example, the noise reduction procedure reveals the limit cycle behavior quite well. 8 The results obtained by applying the method to chaotic data from the "We have not attempted to find the smallest amplitude at which the noise reduction procedure can distinguish quasiperiodic from periodic flow. I 1 11 (b) 3 2 0 -2 L -3 -4 (C)3 (d) 1 1 -2 0 -2-1 -4 I o-2 -2 0 -2 -4 0 0.1 0.2 0.3 0.4 0,5 Figure 4: Phase portraits and power spectra for measurements of wavy vortex flow in a Couette-Taylor experiment. (a)-(b) Phase portrait and power spectrum before noise reduction is applied: ()-(d) after noise reduction; (e)(f) after a low pass filter is applied to che original data. The vertical axis in (b), (d) and (f) is the base-10 logarithm of the power spectral density; the horizontal axis is in multiples of the Nyquist frequency. 12 U Scesses Couette-Taylor fluid flow experiment described in (4] are shown in Fig. 5. Figure 5(a) shows a two dimensional phase portrait of the raw time series at a Reynolds number R/& = 12.9, which corresponds to weakly chaotic flow [4]. The corresponding phase portrait from the filtered time series is shown in Fig. 5(b). Figs. 5(c)-(d) show the power spectra for the corresponding time series. 9 It is difficult to estimate how much noise is removed from the data in this example on the basis of power spectra. One problem is that the transition from quasiperiodic to weakly chaotic fluid flow is marked by a sudden rise in the noise floor in the power spectrum (cf. Fig. 3 in (4]). Hence one cannot determine how much of the noise floor is due to deterministic chaos and how much results from broadband noise. The noise reduction procedure described here has the effect of reducing the power in the high frequency components of the signal. One question therefore is whether reducing the high-frequency noise corresponds to discovering the true dynamics which have been masked by noise. We believe that the answer is yes, based on those cases where there is an underlying low-dimensional dynamical system. However, in chaotic prosome high-frequency components remain, because they are appropriate to the dynamics. 6 * Numerical Experiments on Noise Reduction One important question is how much noise this method removes from the data. The power spectra above suggest that the method eliminates most of the noise, but it is impossible to give a precise estimate for typical experimental data. However, the H1non map [23] provides a convenient way to quantify the noise reduction, because it can be written as a time delay map of the form (3) + #i-I. XI+1= f(z, xi- 1 ) = 1 -X We use Eq. 3 to generate a time series as follows (with the standard parameter values a = 1.4, P = 0.3). We choose an initial condition and discard the 9The time series consists of 32,768 values, from which an attractor is reconstructed in four dimensions. Linear maps are computed using 50-100 points in each ball. Trajectories are fitted using sequences of 24 points. 13 (a) (b) 2 I, 0 -2 .3 .4 )(d) 2 -2 -3 -4 0 8 16 24 32 Figure 5: Phase portraits and power spectra for measurements of weakly chaotic flow in a Couette-Taylor experiment. (a)-(b) Phase portrait and power spectrum before noise reduction is applied; (c)-(d) after noise reduction. The units for the power spectrum plots are the same as those in [4]. 14 40 :1 3i U 3 *I first 100 iterates. The next 32,768 iterates are stored, and a time series is generated by adding a uniformly distributed random number to each iterate. This simulates a time series with measurementnoise, i.e., a time serics where noise results from errors in measuring the signal, not from perturbations of the dynamics. We measure the improvement in the signal after processing by considering the pointwise errorei = lxzi+ -f(xi, zi-t)I, i.e., the distance between the observed image and the predicted one. Let the mean error be E = (F e /V)I1 2 , the rms value of the pointwise error over all N points on the attractor. We define the noise reduction as R = 1- Efitd/Eno1 , where the mean errors are computed for the adjusted and original noisy time series, respectively. The quantity R is a measure of the self-consistency of the time series. (In other words, R measures how much better on the average the output attractor obeys Eq. 3 as one hops from point to point.) When 1% noise is added to the input as described above, the noise reduction (measured with the actual map) is 79%.1o Nearly identical results are obtained wnen the input contains only 0.1% noise. In addition, noise levels can be redtced almost as much in cases where the noise is added to the dynamics, i.e., where the input is of the form {xi+,: xi+1 = f(xi + i7i, xi-.i + i.i_i), mi, ?1i_ random}. When the program is run on noiseless input, the mean error in the output is 0.025% of the attractor extent, which suggests that errors arising from small nonlinearities are negligible when the input contains enough points. 7 I Simplicial Approximations of Dynamical Systems Recent work has shown that simplicial approximations of dynamical systems can reproduce the behavior of the original system to high accuracy [341. (See also [33] for a bilinear approach.) In particular, the fractal structure of the original attractors and basin boundaries is preserved over many scales. Such approximations can yield significant computational savings, especially when the original system consists of ordinary differential equations. 1The pointwise error is measured using Eq. 3. However, the attractor can Le embedded in more than two dimensions when performing the noise reduction. 3 I I 1 15 aa• • •a This approach can be extended in a natural way to generate simplicial approximations of the dynamics on attractors reconstructed from experimental data. Our objective here is to find an approximate dynamical system in a neighborhood of the attractor as follows. A simplex in an m dimensional space is a triangle with m + 1 vertices. Suppose the map is known at each point on a grid. Then there is a unique way to extend the map linearly to the interior of the simplex S whose vertices are grid points. Given a point P in the interior of S, let {bi}!= be its corresponding barycentriccoordinates (see [34] for an algorithm to compute them). Let f(vi) be the map at the ith vertex. The dynamical system at P is iterated by computing m D(P) = _bif(vi). i=o (4) We apply this method to experimental data by finding a linear approximation of Lae dynamics at each vertex vi with the least squares method described above, using a collection of points in a small ball around vi. The maps are stored and retrieved using a hashing algorithm similar to that described in [34]. This yields a piecewise linear approximation of the aynamics from a set of experimental data which can be analyzed with the methods 11 that previously were available only to theorists. We illustrate the approach using a time series of 32,768 values from the H~non map with a = 1.2, 3 = 0.3 using Eq. 3 and adding 0.1% noise as described above. The original attractor is shown in Fig. 6(a). We take a grid of points which are spaced at 1% intervals (this and subsequent distances are expressed as a fraction of the original attractor extent). The time series is embedded in two dimensions, and a linear approximation of the dynamics is computed at each grid point for which 50 or more attractor points can be collected with a ball of radius 0.03; the set of such grid points is shown in Fig. 6(b). We take an initial condition near the original attractor and show the first 3000 iterates using Eq. 4 in Fig. 6(c). Although some defects are visible, the attractor produced by the approximate dynamical system looks almost identical to the original one. "This approach is leps ambitious than that of Crutchfield and McNamara [7], who attempt to find a single set of nonlinear difference equations that creates the observed attractor. 16 3(a) (b) ...... ... ....... / //N ,r... N -I7'M M-1 ... ...... .......... N 3~ ~~~ ........... .____________ ~~~~~~ ______ ....... ..... ..... MMM .1 3 Figure 6: (a) H~non attractor computed from Eq. 3 with a 1.2, i3=0.3. (b) 1%grid on which linear approximations of the dynamnics are computed from the available attractor points. (c) Attractor produced by the simplicial approximations. 17 period D=2 exact I D=3 1 1.793 1.695 1.757 2 2.178 2.199 2.183 4 4.226 4.329 4.051 6 10.38 10.70 9.626 6 10.38 11.32 12.12 8 25.80 24.88 30.25 8 20.02 20.60 20.38 8 17.70 24.32 21.70 Table 1. The largest eigenvalues of the Jacobian of the periodic orbits located using the simplicial approximation of the H6non attractor. One application of simplicial approximations is the location of periodic saddles and the estimation of the largest eigenvalue of the corresponding Jacobian. That is, if x is a periodic point of period p, then we find the eigenvw' e of DfP(x) of largest modulus, where DfP(x) refers to the matrix of paftia, derivatives of the pth iterate of the map f evaluated at x. Given an initial guess for x, one can apply Newton's method using the maps computed at the grid points and Eq. 4 to locate the'saddle using the simplicial approximations. Likewise, Eq. 3 can be used to locate the corresponding "exact" saddle. Saddle orbits up to period: 8 have been computed in this way. In all cases, the saddle point for the simplicial approximation is within 2% of the corresponding saddle point for the H6non map. Table 1 shows the largest eigenvalues of the saddle orbits. (The columns labeled m = 2 and m = 3 refer to the embedding dimension used to reconstruct the attractor.) In most cases, the relative error is only a few percent, and in no case exceeds 25%. (The largest relative error is for the period 8 saddles, where one finds the eigenvailue of the product of 8 Jacobians computed from the least squares.) This method can be extended to experimental data sets. However, there are relatively stringent requirements on the data that can be handled: the time series must be long enough to trace out many trajectories near the principal unstable saddle orbits, and the noise level must be low. (Presumably noisy data can be preprocessed using the approach described in Section 3.) 18 ii 13 3 The current computer implementation uses a large amount of disk space to store the linear map approximations at the grid points. We have constructed a simplicial approximation for an attractor obtained from a Belousov-Zhabotinskii chemical reaction [6, 28]. The attractor is reconstructed in three dimensions from a set of 32,768 measurements of bromide ion concentration. The phase portrait is shown in 7(a). Linear approximations of the dynamics are computed at each point of a grid consisting of 50 intervals along each coordinate axis for which 50 or more attractor points can be located within an 8% radius of the grid point. This produces a database of 59,550 maps. We observe from graphical evidence that many trajectories approach what appears to be a period 3 saddle in the middle of the attractor. Using initial guesses from some of the trajectories, we apply Newton's method to locate the saddle orbit shown in Fig. 7(b). Moreover, we obtain estimates of the Jacobian DF of the map evaluated ac a point on saddle orbit. The eigenvalues of DF are estimated as A1 = 1.14, A2 = 0.102, and A3 = -1.53. These quantitative results confirm that the * orbit is a saddle since A1 > 0 > A3 . (Note that wie expects A2 = 0 for a how generated from a set of differential equations.) * 8 3 Methods for approximating the dynamics of attractors reconstructed from experimental data provide powerful tools. Most of the same procedures that have been so important for theoretical insight, such as Poincar6 maps, unstable fixed points and their manifolds, basin boundaries, and the like, are now available to experimenters, at least in cases where the dynamics are low dimensional. There is little doubt that these tools will lead to breakthroughs in the understanding of a wide variety of physical systems. However, considerable effort is needed before we learn which kinds of systems will benefit Conclusion most from these types of analyses. Significant improvements in technique will certainly extend the applicability of dynamical embedding methods, for example to higher dimensional attractors. 1 19 (a)) Figure 7: (a) The attractor reconstructed from a time series of bromide ion concentrations in a Belousov-Zhabotinskii chemical reaction. (b) The period 3 saddle orbit. 20 I n I g 3 3 I Appendix In this appendix we outline a possible alternative noise reduction method based on the theory of least squares when all the quantities in the regression are measured with error. In ordinary least squares, the variables in the problem fall into two classes: the independent variables, which are known exactly, and the dependent variables, which are observations assumed to be functions of the independent variables. The dependent variables are subject to random errors that are assumed independent and identically distributed (i.i.d.). On an attractor reconstructed from experimental data, we assume that the mapping which takes points in a sufficiently small ball to their images is approximately linear. However, the locations of all the points are subject to small random errors because of the noise. Hence one cannot describe the points as independent variables and their images as dependent variables. The usual least squ-es method produces a biased estimate of the linear map, and this bias does not decrease if more observations are added [15, 10]. The so-called "errors in variables" least squares methods can be used to handle the latter problem. This approach can be used to obtain both an estimate of the linear map as well as estimates of the "true" values of each of the observations. At first this appears to be an underdetermined problem: from n pairs of observations one wants to compute the parameters of the functional relation between them as well as estimates of the n actual pairs. 12 However, it is possible to solve this problem by making some assumptions about the errors [15, 10]. In our case, we assume that the errors in the location of each point and its image are i.i.d. In particular, we let the covariance matrix of the errors in the variables be the identity matrix. This assumption is valid whenever 13 the noise is independent of the dynamics. We illustrate the procedure for the case where we are given a collection of n points (in Rm ) and their images. Following Jefferys [131, we form a set 12 1n the statistical literature, the problem is said to be unidencified. 13Dynamical noise (i.e., each point is perturbed slightly before iterating) yields a covariance matrix which depends on the point. However, as long as the dynamical noise is small, our assumptions about the covariance matrix of the errors should not compromise the accuracy of the method. 21 of n equations of condition given by fi(xi) = x,+i - Ax - bi = x,+/ - L(x) (5) where xi is the ith point, x,,+i is its observed image, A is an m x m matrix, and b is an m-vector. The goal is to find estimates of L (i.e., A and b), together with perturbations ',, such that fi(xi + i)= (X,+1 + i+O- L(xi + =0 and such that the quadratic form 1' -i 1 v SO0 = (6) is minimized. The superscript t denotes transpose and a is the covariance matrix of the observations (which we assume is the identity matrix here). Th'i minimization problem can be solved using Lagrange multiplies (see [13] and [14] for a numerical algorithm). The solution gives A and b together with estimates xi + ,iof the "true" observations. It can be shown [10] under fairly mild hypotheses that the estimates of L and the observations are the best in the class of linear estimators. One way to approach noise reduction is to extend Eq. 5 to include several iterations of the observed points. Given a collection of points in a ball, together with the next p iterates of each point, the method above is used to find a collection of linear maps L1 , L 2 , ... , Lp approximating the dynamics. The method also finds estimates of the actual observations. In this approach, therefore, the calculation of the maps and the adjustmenc of the trajectories is done in one step. Moreover, each point and its image exactly satisfy a linear relationship. Of course, p cannot be too large, because nonlinear effects eventually will become significant when the dynamics are chaotic. On the other hand, Eq. 5 p-ovides a natural way to include quadratic or other nonlinear terms. We have written a computer program to implement this alternative noise reduction algorithni. So far, the results of this approach have not been as good as those from the method described in the main part of the paper, but further refinement should improve them. 22 Acknowledgments Dan Lathrop provided invaluable assistance in finding periodic orbits in the Hnon and BZ attractors. We thank Bill Jefferys for useful discussions and computer software for the errors in variables least squares problem. Andy Fraser, Randy Tagg and Harry Swinney all provided helpful suggestions. This research is supported by the Applied and Computational Mathematics Program of the Defense Advanced Research Projects Agency (DARPA-ACMP) and by the Department of Energy Office of Basic Energy Sciences. References [1] R. P. Behringer and G. Ahlers, J. Fluid Mech. 125 (1982), 219; G. Ahlers and R. P. Behringer, Phys. Rev. Lett. 40 (1978), 712. (2] J. L. Bentley and J. H. Friedman, ACM Comput. Surv. 11 (1979), 397. [3] See for example S. M. Bozic, Digital and Kalman Filtering (London: Edward Arnold Publishers Ltd., 1979). [4] A. Brandsti.ter and H. L. Swinney, Phys. Rev. A 35 (1987), 2207. (5] M. Casdagli, "Nonlinear Prediction of Chaotic Time Series," preprint (Dec. 1987). (6] K. C. Coffman, Ph.D. thesis, University of Texas at Austin, 1987. [7] J. P. Crutchfield and B. McNamara, Complex Systems 1 (1987), 417. (8] J. J. Dongarra, C. B. Moler, J. R. Bunch, and G. W. Stewart, LINPACK User's Guide (Philadelphia: Society for Industrial and Applied Mathematics, 1979). (9] J.-P. Eckmann and D. Ruelle, Rev. Mod. Phys. 57 (1985), 617. (10] W. A. Fuller, Measurement Error Models (New York: John Wiley -Sons, 1987). [11] S. M. Hammel, J. A. Yorke, and C. Grebogi, J. of Complexity 3 (1987), 136; ibid., Bull. Amer. Math. Soc. 19 (1988), 465. 23 I [12] D. Hirst, Ph.D. dissertation, University of Texas, Dec. 1987. (13] W. H. Jefferys, Astron. J. 85 (1980), 177. [14] W. H. Jefferys, Astron. J. 86 (1981), 149. [15] M. G. Kendall and A. Stuart, The Advanced Theory of Statistics, Vol. 2 (London: Charles Griffin & Company Limited, 1961), p. 375. [16] A. Libchaber, S. Fauve, and C. Laroche, Physica D 7 (1983), 73. (17] J.-P. Eckmann, S. 0. Kamphorst, D. Ruelle and S. Ciliberto, Phys. Rev. A 34 (1986), 4971. (18] J. D. Farmer and J. J. Sidorowich, Phys. Rev. Lett. 59 (1987), 845. [19] P. R. Fenstermacher, H. L. Swinney, and J. P. Gollub, J. Fluid Mech. 94 (1979), 103. [20] W. Franceschini and L. Russo, J. Stat. Phys. 25 (1981), 757. [21] A. Fraser and H. L. Swinney, Phys. Rev. A 34 (1986), 1134. [22] E. G. Gwinn and R. M. Westervelt, Phys. Rev. A 33 (1986), 4143. (23] M. H6non, Comm. Math. Phys. 50 (1976), 69. [24] E. N. Lorenz, J. Atmos. Sci. 20 (1963), 130. [25] S. W. MacDonald, C. Grebogi, E. Ott and J. A. Yorke, Physica D 17 (1985), 125. [26] For example, see the papers in Dimensions and Entropies in Chaotic Systems, ed. by G. Mayer-Kress (Berlin: Springer-Verlag, 1986), and references therein, [27] For example, see L. R. Rabiner and B. Gold, Theory and Application of Digital Signal Processing(Englewood Cliffs, N. J.: Prentice-Hall, 1975). [28] J.-C. Roux, Physica D 7 (1983), 57; J.-C. Roux, R. H. Simoyi, and H. L. Swinney, Physica D 8 (1983), 257. 24 I [29] M. Sano and Y. Sawada, Phys. Rev. Lett. 55 (1985), 1082. [30] C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (New York: Springer-Verlag, 1982). [31] R. Tagg, private communication. [32] F. Takens, in Dynamical Systems and Turbulence, ed. by D. A. Rand and L.-S. Young, Springer Lecture Notes in Mathematics, Vol. 898 (New York: Springer-Verlag, 1980), p. 366. [33] B. H. Tongue, Physica D 28 (1987), 401. [34] F. Varosi, C. Grebogi, and J. A. Yorke, Phys. Lett. A 124 (1987), 59. 1 (35] A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica D 16 (1985), 285. 2 12 II November 1989 ACCESSIBLE SADDLES ON I' FRACTAL BASIN BOUNDARIES i £by Kathleen T. Alligood1 Department of Mathematics George Mason University Fairfax, VA 22030 5 and James A. Yorke' 2 Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742 I I I I V 1Research I I. 2 partially funded by a contract Computational Mathematics Program, DARPA. from Research partially funded by a grant from AFOSR. the Applied and ABSTRACT For a homeomorphism of the plane, the basin of attraction of a fixed point attractor is open, connected, and simply-connected, and hence is homeomorphic to an open disk. The basin boundary, however, need not be homeomorphic to a circle. When it is not, it can contain periodic orbits of infinitely many different periods. Certain points on the basin boundary are distinguished by being accessible (by a path) from the interior of the basin. For an orientation-preserving homeomorphism, the accessible boundrv points have a well-defined rotation number. We prove that this rotation number is rational if and only if there are accessible periodic orbits. In particular, if the rotation number is the reduced fraction p/q, then every accessible periodic orbit has minimum period q. In addition, if the periodic orbits are hyperbolic, then every accessible point is on the stable manifold of an accessible periodic point. 1. Introduction and Statement of Main Theorems When a dynamical system has more than one attractor, the boundaries between respective basins of attraction can exhibit very complicated patterns. For invertible maps of the plane, these boundaries can be smooth or fractal, and they can contain infinitely many saddle-type periodic orbits. (By fractal, we mean that the set has non-integer Hausdorff dimension.) Two basins of attraction of the time 2n map of the forced damped pendulum equation are shown in black and white in Figure 1. This picture was constructed by choosing a 960 x 520 grid and, using each grid point as an initial condition, testing where its traject, y goes. The system has two fixed point attractors--one in the white region to which all grid points colored white tend under iteration by the map, and one in the black region to which all grid points colored black tend. The boundary between the black and white basins is fractal, making final state predictability very difficult. In addition, buried within the fractal layers of the boundary are saddle periodic orbits of arbitrarily high periods. Even though the dynamics on the boundary appear to be very complicated, it has been observed (see, for example, [G[)Y I ) that some points on the boundary exhibit regular behavior. p We say that a point )n the boundary of an open set W is accessible from W if there is a path beginning in W such that the path hits. p is the first boundary point which Surprisingly, when the boundary is fractal, most points are not accessible. For the map in Fig. 1, ther: are two points that are saddles of period two (i.e., one period two orbit) which are accessible from the white region, and all other points which are accessible from the white region are on the stable manifold of this periodic orbit. In this paper, we investigate the dynamics of the accessible points on basin boundaries. The paper is strongly motivated by numerical studies that repeatedly conclude there are accessible periodic saddles in the boundary. In fact, we know of no natural case of an area-contracting diffeomorphism having a basin boundary without accessible periodic orbits. We would like to thank J. Mather and H. Nusse for helpful discussions. Throughout this paper, set either in the plane R2 W is a connected, simply-connected open S2 , and or in the sphere F is a homeomorphism (or diffeomorphism, if differentiability is required) of the plane or the sphere. (i.e., F(W) = W). attraction. We assume that W is invariant under Our main examples of such sets will be basins of In particular, the basin of attraction of an attracting fixed point must be such a set. (See Sec. 2.) We assume in addition that Wi is not the entire plane, in which case its boundary more than one point. under F, F, 8W d - a nact: -.. is also an invariant set. Since W 8W is is invariant All connected, simply-connected open sets are homeomorphic to an open disk. On the other hand, the boundary of such a region does not have to be topologically a circle, and examples abound in which the boundary of a basin of attraction is a fractal set. The characterization of a set W I 3 as a topological open disk occurs in the study of the Riemann Mapping Theorem which says that for any such set W there is always a one-to-one analytic map 3, h of an open disk D onto W. The knowledge that the basin is topologically an open disk tells us nothing about the boundary of a basin, and It Is our objective to describe the dynamics on the points in aW that are accessible from W. 3point In the following we say that p is accessible only if it is a of 8W that is accessible from W. Caratheodory [C] investigated the behavior of the map h in the Riemann mapping theorem to see when h could be defined at boundary points of the disk. If r is a (continuous) pph in W which limits on the accessible point p, then h-1 () is a (continuous) path in D limiting on exactly one point r in Si, the boundary of D. We call such points as r trivial circle points; we call all other points on the circle non-trivial circle points. Caratheodory's approach was to construct a compactification of W which is topologically identical to D, the closed disk. (His is not the standard compactification; points in this compactification which correspond to points in the boundary S1 of D are called "prime ends" and are defined precisely in Sec. 5.) We define a map h on points in D and on those points in D that are trivial circle points by hc (x) = h(x) for x in D, and hc (r) = p where p is an accessible point and r is an associated trivial circle point, as defined above. It is clear from the construction that each accessible point is the image of at least one trivial circle point. 3 The map hc is not necessarily one-to-one on trivial circle points. However, once a path r in W (See Sec. 7; in particular, see Fig. 7.) limiting on an accessible point p is specified, then there is exactly one trivial circle point x which is the limit of h-1 (r) We mention two properties of accessible points and the map hc: PROPERTY 1 (DENSITY) The set of accessible points is dense in aW; the set of trivial circle points is dense in S , the boundary of D. PROPERTY 2 (EXISTENCE OF AN INDUCED MAP) There is a map, denoted f and called the induced map, from D to itself such that h cf(x)) = F(h Cx)) c when x is in D or x is a trivial circle point. C If p is an accessible point and r is a path in W ending at p, then F(F) is a path in W ending at F(p). to accessible points. trivial circle points. Hence, accessible points map If follows that f maps trivial circle points to On the set of trivial circle points, f is one-to-one, onto, and order-preserving. Such a map can be uniquely extended to a homeomorphism defined on all of SI. These properties allow us to study the dynamical system on the closed disk, maintaining the dynamics on the accessible points. Since in general aW will include much more than the accessible points, much of aW is lost in this representation. For us, however, the simplification is advantageous since we wish to describe the dynamics 4 on the accessible points. We have important examples in which W is not a basin even though - a dense set of points in W have trajectories tending to an attractor. 'I 3i The following definition allows the inclusion of such examples. We say that aW is unstable In W if there is a neighborhood Bc of aW with the property that the set of points in B leave b B W In n>O.) is dense in such that B xeQ W. (I.e., implies that whose orbits eventually there is a dense set Q in Fn(x) is in W\B for some This definition is easily seen to be satisfied when W is a basin of attraction. It is also satisfied in the very different case where there is a dense orbit in W. 3 Certain types of periodic orbits in S 1 merit particular attention. p 6 S be a periodic point of period k. We say p is Let attracting on at least one side (of SI) that I x * p and lim such The following key theorem is proved in Sec. 5: that for each I x e S1 fnk(x) = p. THEOREM 1.1 (ATTRACTING LEMMA). * if there exists k Assume that aW unstable in W and the fixed points of Fk are isolated. Then each periodic circle point that is attracting on at least one side is a trivial circle point. An orientation preserving homeomorphism of the circle can be classified according to its rotation number--a number p, with I 0 5p <1, which represents the average rotation of points under the map. (A precise definition is given in Sec. 5.) is independent of the choice of point on S I. The rotation number The idea of associating a single rotation number with each orientation preserving Such a homeomorphism of the circle originated with Poincar6. homeomorphism will have a periodic point if and only if its rotation number is rational. It will have a fixed point if and only if its rotation number Is 0. We define the rotation number p(aW,F) of F on the accessible points of aW to be the rotation number of the induced map f on S1 . If W is a connected, simply-connected open set in R , if F Is orientation preserving, and if the closure of W is invariant under F, then W has a rotation number. In particular, if p is an 2 Isolated, attracting fixed point in R , if its basin W is not all of R , and If F is orientation preserving, then aW has a rotation number. (See Sec. 2.) G.D. Birkhoff recognized that the set of accessible points is dense in the boundary of an invariant region and that their dynamics can be characterized by their rotation number. He used this idea in [B] to construct a map of the annulus into itself with an unusual Invariant set J. On one hand, J resembles a closed Jordan curve in that each of its points is on the boundary of both an interior region Sint (conti.ning one boundary circle of the annulus) and an exterior region Sex. tcontaining the other boundary circle). On the other hand, J is "remarkable" in the sense that it contains a dense set of points accessible from Sin t with one rotation number and a dense set accessible from Sex t with a different rotation number. To compare this situation with our hypotheses, notice that such a map has an inverse for which J is unstable (in Sint and in Sext ) and J is the boundary between the points which go outward and those which go inward (under the inverse). Cartwright and Littlewood further developed these ideas in [C-L1], where they prove the existence of and determine the stability of periodic orbits for a certain class of second order differential equations in the plane. More recently, J. Mather" has given purely topological proofs of some of the topological results of Carath~odory in [M1] and has used the theory to study Invariant sets for area-preserving homeomorphisms of the annulus (M2],(M3]. We rely on the proofs in the above references of Cartwright-Littlewood and Mather I for much of the material on prime ends given in Secs. 5, 6, and 7. A general reference for Carath~odory's theory is (C-Lo], Chapter 9. The following argument explains the significance of the Attracting Lemma. Assume that the rotation number of f on S rational (say the reduced fraction p/q). is Then SI will have at least 1 fixed point under fq, (i.e., a periodic point of period q). If a trivial circle point x is not fixed under fq, then its orbit converges to a fixed point r under iterates of fq. is necessarily a trivial circle point. accessible point p on aW. By the Attracting Lemma, r Corresponding to r is an By Property 2, p is fixed under F we have the following result: 7 Thus THEOREM 1.2. Assume that aW is unstable in W and that for each k the fixed points of Fk are isolated. number p(W,F) is p/q (resp., 0). Assume further that the rotation Then there Is an accessible fixed point of Fq (resp., F) on aW. In Sections 6 and 7 we describe the dynamics on the set of accessible points under the hypotheses that p is rational, F is a diffeomorphism, and periodic points in the boundary are hyperbolic. (A periodic point p is hyperbolic if the Jacobian matrix no eigenvalues with absolute value 1.) DF(p) has By the Inverse Function Theorem, a hyperbolic point is isolated from other periodic points of the same period (or smaller period). In the following theorem, which is a special case of Theorem 6.1 in Sec. 6, we assume that W is a basin of attraction: i.e., there exists a compact set K in W such that the "w-limit set" of the orbit of each point x in W is non-empty and is contained in K. (Given a point x, the point z is in the w-limit set of the orbit of x, if there exists a sequence {tn }, with tn --) co, t such that f n(x) -- z.) If the orbit of each point in W is bounded, then there exists a compact set Sec. 2 for definition) [BS]. K' S K which is Liapunov stable (see - 3 I I I I THEOREM 6.1'. Assume that the periodic points of F in aW are hyperbolic and that W is a basin of attraction. If the rotation number p is rational, then every accessible point either is a periodic point or is In the stable manifold of an accessible periodic point. Theorems 1.2 and 6.1' do not mention the minimum period of an accessible periodic orbit. Degeneracies can occur due to the fact that the map hc is not necessarily one-to-one on trivial circle points, so that the period of the accessible points can strictly divide the period of the orbit on S1 . In Sec. 7 we prove that such degeneracies are ruled out for homeomorphisns of the plane, although they can still occur for homeomorphisms of the sphere. following two results. We use the The first, a converse of Theorem 1.2 for planar maps, implies that the period of an accessible periodic point cannot be strictly smaller than the period of a trivial periodic circle point. The second implies that it cannot be strictly larger. PROPOSITION 7.3. Let F be a homeomorphism of the plane R2 . there exists an accessible fixed point on aW, then PROPOSITION 7.4. p(aW,F) If is 0 If p = 0, then every accessible periodic point in aW is a fixed point. 9 COROLLARY 7.5. Let F he a homeomorphism of the plane R2. If Is the reduced fraction p # 0 p/q, then every accessible periodic point in aW has minimum period q. The next corollary (a special case of Cor. 7.6) follows, although not directly, from Prop. 7.3, Prop. 7.4, and Thm. 6.1'. In particular, it remains to be shown that if the orbit of an accessible point converges to a fixed point in aW, then the fixed point is accessible. We point out that this corollary does not mention the rotation number p. COROLLARY 7.6'. Assume the following conditions hold: (1) F is a diffeomorphism of the plane 2; (2) the periodic points of F in aW are hyperbolic; (3) W is a basin of attraction; and (4) either (I) there exists an accessible period point of minin~um period q, or (ii) there exists an accessible point which converges (under fq) to a periodic point of minimum period q. Then every accessible point in aW either is a periodic point of minimum period q or is in the stable manifold of such a periodic point. 10 In Sec. 2 we define a general class of connected, compact attractors and show that attractors in this class have connected, simply-connected basins. In Sec. 3 we study the orientation-reversing case, and in Sec. 4 we apply Theorem 1.2 to a class of chaotic i attractors, viewed as boundaries for the inverse of the map F. Figures 1 through 4 were made using Dynamics [Y]. i i I i i I I i I U I 2. Attractors with Simply-Connected Basins If A is a hyperbolic fixed point, then A has a connected, simply-connected neighborhood which contracts to it under iteration by F. In this case, the entire basin of A (see Sec. 1 for definition) is connected and simply connected. Here we look at a more general class of attractors and show that their basins are connected and simply connected and thus satisfy the hypotheses of Theorems 1.1 and 1.2. (The hypothesis that the boundary aW is unstable in U is trivially satisfied if either the attractor A does not intersect aW or if A has a dense orbit and is not a subset of For a closed set S, let S is the set of points y such that W.) be the c-neighborhood of S; i.e., S min U x - y I < c , where xcS 1111 denotes the Euclidean norm in-R . We say a set A is a regular attractor if A satisfies the following three properties: (2.1) A is compact and connected; (2.2) A is Liapunov stable; i.e., for each neighborhood Y of A there exists c>O such that A c Y, and if x e A then Fn(x) for all nl; (2.3) The basin of A contains an open neighborhood of A. in the following proposition, "area-contracting" means specifically that there exists a number g, where Idet DF(x)I < 9 , for all x in R2. - am. -- •m • . . m•m •• 12 •iI g < 1, such that Y, Il PROPOSITION 2.4. Let F be an area-contracting map of the plane. If A is a regular attractor, then the basin U of A is open, connected, and simply connected. Proof. Let given such that such that Y c U be an open neighborhood of A. Let Fn(A )C Y for all Fn(A)cA n t O. c Yc U, for all c > 0 be Select 6, 0 < 6 < c, nt J. Such c and Sexist, since A is Liapunov stable. Let x e U be given. I' Choose k > 0 such that Fk(x) e A . Since AC is open, there is an open neighborhood VX of x which maps into AC under Fk. Thus each point x in U has an open neighborhood V in U, and U is open. Let x I and x2 in U be given. Choose integers P>O and Q>O such that F (X ) E A8 and FQ(x ) e A Define m = max PQ. Then 1 2 a Fm(X ) and Fm(x ) are in AC . Since A is connected, Ac is connected for each c>O. Hence, A is an open, connected set. Since open, connected Cm sets are path connected, there is a path r in Y connecting Fm(xI) and Fm (x) Thus F-m(F) lies in U and connects x to x Therefore, U is connected. 31 It remains to show that U is simply connected. Suppose that U is not simply connected, and let C be a simple closed curve in U which bounds a region D containing a set S (consisting of one or more points) that is not in U. This implies that the distance between SFn (S) and that A is at least c, for all Fn (A ) c AS, for all n 0. Select a, 0 < a < 6, such n : 0. Since C is compact, there exists 13 an integer rJ(C) c A., J(c) > 0 such that FJ(c)(C) c A for all J a j((c). We conclude that the distance between Fn (S) and Fn(C) is at least c-8 for all . n ? J(c). Therefore, On the other hand, since F is area contracting, the distance between Fn(S) and Fn(C) converges to zero as n 4 w. This contradicts the fact that Therefore, U is simply connected. c-6 > 0. 3. Continuation and Orientation-Reversing Maps Let FA be a homeomorphism of R2 depending on a scalar parameter A. We assume that FA has a fixed point regular attractor 3A., which depends continuously on A, for each A. We define the maximal basin W to be the largest open set having a dense set of -A points that are attracted to A WA; under FA. Let B and let p. be the rotation number of F points in BA' be the boundary of on CA , the accessible For a parametrized homeomorphism on a circle, the rotation number varies continuously with the parameter (see, for example, (D]). Unlike the circle case, however, pA is not necessarily continuous in A. In fact, the boundpry BA can jump discontinuously, even when there is no change in the attractor. It was shown in [HJ] (see also [GOY] and [ATY]) that when the stable and unstable manifolds of an accessible saddle on the boundary become tangent at then cross for A = A., and A > A,, the stable manifold jumps a positive distance c (not dependent on A) into W.. for each A > A,. Figure 2 shows in black the basin of attraction of infinity for three different values of the parameter A in the Henon map FA b(x,y) = (A - x2 -by, x) 3 (3.1) where b is fixed at 0.3. There is a period two attractor in the white region to which the orbits of almost all white points tend. 3 experiments indicate that for Numerical X=1.39 (in Fig. 2a), a period-four saddle orbit and its stable manifold are the only boundary points I I 15 accessible from the white region. There is a tangency of the stable and unstable manifolds of consecutive points in this orbit at z 1.395. X = X. Specifically, if we number the four points in the orbit x1 9....x4 consecutively (in the counter-clockwise direction around the basin boundary), and if we set is a periodic trajectory. xn = Xn(mod 4) for n>4, then {Xn I At A=A, the unstable manifold of x. is tangent to the stable manifold of x + 1. For each A > A,, black points appear in what was the interior of the white region. addition, it has been numerically observed that for each In A > A. (near X,), the set CX of accessible boundary points is composed of a period-three saddle and its stable manifold. Fig. 2bc show in black the basin of attract.-n of infinity at A = 1.4 and A = 1.42,respectively, with the accessible period-three saddle. A numerical investigation of rotation numbers for the orientation-preserving, area-contracting Henon map appears in [AS]. When f is an orientation-reversing homeomorphism, the possible dynamics on accessible orbits are limited. For a connected, simply connected basin of attraction W, an orientation-reversing homeomorphism on W restricts to an orientation reversing-homeomorphism on aW. Again, we study the dynamics on aW through its association with the circle. have fixed points. two. An orientation-reversing homeomorphism f of S1 must It may or may not have periodic points of period Notice, however, that f can have no periodic points of minimum period greater than 2. The map f2 is orientation preserving and has rotation number 0 since it has fixed points. But an orientation-preserving homeomorphism of the circle with rotation 16 number 0 has no periodic orbits of minimum period greater than 1. i 3 Suppose f has a periodic orbit of minimum period k, kL3. a periodic orbit of minimum period k/2, if k is even, or of minimum period k, if k is odd. 3 Then f has Thus f has only periodic points of period one or two. We have the following restatements of Theorems 1.1 and 1.2 for * i orientation-reversing maps: 3 THEOREM 3.2 (ATTRACTING LEMMA). Let F be an orientation-reversing homeomorphism of the plane. i unstable in W. Assume further that the fixed points of F2 i aW are isolated. I Assume that aW is Then each circle point that is fixed under f2 and is attracting on at least one side is a trivial circle point. THEOREM 3.3. Under the hypothesis of Theorem 3.2. there is an accessible fixed point of F2 on aW. I Let F be a one-parameter family of orientation-reversing homeomorphisms. occurs for FV From Theorem 3.3, we observe that if a metamorphosis then BA must jump to different fixed points of F2 I I 17mm m~n Example. The Henon map (3.1) is orientation reversing for b<O. It is easily verified that F ,b can have at most 2 fixed points and at most one periodic orbit of minimum period two. In this situation the possible metamorphoses are severely limited by Theorem 3.3. As long as the period-two orbit and one of the fixed points is in the attractor (and the hypotheses of Thm. 3.2 are satisfied), no metamorphoses will occur. If, however, the basin becomes disconnected, as shown in Fig. 3, then the theorem no longer applies and the boundary can be fractal. Fig. 3 shows in white the basin of a two-piece attractor (which is also plotted in the white region). A metamorphosis has occurred, and there is no longer an accessible fixed point on the boundary. Now the accessible iddle has period six. The existence of periodic orbits in the maximal basin of the attractor but not in the attractor itself is also restricted by Thm. 3.2. Suppose (3.1) has a regular attractor A Properties (2.1)-(2.3)). (i.e., A satisfies If A contains a fixed point and an orbit of period two or if (3.1) is in a parameter range where there is no period-two orbit and A contains a fixed point, then the basin U of A is necessarily bounded by the stable manifold of the (other) saddle fixed point p. (For every choice of parameter values, the orbits of some points in the plane go to infinity; boundary.) thus the basin U has a In particular, under these hypotheses, there are no periodic orbits in the region containing A and bounded by W s(p) except those in A. 18 4. Rotation Numbers for Chaotic Attractors. Here we look at a class plane: an attractor B A is in of non-periodic attractors in the if 8 4 is compact, connected, invariant under F, and contains more than one point. In order to apply Theorem 1.2, we show how to assign a rotation number to an attractor in the class 4, assuming that F is an area-contracting nomeomorphism of the plane. This approach is reminiscent of Birkhoff [B] and also of Cartwright and Littlewood [C-L2] and Levinson [L] who studied attractors in forced 'an der Pol type equations. In looking at the Poincare map of such equations, Cartwright and I 3 3 Littlewood showed that there are invariant annuli which have unequal rotation numbers on the boundary circles and which possess strange attracting sets. Each such attractor is the boundary of the inside contracting and outside contracting parts of the annulus. The existence of different rotation numbers inherited from the boundary I circles was evidence to them of a continuum attractor which was not I homeomorphic to S . Levinson gave a careful analysis of the attracting invariant set of a piecewise-linear version of this map in I [Lni. His work set the stage for the discovery of the horseshoe map by Smale. I I I I See also Levi's analysis of forced van der Pol type equations in(Lil]. Let 2 Z = F u {ic} be the one-point compactification of R F extends to a homeomorphism of Z by setting F({o}) = {co}. 2 Then LEMMA 2 plane R2 . 4.1. Let F be an area-contracting homeomorphism of the If 6 Is in 4, then Z - 6 is connected and simply-connec'ed in Z. Proof. Since 8 is connected, each component of connected in Z. Z - 8 is simply (This simple fact follows most clearly from Alexander Duality with Cech cohomology. See, for example, (Do].) compact, only one component D of R2 8 - Since e is has infinite area (in R2 ) and, given any bound 7),there are only finitely many other components with area larger than n. Let D be a component of H maximum finite area in R2. of D. R2 -6 a homeomorphism. Thus 8 with Since F " is area-expanding and components map onto other components of But F -I also maps D R2 R-2 6, F "1 maps DM ont onto D , contradicting the fact that F" Z -e is is connected and simply connected in Z. 1 Now we can apply Theorem 1.2 to 6, which is the boundary of the open, connected, simply-connected region Z - 6. By looking at F-_ instead of F, it can be shown that G is unstable in follows. in C Let 6 n (Z-G) Z - e, as be an c-neighborhood of 0, and let D be an open set Since F-1 is area-expanding, the area enclosed by the boundary of D becomes unbounded under iteration by F_ easily be shown that almost all points in mapped out of G Z - 6 under F - -1 under iteration of F ; D n E eventually will be hence, G is unstable in Theorem 1.2 provides the following result: 20 It can I 3 3 I 3 3 PROPOSITION 4.2. Let F be an area-contracting homeomorphism of the plane, and let 8 be in the class A of attractors. for each k, the fixed points of Fk are isolated. number p(6,F) Assume that, If the rotation isthe reduced fraction p/q, then there is an accessible fixed point of F on 8. Figure 4 shows an attractor for the Ikeda map with an accessible period 6 orbit. For a typical area-contracting diffeomorphism depending on a parameter A, we conjecture that the rotation number p(A) will vary continuously, except possibly at a discrete set of values of A, and that p(A) will 3 I I I I I I I I of measure 0. le irrational for a non-empty set of A 5. Froof of the Attracting Lemma. Let F be an orientation-preserving homeomorphism of Z = R2 u {}, the 1-point compactification of the plane. A simple arc Q in W with end points q1 and q2, 1q q2 0 on aW and no other points on aW is called a crosscut of W. Each crosscut divides W into 2 subdomains, since W is simply connected. Let {Qn} be a sequence of pairwise disjoint crosscuts such that Qn separates Qn+ from Qn-l' Then there is a corresponding sequence {V n } of subdomains of W such that Vn contains Qn+ except for its endpoints. See Figure 5. The VI D V2 D V 3 D ... is called a chain. sequence V, = {V')} If V ={V n} and are two chains, we say V divides V' if for each i, there is n a j such that V' 9 V the other. We say V and V' are equivalent if each divides Under this relation, an equivalence class of chains is called an end. A chain V is called prime if any chain which divides it is equivalent to it. A prime end is the equivalence class of a prime chain. only if For the unit disk D in R2 a chain nvn {Vn } is prime if and n is a single point (necessarily on the boundary S') general, if there exists a sequence In Qn } of cross-cuts defining an end V such that {Q n} converges to a point in aW, then V is prime (see, for example, [MIl). Let (Vn} be a representative chain in a prime end V. Since each Vn is connected and W is compact in'Z, n vn is a connected, compact, non-empty subset of Z. Thus it is either a single point or a continuum. We call I(V) = nV nEIn 22 the impression of the end V. The I impression of V is independent of the defining chain in V. (However, I two prime ends can have the same impression. * that there are two prime ends corresponding to non-trivial circle 3 points and that beth have impressions that equal aW.) In [C], Carathdodory presents an example of a domain for which the impression of each prime end is a continuum; I In Fig. 1, it appears i.e., none is a single point. A point p in I(V) is called a principal point of V if there exists a sequence {Qn of crosscuts (defining a chain in V) such that {Q } converges to p, i.e., p is the only limit point of this sequence. The set of all such points is called the principal set of V. Finally, we say a point r in aW is accessible from W if there is an embedding n of I 3 I (0,1] into W such that these definitions. In Fig. 6, wp illustrate The following lemmas appear, for example, in [Mi] (as Theorem 17.1 and Corollary 15, resp.): LEMMA 5.1. I limt_>o+q(t) = p. The principal set of V has only one point e if and only if e is accessible from W. LEMMA 5.2. The principal set of V is compact, connected, and 3 non-empty. I 3 Now we describe a topology on the set of prime ends. open set in W. We say an end V is contained in U (i.e., Let U be an Ve 'U) if there exists a chain {V } in V all of whose elements are subsets of U. n Let W* = W u 0, where 0 is the set of prime ends of W. W is open if and only if U1 n W is open (in W) and 23 A set U* in "O = {V: V contains a chain all of whose elements lie in W}. With this topology, a sequence {E n) n of prime ends "converges" to a prime end E, represented by {V n}, if for every m, there exists such that En V for every n > N(m). N(m) e N We call WO together with this topology the Prime end compactification of W. Central to the theory of boundary sets is the following theorem of Carath6odory (see, for example, [C-Lo]): THEOREM 5.3 (Caratheodory). Let W be a connected, simply connected open set. Assume that aW has more than one point. Then W* is homeomorphic to a closed disk, where points in W correspond to points in the interior of the disk, azv- the prime ends correspond to points in S1, the boundary of the disk. Furthermore, if F is a continuous map on Z with W Invariant under F, then there is map FO on W* so that F* = F on W. With this theorem, we are able to learn about the dynamics of F on the boundary of W by studying the corresponding dynamics o" Si, the boundary of D. Prime ends "map" to prime ends under F; induces a map F' on W'. Let T hence F be a homeomorphism from W* to D, the closure of D. Then the circle S I is invariant under the induced homeomorphism f = ToF'of circle is classical. of D. The study of homeomorphisms of the Here we mention briefly some facts about these maps which are needed in the arguments that follow. A reference for this material is (D]. 2 24 ! Poincar6 showed that associated with each orientation preserving homeomorphism I of the circle is a "rotation" number, an asymptotic measure of the rotation of points on the circle under iteration by 1. 3 I In order to define this number, it is convenient first to consider a "lift" of T. A map G of R is called a lift of 7 if noG = yon, where n is the covering map from R to S I 3 I * 3 I i.e., ir(x) = exp(27rix). Let G(x) = limn) GCny)/n, for x in SI and y in R such that n(y) = x. (The value of independent of the choice of y.) We define the rotation number r of I to be the unique number In (0,1) such that pG(x) is pG (x) - r is an integer. This value is well-defined: FACT C1. The value r = r(j) is independent of both x and the particular lift G of 1. The dynamics of I are, to a large part, described by the rotation number r(T): I 3 3 FACT C2. A map z of the circle has points of minimum period q if and only If r(j) is an irreducible fraction of the form p/q, for some positive integer p. The map 1 has fixed points if and only if r(4) = 0. 25 Thus, if z has periodic points, they must all have the same period. FACT C3. If x has a periodic point of period n, then every point on the circle is either a fixed point of yn or is asymptotic to a fixed point under iterates of yn In the following set of definitions, we describe various notions of stability for periodic points and periodic prime ends. We often mention only fixed points, but the definitions and lemmas which follow carry over to periodic points by considering the appropriate iterate of f: a periodic point of period n is a fixed point of fn A fixed point p on S1 is called: (1) attracting on one side If all nearby points on one side of p converge to it under Iteration by f; (2) repelling on one side if all nearby points on one side of p converge to it under iteration by f-1. The analogous definitions hold on the space of prime ends if the word "point" is replaced by the term "prime end", and if "f" is replaced by "FO".) By Fact C3, an isolated fixed point p on SI is either attracting or repelling on each side. If p is attracting (resp., repelling) on one side, then by Carath~odory's Theorem, the associated prime end P is attracting (resp., repelling) on one side. A prime end P fixed under F* is called weakly contains a chain {Vn } such that 26 F() I ' table from W if 91 for every i. The following lemma follows easily from the definition of aW being unstable I I I in W (see Sec. 1): LEMMA 5.4. weakly stable from W. 3 I 3 I If aW is unstable In W, then no fixed prime end is The following three lemmas are important In relating fixed points of F on aW to fixed points of f on S1. Although there is a fixed prime end corresponding to each fixed point on the circle, It is not the case that a prime end which is fixed under FO necessarily contains a point which is a fixed point of F. Lemma 5.5 appears in [C-Li]. l LEMMA 5.5 (Cartwright-Littlewood). SF', Let ? be a fixed prime end of and let {Q I} be a chain of cross cuts converging to a point q (necessarily a principal point) of P. If, for every I, F(Qi ) has at least one point In common with QV then q Is a fixed point of F. LEMMA 5.6. If aW is unstable in W and if a fixed prime end P is attracting on one side, then all principal points of P are fixed under F. Proof. Suppose P is attracting on one side. principal point of P. In Let z be a By Lemma 5.2, there exists a sequence 27 Q} of cross-cuts converging to z. Let {Vn } be the chain defined by these crosscuts. By throwing out elements of the chain where necessary, we can assume that either F(Q) A QI * o, for all i, or that F(QI ) is disjoint from QV for all i. In the former case, z is fixed, by Lemma 5.5. Suppose that F(QI ) is disjoint from QV for all i. Then (OF)(Q I ) is disjoint from r(QI), for all i, and T(M) = p is attracting on one side. Let a- on S I be the end point of T(QI ) which is on that side of p. Then for I sufficiently large, fn ( I)--+p, as Since r(QI ) and (CoF)(QI ) are disjoint, we then have that But then F(VI) C Vi. for all i, contradicting (roF)(QI) C r(V). n--w. Lemma 5.4. Thus z is fixed under F. Proof of Theorem 1.1 (Attracting Lemma). a Suppose that x is a periodic circle point of period n and that x is attracting on one side. Then the corresponding prime end P is fixed under (F*)n and attracting on one side. fixed under Fn. connected. By Lemma 5.6 all principal points of P are By Lemma 5.2 the set of principal points is Since fixed points of Fn are isolated, there can be only one principal point, say p. By Lemma 5.1 the point p is accessible. For a given curve r in W limiting on p, the corresponding curve h-(r) (by definition) limits on a trivial point r in Si. 28 6. Hyperbolicity In this section we describe the dynamics on the set of accessible points under the hypotheses that F is a diffeomorphism of either the plane or the sphere and that periodic points in the boundary are In addition, we either assume that W is a basin of hyperbolic. attraction (see Sec. I for definition) or we add a condition on the map F at w. there exists We say that c is repelling in W r2 > 0 such that if Ixi< rI , if, for each then r1 > 0, IFn(x)l < r2 for all x in 1 and n a 0. THEOREM 6.1. Assume that the periodic points of F in aW are hyperbolic, and that either aW is unstable in W and w (i) W is a basin of attraction, or is repelling in W . (ii) If the rotation number p is rational, then every.accessible point either is a periodic point or is in the stable manifold of an accessible periodic point. The following lemmas are used in the proof of Theorem 6.1. each, the hypotheses of Theorem 6.1 are assumed. For Let S be a (finite) periodic saddle of F in aW, and let Ws (resp., Wu ) represent either branch of the stable (resp., unstable) manifold of S, excluding S. LEMMA 6.2. If8W Intersects Ws , then Ws and W are disjoint. 29 Proof. disjoint. If W is a basin of attraction, then clearly W s and W are Suppose therefore that aW is unstable in W, that repelling in W, and that both 8W and W intersect WS . Let Q crosscut in wnws, three components. and let Q2= F(Q1 ). Then W- One component meets both Q1 and w is be a j {Q1 U Q2 ) has %2 . Let D 1 be the component that meets only Q1, and let D2 be the component that meets only Q2 . Then Since v D2 = F(DI). is repelling in W exists a compact set K such that and W is invariant under F, there F(K nW) and an open neighborhood of S is in K. exists a sequence {D n } of open sufficiently large Dn nK is contained in K 0 w Iterating D 1 forward, there ets in W intersecting Ws such that {D n } approaches Wu (locally), as n-w. Given c > 0, choose j u. that Dn intersects K and there is no c-disk in for all n > j. (This is possible since K includes an open neighborhood of S and there are only a finite number of c-disks inside K.) Then for n > J, every point in D n n K boundary, contradicting the hypothesis that LEMMA 6.3. is within c of the aw is unstable in W. * If p c S' is a trivial fixed point, then it corresponds to an accessible fixed point S in the boundary aW. a repeller, then so is p. 30 I If S is I I 3 Proof. Corresponding to p Is an accessible point S in 8W. The point S is necessarily a fixed point since accessible points map to accessible points and S is the only accessible point corresponding to the prime end p. Suppose that S is a repeller. 3 g 3 Since the boundary 8W is connected and more than one point, each circle centered at S of sufficiently small radius must intersect aW. Let 7 be an "accessing" path in 11 which limits on S (corresponding to a path in the disk which limits on p), and let {QnI be a sequence of crosscuts converging to S such that (1) Qn is an arc of a circle of radius 1/(n+N) for some fixed integer N t 1, and (2) 1 intersects Qn an odd number of times, for each n. As I 3 described in Sec. 5, since the sequence (i.e., {Q n} the point S ), it defines a prime end. converges to one point Since this prime end has accessible point S with accessing path 7, it is represented by p on S1 By the construction, p is a repeller on S I 3 We say that two iccessing paths can be homotoped to z 0 and via a continuous family of paths that remains in W, all having the same endpoint S, (i.e., I 3 3 3 continuous family g(0) = S, for all 1 are equivalent if 0 gt:I --> W such that if there exists a g0 (I) = lot g1 ) = T, and t c I). Notice that if S has two non-equivalent accessing paths, then it corresponds to (at least) two different circle points under hc . In the next two lemmas, we assume the following additional hypotheses: (1) S is an accessible fixed point saddle; and (2) S has an associated trivial circle p which is attracting on at least one side, (i.e., lim there exists a point z c S1 , z ; p, such that fn Cz) = p). For c small, let M C be the union of the segments of the stable and unstable manifolds that connect S to the boundary of BC (S), the c-ball around S. We can assume that c is small enough that the segments of the stable and unstable manifolds in MC intersect only at S. LEMMA 6.4. Let 1 be an accessing curve to S. T)-n 7 is equivalent to an accessing curve that does not intersect MC . Proof. Suppose that I is not equivalent to an accessing curve that does not intersect MC . Since W is open, it must be the case that I intersects at least two components of and the boundary aW intersect Ws BC (S) - Mc n M.or and that both z both intersect Wu case in which both intersect Ws is ruled out by Lemma 6.2. that both intersect W . Let uW{Q n} n M. The Suppose be a sequence of crosscuts converging to S such that Qn is a closed interval on Wu and Qn intersects 7 an odd number of times, for each n. (Since the endpoints of Qn are the only points of Qn on the boundary aW, we can assume in fact that Qn intersects 7 only once.) Q} n The prime end determined by is represented by p on the circle. 32 In- this case, p must be a repeller, a contradiction. In the following, let c > 0 and let 7 be an accessing path to S such that there is a unique component of i. Call this component QC Lemma 6.4.) B C(S) MC - (The existence of Q that intersects is guaranteed by Since S is hyperbolic, we can further assume that B (S) is a neighborhood in which F is smoothly conjugate to a linear map, that S is the origin, and that QC is an (open) quadrant in R2. LEMMA 6.5. The romponent Q , as defined above, contains no points of the boundary aW. Proof. Suppose that Q contains a point of aW. Let ea, aeR, be a family of ("hyperbolic-like") invariant curves in Q boundary is connected, there is a connected component of Since the 8W n Q. containing S and a point b of e , for e sufficiently close to S. a a a Assume ea is sufficiently close to S that I extends from S to a point ga on ea. is above). Assume ga is below ba on ea (the argument is similar if it Assume further that F(g a ) is above b . (Otherwise take a higher iterate.) Then F(g a ) is between ba and F(b on a) e Since 7 and f(T) are both accessing curves to S (and they I- correspond to curves in the disk limiting on the same circle point), ga and F(g) can be joined by a curve contained entirely in W so that 33 the resulting loop g is null-homotopic in W. This is a contradiction since either b or F(b ) is contained in (. a Proof of Theorem 6.1. We assume that the rotation number is 0. (If the rotation number is p/q the proof.) with p*O , then replace F by Fq in Let x be an accessible point in aW which is not a fixed Corresponding to x is a trivial circle point z. By Lemma 6.3, point. z is not a fixed point. fixed point p on S1. point. Then the forward orbit of z converges to a By the Attracting Lemma, p is a trivial circle Corresponding to p is an accessible point S in aW. 6.3, q is a fixed point. aW * a By Lemma Since either W is a basin of attraction or is unstable in W, S cannot be an attractor, and again by Lemma 6.3, S is not a repeller. Thus S is a saddle, and the hypotheses of Lemmas 6.4 and 6.5 are satisfied by S, since p is attracting on one side. By Lemmas 6.4 and 6.5, there is at least one component Q BC (S) - MC which is in W and contains no boundary points. are boundary points in another component of of If there B (S) - MC, then they are in connected components of aW which intersect both invariant manifolds bounding that component. If exactly one component is free of boundary branch W s points and is in W, then there are accessible points on one of the stable manifold and one branch Wu of the unstable manifold. By Lemma 6.2, each point on this branch of Ws is an accessible boundary point. Thus points on one branch of the stable manifold of S are in one-to-one correspondence with points of S1 on one side of p. Let g and g2 refer to the segments on either side of p 34 consisting of points on the circle between p and the closest fixed points on either side. Let (If p is the only fixed point, then 9, = 92' be the segment which corresponds to W s . Necessarily, 9 l is part {Q n} of the stable set of p. Let be a sequence of crosscuts converging to S such that one endpoint of Qn is on Wu and one is on W for each n. Since accessible boundary points on W F- to S, given a point y in WU fn aW converge under (necessarily accessible) and n>O, all but a finite number of points in the forward orbit of y under F will be in Vn' n, the closure of the domain determined by Qn and S. In this case p, which corresponds to the prime end determined by is repelling on 02" Qn , Since the forward orbit of z converges to p, z must be on gi, and thus x is in the stable -inifold of S. * The argument given in the previous paragraph holds in all cases in which a sequence {Q n} of crosscuts in W converging to S (i.e., a sequence which defines the prime end represented by p) has the I property that one endpoint of Qn is in W n 3 u and one is in W s , for all 0. The case in which there are exactly three components of B (S) - M in W this case. If the crosscuts do not have this property, then there are which are free of boundary points also reduces to necessarily exactly two or exactly four components in W. In these cases, both endpoints of a crosscut are in one or the other branch of the stable manifold of S. 3 (Since the fixed point p is attracting from at least one side on S I, the case in which only the unstable manifold of S intersects the boundary is ruled out by an argument similar to I that in the proof of Lemma 6.3.) In this case, p is necessarily attracting on the circle, and points on both g I 35 and 02 are in one-to-one correspondence with points in the stable manifold of S. Thus x is in the stable manifold of S. x The following corollaries follow from the proof of Theorem 6.1. The first extends Theorem 1.1 (the Attracting Lemma) to all points of S1 not just periodic points. The second shows that the map hc , the accessible-point extension of the Riemann map h (described in Sec. 1), is continuous on stable manifolds of periodic points of S I (up to and including the periodic point). For a trivial circle point r, we let r denote the corresponding accessible point in aW. We assume the hypotheses of Theorem 6.1. COROLLARY 6.6. Assume that p is rational. If a point r in S' is not a periodic point, then r is a trivial circle point. COROLLARY 6.7. Let p in S' be a periodic point of f, and let {r n } be a sequence of points in SI converging to a point r. If rn is in the stable manifold of p, for each n, then the corresponding sequence {r } of accessible points in aW converges to r in 8W. 36 36I I ;I 7. Minimum Periods of Accessible Periodic Orbits Unfortunately, although a rational rotation number p/q implies Il that f has a periodic orbit of minimum period q on SI, we cannot claim that F has a periodic point of minimum period q. See, for example, the boundary depicted in Fig. 7, where p(CW,F) is 1/3, and F has an *I accessible fixed point on the boundary but no period three orbit. Recall that hc is the accessible-point extension of the Riemann map h. If the rotation number p of f is rational (say p/q), but not 0, then trivial circle points which are periodic (necessarily of minimum period q) can map by hc to periodic points in the plane of smaller minimum period. This situation is illustrated in Fig. 7, where all points in one orbit on the circle coalesce to a fixed point Surprisingly, Cartwright and Littlewood [C-Li] showed on the sphere. that this type of example is the only possible one when accessible points coalesce: THEOREM 7.1 (Cartwright-Littlewood). If p # 0, then 8W contains at most one accessible fixed point. It is easily seen that this theorem rules out coalescing to an orbit of minimum period strictly between 1 and q. Suppose that a trivial periodic orbit of minimum period q on the circle maps (under hc) to a periodic orbit of minimum period k on 37 W, where k#1 and kaq. Then k = q/r for some divisor r of q (r*1), and Fk has k accessible fixed points on aW. But the rotation number of the induced map fk on I the circle is non-zero, contradicting the theorem. The situation illustrated in Fig. 7 can be largely overcome by using Theorem 6.1 and assuming that the accessible periodic points are saddles. PROPOSITION rotation number 7.2. p * 0 Assume the hypotheses of Theorem 6. 1. is the reduced fraction p/q, where If the q 0 2, then every accessible periodic saddle in aW has minimum period q. I Proof. Suppose there exists an accessible orbit of period k on k aW, where 1<k<q. Then Fk has at least k fixed points, but the I rotation number of the induced circle map fk Is non-zero, contradicting Theorem 7.1. fixed point saddle z on aW. Hence we assume there is an accessible Given a path r in W limiting on z, let y be the trivial circle point which is the limit point of h -1 (r). Either y is a periodic point of period q, or the forward orbit of y under fq converges to a periodic point r. By Theorem 1.1, r is a trivial circle point. By Corollary 6.4, the trivial circle point r corresponds to the accessible point z (i.e., hc (r) = z ), as do each of the q points I r = rI , r2 ..... rq 38 in the orbit of r. U Let 0 be the center of the disk which S 711-...7q Then 3 3 r be line segments joining 0 to bounds and let rl,.. .rq, respectively. h(I11J.... h(q) are paths in W, all of which limit on z. Let and r be adjacent points on the circle. Since r1 ,..., rq represent distinct prime ends on S1 , the closed loop formed by h(v ), and Therefore, by Lemma 6.5, manifold of ) z necessarily contains boundary points in its interior. These boundary points are connected to If q h is 3 or z q 4, z within the loop. can be at most 4. then at least one branch of the stable is in W, and r is necessarily attracting on at least one side of the circle under fq --as is each of the q points in i the orbit of z. Each of these stable sets must correspond to a branch of the stable manifold of 3 I 3 r in the disk connecting a point in the stable set of a path a point in the stable set of TI z. On the other hand, there exists or xj r ri to which crosses one of the segments exactly once and intersects none of the other segments. Hence, in W, h(f) crosses h(iI) (or h( )) exactly once and intersects none of the other "accessing" paths, a contradiction for q >2. i For a map of the sphere, two types of degeneracies are possible when p = 1/2, even with the hypothesis that accessible orbits are hyperbolic saddles. In Fig. 8a, I p = 1/2 These possibilities are illustrated in Fig. 8. and there is an accessible fixed point saddle p I on aW. In Fig. 8b, aW Is a line segment. The basin W complement of B) is simply connected on the sphere. p = 1/2 (the In this case, I and there is an accessible fixed point saddle p and an accessible saddle orbit {r,r 2 } of period two. I I The situation is greatly simplified when we look at homeomorphisms of the plane. We use the following converse of I Theorem 1.2 for planar maps: PROPOSITION 7.3. '.et F be a homeomorphism of the plane R2. there exists an acce!aile fixed point in aW, then If p = 0. 1 1 I Suppose Proof. and let p c S1 0 0. Let 2 N > I M-I (f(p), f (p)]...., [f' Let I be an accessible fixed point, be a corresponding trivial circle point. p e 0, we can choose extension h x such that the intervals N (p), f (p)] of the Riemann map, and 6 cover S I . Since [p, f(p)], By Property 2 of the hc(fi(p)) = x, for be paths beginning at a point 0 i z 1. in D and r be the closed loop ending at p formed by h(z), h(6), h(O), and x. Choose a preferred direction, and f(p), respectively. Let clockwise or counterclockwise, so that the accessible boundary points corresponding to trivial circle points between p r. Let G be r together with its interior. Since the accessible 40 . and f(p) are in 1 I IN points are dense in compact set aW, the entire boundary is contained in the K = U f (G) .. The complemeint of D is contained i=1 entirely in W or entirely in R2\W. since W is simply connected. But then of 3 The former case is ruled out h(7) c W is in the boundary K, a contradiction. Now assume that p is the reduced fraction p/q. Assume further that there is an accessible periodic orbit of minimum period in 3 aW Since (in the plane). Fr r, F induces the map has a fixed point, the rotation number of Prop. 7,3. I The iterate Thus all periodic points c.-* which Implies that q divides r. in S1 f fr f on S is 0, by are fixed points of The next proposition shows that q must equal r. I PROPOSITION I 7.4. If p = 0, then every accessible periodic point in aW isa fixed point of F. I Proof. Assume there is an accessible periodic point period q, q > 1. Let I p and let D, let 7 hc, p is not a fixed point. be the line segment from be the line segment from I I of be a trivial circle point corresponding to x. By Property 2 of the map be the center of x 41 0 to 0 Let to 0 p, f i(p), for each i, Then 1 S i S q+1. form a closed curve Since of r which, except for p, ii contained in W. p = 0, f(p) the circle and is between By Prope rty 2 of the map is inside on one side of heCf(p)) F hc (fq+l(p)) I (hc(p)) = hc (fq ()). I is i,% r, a M contradiction. COROLLARY 7.5. n q c o and Thus one r and the other is outside. F(h C(p)) = Fx) =Fq+1() But only one of p # 0 fq(p) p and fq+ 1 (p) is between them on the other side. Ac(?1) and hc(q+1 hc(f(p)) p, and they both contain h (Xq) and h(z Let 1 be a homeomorphism of the plane R2 If is the reduced fraction p/q, then every accessible periodic point in 8W has minimum period q. Proof. Suppose there is an accessible periodic point with period r. By the discussion following Prop. 7.3, q divides r. It follows from Prop. 7.4 that since the rotation number of fq (on S1 ) is 0, q must equal r. I The final corollary puts together the previous results with the assumption of hyperbolocity to obtain a statement that does not mention the rotation number p: I 42 I COROLLARY 7.6. Assume the following set of hypotheses: (1) F is a diffeomorphism of the plane R2; 3 (2) the periodic points of F in aW are hyperbolic; either (I) W is a basin of attraction; i(3) or (ii) aW is unstable in W, and w is repelling in W; 3 and (4) either (i) there exists an accessible periodic point of minimum period q, or (ii) there exists an accessible point which converges (under fq) to a periodic point of minimum period q. I 3 Then every accessible point in W either is a periodic point of minimum period q or is in the stable manifold of such a periodic point. I I 3 We need to prove that if an accessible point Proof. fq under show that z, then to a periodic point p z exposition, we assume that q is 1 and that I 3 3 Otherwise, replace F with F of the stable manifold of and gI 'be paths from y U .) By hypothesis, z. Let to z x y converges First we (For ease of is a fixed point. x s is on one branch W be a point in W, and let and from y to go F(x), respectively. By Lemma 6.2, W and Ws are disjoint. on any assumption about p.) I is accessible. Assume otherwise. is rational. x unmn mu nuum (This lemma does not depend Therefore, there must be accessible 43m points in the region between x and F(x). V bounded by g0 , In fact, since and the portion of W s g, is irrational, there must p be points in the orbit of the accessible point But then W s must enter V. 91, 91 p or W5 , x in the region The only way Ws can enter V is through all of which are impossible. (In particular, are in W, which by Lemma 6.2 does not intersect Ws.) ir rational, and by Theorem 6.1, an accessible periodic point, namely Since z V. g g0, and Therefore, is in the stable manifold of x z. has minimum period q and p is rational, by Cor. 7.5, eve-y accessible periodic point has minimum period q; the result follows from Theorem 6.1. I I II 1 44I U I 3 References [AS] K. Alligood and T. Sauer, "Rotation numbers of periodic orbits in the H~non map", Commun. Math. Phys. 120 (1988), 105-119. [ATY] K. Alligood, L. Tedeschlni-Lall, and J. Yorke, "Metamorphoses: sudden jumps in basin boundaries", preprint. [B] G.D. Birkhoff, "Sur quelques courbes fermees remarquables", Bull. Soc. Math. France 60 (1932), 1-26. [BSI N.P. Bhatia and G.P. Szeg, I 3 Stability Theory of Dynamical Systems, Springer-Verlag, Heidelberg, 1970. [C] C. Caratheodory, "Uber die Begrenzung einfach zusammenhangender Gebiete", Math. Ann. 73 (1913), 323-370. [C-L1] M.L. Cartwright and J.E. Littlewood, "Some fixed point theorems", Ann. Math. 54 (1951), 1-37. [C-L21 "On non-linear differential equations of the secorA order: I. Th- equation y"-k(1-y 2 )y'+y = bAkcos(At+m), k large", J. London Math. Soc. 20 (1945), 180-189. I 3 __ (C-Lo] E.F. Collingwood and A.J. Lohwater, Theory of Cluster Sets, Cambridge Tracts in Mathematics and Mathematical Physics, No. 56, Cambridge Univ. Press, 1966. [D] R.L. Devaney, An Introduction to Chaotic Dynamical Systems, Benjamin/Cummings Publishing Co., Menlo Park, 1986. [Do] A. Dold, Lectures on Algebraic Topology, Springer-Verlag, Heidelberg, 1972. I 3 [GOYa] C. Grebogi, E. Ott, and J. Yorke, "Basin boundary metamorphoses: chanees in accessible boundary orbits", Physica 24D (1987), 243-262. [GOYb] , "Critical exponent of chaotic transients in nonlinear dynamical systems", Phys. Rev. Lett. 57 (1986), 1284-1287. 3 3 [HJI S. Hammel and C. Jones, "Jumping stable manifolds for dissipative maps of the plane", Physica 35D (1989), 87-106. [Lil M. Levi, Qualitative Analysis of the Periodically Forced Relaxation Oscillations, Mem. AMS 214, 1981. [Ln] N. Levinson, "A second order differential equation with singular solutions", Annals of Math. 50, no. 1 (1947), 127-153. 45 [Ml] J. Mather, "Topological proofs of some purely topological consequences of Caratheodory's theory of prime ends", Th.M. Rassias, G.M. Rassias, eds., Selected Studies, North-Holland (1982), 225-255. [M2j J. Mather, "Area preserving twist homeomorphisms of the annulus", Comment. Math. Helvetici 54 (1979), 397-404. [M3 , "Invariant subsets for area-preserving homeomorphisms of surfaces", Advances in Math. Suppl. Studies, Vol. 7B (1981). _ [Y] J.A. Yorke, "Dynamics, a Program for IBM-PC Clones", 1987. (Available from J. A. Yorke.) 46 I I I 6 * :'. - " "w.l . 1.2. -a. I • f I -. -2 -3.-2- .1 1 •O1• .3 .. FIGURE 1. Two basins of attraction of the time 2Tr map of the forced damped pendulum equation 0" + .10' + sine = 2cost are shown in black and white. The black and white regions are connected on the cylinder. !A +1 FIGURE 2. A portion of the basin of infinity of the Hnon map (3.1) is shown in black for b fixed at 0.3 and each of three values of the parameter X. The x and y values shown are in the rectangle L-2,21 x [-2,11] In (a) at X = 1.39, the set of accessible points consists of a period-four saddle and its stable manifold. Crosses show a period-three saddle to which the boundary jumps at a boundary metamorphosis at X z 1.395. In (b) and (c) at = and X = 1.42, respectively, set of accessible points consists1.40 of this period-three saddle and the its stable manifold. I - I I I U I I I I I I I I I I .:..* .. - . - I I I I -, *~l(~) I I I I I 1 I 1 I I I 1 I I I I I I U I I I I I I U II I m FIGURE 3. A portion of the basin of infinity of the orientationreversing H6non map (3.1) is shown in black. There is a twopiece attractor whose basin is not connected, and the basin boundary is fractal. 4.A caoti oftheIked ma FIGUE atractr 09(xsnT ycoT)I (97 -sinT, +0.9xcos f~x~) isshow. Tere sa T.4 x+y2)-6.0(1.0+ wher accesibe (1rbi on peiod6 hettratoI F's x FIGURE 5. Sequences of crosscuts and subdomains defining a prime end are illustrates. Ii I /Qn _ _ _ _ _ _ _ _ _ A _ _ B I (b) __ _ __ A _ __ P _I I'BI (c) FIGURE 6. Each figure represents an open, simply connected setI (the interior of the rectangle minus the line segments). In each case, segment AB is the impression of a prime end. In (a), each point of AB is a principal point, and there are no accessible points in AB. In (b), segment CD is the principal set of AB, and there are no accessible points in AB. In (c), P is the only principal point and the only accessible point of AB. I U I, I p=f (PI 1P I 3 ~P3f 2 (pl) I P=f (PI) II (a) I i I I I I (hi FIGURE 7. In (a) the rotation number on the boundary circle is 1/3. The circle maps to the boundary in (b), however the boundary in (b) does not have an accessible periodic point of period 3, but rather has an accessible fixed point. This example is realizable on the sphere, not in the plane. Ft, 7 o I I I I ' / cb) / I ,r'/l / FIUR S.Totpso|eeea so teshri hc h iiu periods of accessible saddles do not equal the periods of orbits of then associated circle map, are shown. By Prop. 7.2, the rotation numbers in each case is 1/2. in (a), the boundary DW has a fixced point saddle p. in (b), W is the complement in the sphere of the line seIi-nt 31 from rI to rg; thus 3W is the boundary of a simply connected set v on the sphere. Agai, the rotation number is 1/2 ad there is an accessible mixed point saddle p and an accessible saddle orbit r,rl 2 of period two.s of h I 5 Figure 1 Two basins of attraction of the time 21r map of the forced damped pendulum equation )" + .16' + sinO = 2cost are shown in black and white. The black and white regions are connected on the cylinder. 1 Figure 2 A portion of the basin of infinity of the H~non map (3.1) Is shown in black for b fixed at 0.3 and each of three values of the parameter A. The x and y values shown are in the rectangle [-2,21 x (-2,11]. In (a) at A = 1.39, the set of accessible points consists of a period-four saddle and its, stable manifold. Crosses show a period-three saddle to which the boundary jumps at a boundary U metamorphosis at A = 1.395. In (b) and (c) at X = 1.40 and A = 1.42 , respectively, the set of accessible points consists of this perpod-three saddle and its stable manifold. I I Figure 3 A portion of the basin of infinity of the orientation-reversing H6non map (3.1) is shown in black. There is a two-piece attractor whose basin is not connected, and the basin boundary is fractal. Figure 4 A chaotic attractor of the Ikeda map f(x,y) = (.97 + 0.9(xcosT - ysinT), 0.9(xsin'r + ycosr)), where 2 period 6 orbit on the attractor. I 2 r = 0.4 - 6.0/(1.0 + x + y2), is shown. There is an accessible Figure 5 Sequences of crosscuts and subdomains defining a prime end are illustrated. Figure 6 Each figure represeits an open, simply connecteu set (the interior of the rectangle minus the line segments). In each case, segment AB is the impression of a prime end. In (a), each point of AB is a principal point, and there are no accessible points in AB. In (b), segment CD is the principal set of AB, and there are no accessible points In AS. In (c), P is the only principal point and the only accessible point of AB. Figure 7 In (a) the rotation number on the boundary circle is 1/3. The circle maps to the boundary in (b), however the boundary in (b) does not have an accessible periodic point of period 3, but rather has an accessible fixed point. This example is realizable on the sphere, not in the plane. Figure 8 Two types of degeneracies on the sphere, in which the minimum periods of accessible saddles do not equal the periods of orbits of the associated circle map, are shown. By Prop. .7.2, the rotation number in each case is 1/2. In (a), the boundn-y aW has a fixed point saddle p. In (b), W is the complement in the sphere of the line segment aW from r1 to r2 ; thus aW is the boundary of a simply connected set W on the sphere. Again, the rotation nnber is 1/2 and there is an accessible fixed point saddle p and an accessible saddle orbit {r,r 2 } of period two. I| WHEN CANTOR SETS INTERSECT THICKLY 3 Brian R. Hunt'. Ittai Kan2. and James A. Yorke July 3, 1991 Abstract The thickness of a Cantor set on the real line is a measurement of its "size". Thickness conditions have been used to guarantee that the intersection of two Cantor sets is nonempty. We present sharp conditions on the thickness of two Cantor sets which imisly that their intersection contains a Cantor set of positive thickness. 1 - Introducticn Newhouse defined [5] a nonnegative quantity called the "thickness" of a Cantor set in order to formulate conditions which will guarantee that two Cantor sets intersect. (All Cantor sets considered in this paper lie in R 1 .) These conditions have been used [5, 6, 7, 8, 9] in the study of two-dimensional dynamical systems to deduce the existence of tangencies between stable and unstable manifolds whose one-dimensional cross sections are Cantor sets. I Thickness may be thought of as a measure of how large a Cantor set is relative to the intervals in its complement. Henceforth, these intervals will be referred to as gaps; the two unbounded intervals in the complement are each included in our use of the term gap. Newhouse's result [5, 7, 8] is that two Cantor sets must intersect if the product of 3 3 their thicknesses is at least one, and neither set lies in a gap of the other. When this latter condition is satisfied, the sets are said to be interleaved. In [10], Williams observed the surpising fact that two interleaved Cantor sets can have thicknesses well above one and still only intersect in a single point. 'Code R44, Naval Surface Warfare Center, Silver Spring, MD 20903-5000 of Mathematical Sciences, George Mason University, Fairfax, VA 22030 3Institute for Physical Science and Technology and Department of Mathematics, University of Maryland, College Park, MD 20742 The first author was supported by the ONT Postdoctoral Fellowship Program administered by ASEE, by ONR, and by the NSWC Independent Research Program. All three authors were partially supported by Applied and Computational Mathematics Program of DARPA. 2 Department Sthe One might hope that under sufficiently strong -'L3 -G, G3 GR R3 L2 G2 , R2 Figure 1: Constructing a Cantor set thickness conditions, the intersection would be a Cantor set. However, the intersection of two arbitrarily thick interleaved Cantor sets can contain isolated points, so Williams posed the question of what conditions on the thicknesses of two interleaved Cantor sets will guarantee that their intersection contains another Cantor set. Williams obtained such a condition, though" it is not sharp. In this paper we obtain the sharp condition. More precisely, we find a curve in (T, 'r2 )-space such that if the ordered pair (T 1 , "2 ) of thicknesses of two interleaved Cantor sets lies above the curve, their intersection contains a Cantor set, but if the pair of thicknesses lies below the curv,_ here exist examples for which the intersection is a single point. Kraft [2] has independently arrived at this condition. We further show that if the thickness pair lies above the curve, the intersection must contain a Cantor set of positive thickness. This is the only result that addresses in terms of thickness how large the intersection of two Cantor sets must be. There are well known probabilistic results concerning the Hausdorff dimensions of intersections of Cantor sets (c.f. [1, 3, 4]). One may think of a Cantor set as being constructed by starting with a closed interval and successively removing open gaps in order of decreasing length. Williams' formulation of the thickness of a Cantor set may then be thought of as follows. Each gap G, is removed from a closed interval I,, leaving behind closed intervals L, the left piece of I, - G,, and R, oil the right (see Figure 1.) Let p, be the ratio of the length of the smaller of L, and R" to the length of G,. The thickness of the set is the infimum of p, over all n. We consider as an example the "middle-third" Cantor set, constructed as follows. Start with "he'closed interval [0,1], and remove the open interval (1/3, 2/3), the middle third of the original interval. Then from each of the two remaining intervals, remove their middle thirds; repeat this process infinitely often. Each gap G,, is the same length as the adjacent intervals L,, and R,, so p,, = 1 for all n. Thus the thickness of the middle-third Cantor set is one. There is a connection between the thickness of a Cantor set and its fractal dimension, which depends in part on how the ratios p,, are distributed as n -* o. However, two largc gaps close together make the thickness of a set very small, while its dimension can still be large. It was shown in [7] that the Hausdorff dimension of a Cantor set with thickness r is bounded below by log 2/ log(2 + 1/r). This lower bound is sharp for the middle-third Cantor set (whose dimension is log 2/log 3.) We offer here a new formulation of the definition of thickness which we state for all 2 SI: ib, I a, l C2 2: b2 Figure 2: Non-intersecting interleaved sets compact sets, not just Cantor sets. (The results in this and previous papers are found to be valid for all compact sets.) We define non-degenerate intervals to have infinite thickness, while singletons are defined to have thickness zero. In fact, any set containing an isolated point will be seen to have thickness zero. To define the thickness of a compact set S which is not an interval, we consider a type of subset of S obtained by intersecting S with a closed interval. We call such an intersection P a chunk of S if P is a proper subset of S and has a positive distance from S - P, the complement of P in S. (Notice that for P to be a chunk both P and S - P must be closed and nonempty.) We then define the thickness of S to be the infimum over all chunks P of the ratio between the diameter of P and the distance from P to S - P. In the case of the middle-third Cantor set, the given ratio can be shown to be smallest when the chunk P is obtained by intersecting S with an interval L" or R, in which case the ratio is one. in Section 2 we will show that our new definition is eq"'valent to the old one for all Cantor sets. The reason thickness is an appropriate quantity for determining when one can guarantee that two compact sets intersect is illustrated by considering an example where each of the two sets is a union of two disjoint intervals. For i = 1,2 let 5, consist of closed intervals of lengths ai and b, with a, < b,, separated by a distance ci. Then each Si has only two chunks, and is found to have thickness a,/c,. If the product of the thicknesses ala 2 /cIc 2 is at least one, then either a, > c2 or a 2 >_c1 (or both); assume a, >_c2 . Then since bi >_a1, neither interval of S, can lie in the gap of S2; hence if the two sets are interleaved, they must intersect. If on the other hand ala 2 /clc 2 < 1, then with an affine map we can scale the sets so that a, < c2 and a 2 <c, and position them so that the component of S, with length a, lies inside the gap of S2 , and vice versa. The two sets are then interleaved, but they do not intersect (see Figure 2). This example could of course be made to involve Cantor sets by constructing very thick Cantor sets in each chunk of each S,. An important point which is apparent in the above example is that the union of two sets can have a smaller thickness than either of the original sets. In other words, adding points to a set can decrease its thickness. By the same token, one may be able to increase the thickness of a set by removing appropriate subsets. This observation is useful in the following way. No matter how thick two interleaved compact sets are, their intersection may have thickness zero because it may contain isolated points, or arbitrarily small chunks which are relatively isolated from the rest of the intersection. Nonetheless we are able to show that 3 6 5 = f(71) 4C g( 2) 'W'7 t 1 3 2 B " = /77 A 0 0 1 2 3 4 5 6 7l-- Figure 3: The intersection of two interleaved compact sets with thicknesses T1 and 72 can be empty for (7"1 , r2 ) in region A, must be nonempty but can be a single point in region B, and must contain a set of positive thickness in region C. if the original sets are thick enough, then by throwing out the relatively isolated parts of their intersection we can obtain a set of positive thickness in the intersection. To define the set C of thickness pairs (r 1 , 72 ) for which a Cantor set of intersection can be guaranteed, we make use of the functions T2 + 3r + 1 r2 fjr) (27 + T3 g(T)= Let C be the set of pairs (71, r- 72) 1)2 for which one of the following sets of conditions holds: - 2, f(r2), and r2 > g(rl) (1.1) r2 > f(r 1 ), and 'r > g(r2) (1.2) 71 > or r 2 >_r,- (see Figure 3.) Our main result is as follows. 4 Theorem 1 There is a function (r, -r2) which is positive in region C such that for all interleaved compact sets S1,$ 2 C R with T(S1 ) >!ri and r(S2) >_r2 , there is a set S C SfnS2 with thickness at least p(r, r 2 ). Notice that a compact set with positive thickness can have no isolated points, and thus must either be a Cantor set or contain an interval; either way it contains a Cantor set. We remark that (ri, r2) is in C if both thicknesses are greater than V2 + 1. This is the critical value Williams found for the case of interleaved Cantor sets with the same thickness. Also, no matter how small one thickness is, the other thickness can be chosen large enough so that the pair lies in C. Our results and the results of Newhouse are summarized in Figure 3. In Section 2 we give a proof of Newhouse's result, which will illustrate some of the methods to be used later. Then we present for all pairs (r,,r2) not in C an example of interleaved compact sets with thicknesses 71 and 72 whose intersection is a single point (except when (r-, r2 ) is on the boundary of C, in which case our example gives a countable intersection.) This example shows that Theorgm 1 is sharp in that its conclusion cannot hold for any larger set of thickness pairs (r1 , r2). In Section 3 we prove Theorem 1, and in Section 4 we discuss some further properties of S1 n S2 . The positive thickness set S E S n S2 constructed in Section 3 need not be dense in S, n 52 ; however we find that there are subsets with thickness at least p(rl, r7 2 ) arbitrarily near any accumulation point of S, n S 2. In additioa, we find bounds on the diameter of S which allow us to obtain thickness conditions that imply thac the intersection of three Cantor sets is nonempty. 2 Preliminaries Let us define precisely the concepts and notation we will use. Definition 1 We say two sets S1, S2 C R are interleaved if each set intersects the interior of the convex hull of the other set (that is, neither set is contained in the closure of a gap of the other set.) We define the distance between two nonempty sets S1, 32 to be d(Si,S 2) = inf{Ix - yjx E Si, y ES 2}, and write S2 - S1 for the intersection of 52 with the complement of S1. We say that a set S is a chunk of a set S,, and write S1 0( S2 ,if S1 is the intersection of a closed interval with S2, is a proper subset of S 2 , and d(S 1 , S 2 - S1) > 0. Notice that a closed set S has a chunk if and only if it is not connected. We denote the diameter of a ret S (the length of its convex hull) by IS1. Lk Gk Rk L, G. R. Figure 4: Chunks and gaps of a Cantor set (k < n) Definition 2 Given a compact set S C IR, we define the thickness of S to be 'r(S) = inf IPI P=S d(P,S - P) (2.1) prozided S has a chunk. Otherwise, we let r(S) = 0 if S is empty or consists of a single point, and r(S) = oo if S is an interval with positive length. The following simple proposition demonstrates that Definition 2 agrees with Williams' definition of thickness for Cantor sets [10]. Proposition 2 Let S be a Cantor set, and define the ratios p,, as in the introduction. Then the quantity r(S) given by (2.1) is equal to the infimum of p,n over all n. Proof The intervals L,, and R, defined in the introduction are the convex hulls of chunks A, = L, nl S and B,, = Pn n S of S. Since the gap Gn is not larger than any previously removed gap Gk, k < n, it follows that d(A,,,S - A) = d(B,,,S - B,) = IG,, (see Figure 4.) Thus for all n, m i'L I IR,,I ( IAI IBnI (S). = i-,['IG,,I, =mI \d(A,-S--A)' d(B,,S-Bn) ) Next, if P is a chunk of S, it must be bordered on each side by a gap of S; let Gn be the smaller of these two gaps. Then IG,,I = d(P,S - P) and IPI > min(IL,,, IR,I). Therefore r(S) inf IPI ?osd(, S - P) - nn which completes the proof I We now prove Newhouse's result in a way that will motivate our later examples and methods. Proposition 3 If S1 and S2 are interleaved compact sets with r(S1).r(S 2) _ 1, then S, n S 2 ts not empty. 6 P3 P1 PO 1 S2-H P2 XO 3 X2 P I-H X4 Figure 5: The points x,, and chunks P,, Proof Let S1 and S2 be as above, and let xo=max infx, xES2 infx) \XESI the greater of the leftmost points of S and S2 . Assume without loss of generality that xo E S2 . We will show that S1 n S2 is nonempty by looking for the leftmost point of this set. Let x, be the leftmost point of S which is at least as great as x0 . Since S and S2 are interleaved, x, must exist (otherwise S would lie entirely to the left of S2; see Figure 5.) Next, let X2 be the leftmost point of S 2 greater than or equal to xj. Once again the interleav:ng assumption implies that X2 exists, for otherwise S2 would lie inside a gap of S. We similarly define X3, X4, ... ; if each of these points can be shown to exist, we claim to be done. Then {x,} will be a nondecreasing sequence which is bounded above (since 51 and S 2 are bouided), so it approaches a limit. This limit must belong to both S and S2 since these sets are closed and the odd numbered terms of {,} belong to S1, the even ones to S. If at any step x,, exists and equals X,-1, then Xn+l,Xn+2, .. .will also equal X,,_, and we will have found a point in S1 nl S2. Henceforth we assume £o < x, < ... as long as they are defined. We know at least that x0, £1, and X2 exist, so there is a chunk PO of S2 which lies in [xo, xj), whose diameter is thus less than x, - xo, and whose distance from the rest of S 2 is greater than £2 - X1 (see again Figure 5.) Then X1X X2 - > x1 > (S2). (2.2) d(Po, S2 - Po) Let P be the largest chunk of S, which lies in [X1, X2). If X3 did not exist, in other words if all points in S1 were less than X2, then S - P1 would lie to the left of P1 , and the distance between these sets would be greater than x, - xo. But then using (2.2) and r(S).T(S2) > 1 we woul t. have d(P,S, - P) x, - o (S2 ) contradicting the definition of the thickness of S1. Thus X3 exists, and similarly to (2.2) we obtain > X2-X> X3 - X2 (S). (2.3) d(P 1 , S1 - P ) Likewise (2.3) car be used to show the existence of by induction. I 7 £4, and so forth. The proof is completed One could similarly find the rightmost point in $n S 2 , but as Williams observed it may coincide with the leftmost point, even if both thicknesses are significantly greater than 1. We next present an example which will give a single point of intersection for thickness pairs ('r-, r2 ) not in the closure .f region C, and a countable intersection for (ri, r2 ) on the boundary of C. In our example both sets are countable unions of closed intervals, but they could be replaced by Cantor sets with the same thicknesses by constructing a very thick Cantor set in each of the closed intervals. Let r be a positive constant, and define the intervals A0 = [r2 +3r+ 1,(2T + 1)21, Bo = [r,r2 + 3r + 1], An = 2 Bn = 2r +1 r +Y Ao, Bo, where multiplication of a set by a scalar means the set obtained by multiplying each element of the original set by the given scalar. Let S,= (0A) U {, B(C U{O . S2 = Notice that Bn is the closure of the interval between An and An+ 2 for all n, and An is the closure of the interval between B.-2 and Bn for n > 2. Thus S, n 82 is countable, containing only the point 0 and endpoints of the intervals An and Bn. Furthermore, the intersection could be reduced to only the point 0 by shrinking the intervals which make up one of the sets by a factor arbitrarily close to one. Let us compute the thicknesses of the sets S and 82. Observe that IA, I d(B-._. 2 ,B,) = (2Ti- IBnI d(Am,An+ 2 ) = (2r ) i r(3r + 1), ) (37+ 1). The intervals An are ordered from left to right A 1 , A 3, As,.. , A 4, A2 , Ao, so any chunk P of S which does not contain 0 must be a finite union of consecutive even or odd numbered A,. Let An be the interval in P with the largest index; then IA,,I 1PI d(P,S, - P) - d(A,,,An,+2 ) 8 with equality holding when P = A,. On the other hand, if a chunk P of S 1 contains zero, let n be the larger index of the leftmost and rightmost Ak in P. Then P must contain A,,- 1, and since P is not all of S, n > 2, so (r/(2r + 1))n-l(3r + 1)(2r + 1) IPI > IA, U A.-I.I (r/(2r + 1))n- 2 (3r + 1) d(P, S1 - P) - d(An,A n- 2 ) Therefore the thickness of S is T. Similarly, if P is a chunk of S 2 , then for an appropriately chosen B,, either _ (2r+ 1)2 IBI IPI g(r). 73 d(B,, Bn+2 ) d(P,S2 - P) or - PI > d(P, S 2 - P) - Bn UB,- II d(B.,Bn_2 ) (r/(2-r + 1))n-1((3-r + 1)/(27 + 1))(72- + 3-r + 1) (7r/(27r + 1))n 72 2 ,(37 + 1) + 3-r + 1 2 7 ff(r). Thus r(S 2 ) = min(f(,r),g(-r)). As we pointed out before, by reducing the thickness of S2 by an arbitrarily small amount we can shrink the intersection of S, and S2 to a single point. Let 7r1 denote the thickness of the set S1 , and let r2 be the thickness of S2 . Then up to a change of indices, the above construction demonstrates that a single point of intersection can be obtained when either 71 < min(f(T2 ),g(r 2 )) (2.4) r2 < min(f(71 ),g(7.1 )). (2.5) or Also, if either (2.4) or (2.5) is an equality instead, the intersection can be countable. (Kraft [2] has analyzed this borderline case and determined when the intersection can be finite.) Therefore we can only hope to guarantee an uncountable intersection if 7ri > min(f(T2 ),g(-r2 )) (2.6) -r2 > min(f(7r1 ),g(Tij)). (2.7) and One may check that g(7) > f(7) > v' + 1 for r < v + 1 and g(7r) < f(r) < v2 + 1 for r2, and to r > vf2 + 1. Therefore (2.6) and (2.7) are equivalent to (1.1) in the case r" 1 >__ (1.2) when r2 > r.. 9 Case 1 s .<------------------.. 1. = Io S2:77- Case 2 J< SI:: <--...... -- ... S:<----1J*J . .......... -----> 11 Figure 6: Cases in the construction of I. and J. 3 Proof of Main Result We now prove Theorem 1 by constructing a set S with positive thickness in S1 n S 2. Proof of Theorem 1 Let S, and S 2 be interleaved compact sets with r(SI) >_-rand r(S 2 ) >_r2 for some (rl, -r2) in region C of Figure 3. Let the gaps of S be lo, I1, 12, ... , with Io and I unbounded, lo to the left of I,,and !1f2 ? 1131 >_ "'.For S2 we define JoJ 1J/ , 2, ... similarly. We refer to the intervals I, and J,, collectively as the "original gaps". Our goal is to construct the complement of S as a union of disjoint open intervals Ko, Yi, K 2 , ... with Ko and K, unbounded, and with every original gap contained in some K, (whence S C SnfS 2.) To get a lower bound on the thickness of S,observe that every chunk P of S is bordered on each side by a gap of S,with at least one of the bordering gaps being bounded. Pick a chunk P, and say P is bordered by Km and K, with m > n and m > 2. Then IPI d(P,S- P) = d(K, I) min(lI,,I 1'.I) - d(K, ,') IIimI The theorem will therefore be proven when we show for some t(rl, r2) > 0 that whenever m>n andm >2, d(KmKn)> ( 2) (3.1) We begin by finding a pair of original gaps I. and J.between which S will lie; that is, I. and J.will be contained in Ko and K 1.The properties we desire of I. and J.are that they are a positive distance apart, that all gaps of S with an endpoint between the closures of I. and J. are bounded and no larger than I., and likewise (in comparison to J.) for gaps of S2 between I. and J.. We will show later that once I. and J. have been determined, the diameter of S can be bounded below by a constant depending on r, and r 2 times the distance between I. and J.. Assume without loss of generality that Jo C lo. If I C J, 1 (Case 1 of Figure 6), then 1. = Io and J. = J1 have the above properties; they must be separated by a positive distance 10 1: J----> F---I S2: H------H -------I S: < ....................................... gfo Figure 7: The construction of Ko I II since S and $2 are interleaved. If J, C 1 (Case 2 of Figure 6), let J. be the largest gap of S2 with an endpoint between Io and 1, and let I. be whichever of Io and 1 is farthest from J.. At least one of Io and 1 must be a positive distance from J. since S and $2 are interleaved. Next, let t be a positive constant whose precise value will be chosen later; for now we assume that t < (rir2 - 1)/(r, +r 2 +2) < min(rl, r2). Assume without loss of generality that I. lies to the left of J.. We begin constructing K0 by requiring that it contain I.. We then require that K0 contain the rightmost bounded J, with d(I., J,) < tIJ, I (we will verify that there is a rightmost gap satisfying this condition when we later examine our construction in more detail.) If there does not exist such a J, that is not already contained il I., we stop the construction and let "0 = I.. Otherwise, we further require that No contain the rightmost bounded I, that is within t times its length of the previously added J,,. Again, if this requirement does not extend Ko any farther rightward, we stop the construction. If' not, we then add to Ko the rightmost J which is within t times its length of Im and is at most as large as J, (see Figure 7.) If a next step is necessary, we consider gaps of S1 which are no larger than 1,, and so forth. We may have to continue this process infinitely often, but if so we must converge to a right endpoint for K0 , sinc3 there is no w,.y this construction can extend past the rightmost point in Sl U 52. We define K, similarly, starting with the requirement that K contain J. and extending K1 to the 1cft if necessary in the same way we constructed Ko. Next, to construct K 2 we first require that it contain the largest original gap (choose any one in case of a tie) not contained in K0 U K1 (if no such gap exists, we leave K2 undefined and let S be the complement of Ko U K.) Then we extend it it on both the left and right in the same manner as before, but considering only gaps that are at most as large as the one we started with, to obtain the endpoints of K2. \Ve next start with the largest original gap not contained in Ko U K1 U K 2 , proceeding similarly to define K3, and so forth. Any given original gap must eventually be contained in some K, because there can be only finitely many original gaps that are as large or larger than the given one. We do not yet know that the KN are disjoint from each other; this .will follow when we prove (3.1), though. Let us now examine our construction more closely. Define 1(1) and r(I) to be respectively !1 the left and right endpoints of an interval I. For a given K,, let Go be the gap we started with in its construction, which for n > 2 must be the largest original gap it contains (or at least tied for the largest.) For simplicity we assume here that Go is a gap of S1 . Consider the collection E of all Jn with IJI - IGol, r(Jn) > r(G,), and d(Go,J) < tlJ,,I. We claim that the members of E (if any) are increasing in size from left to right. If J, J" E E with Jm to the left of J,, then since r(J.) < r(J,,), it follows that d(J, Jn) < d(Go,.J) < tlJi.. Since t < r 2 and d(J,,,, J) > r 2 min(IJm[, IJI), it must then be the case that IJI > IJ l Thus if E is not empty, it must have a rightmost member, which we call G, (notice that G, is also the largest member of E.) If E is empty, we let G1 be empty, but in order to facilitate future formalism, we define IG,I = 0 and r(GI) = r(Go). One must keep in mind this degenerate case in verifying the assertions and formulas that follow. We likewise define G 2 to be tht rightmost gap of $ which is at most as large as Go and lies within t times its length of G1 ; again if no such gap exists with r(G2 ) > r(GI) we say that IG21 = 0 and r(G 2 ) = r(GI). Next, to define G3 we consider only gaps which are at most as large as G 1,for G 4 we look only at gaps no larger than G 2 ,and so forth. Define G-1 ,G- 2, ... similarly to be the leftmost (and largest) gaps added to K, at each stage of the process of extending If,, leftward. Then we may think of the open interval K, as being defined by l(I,,) = lim l(Gm), r(K,,) = lim r(Gm). Each limit exists becausc; it is the limit of a bounded monotonic seq.aence. In the above construction, the even-numbered G. are gaps of S, and the odd-numbered ones are gaps of S2, but if Go had been a gap of S2 it would be the other way around. In any case. Go is the largest even-numbered G and either G 1 or G- 1 is the largest oddnumbered one. Also, the even-numbered G,, decrease monotonically in size as one moves either rightward or leftward from the largest, and the same statement holds for the oddnumbered Gm. We call a given Gm either a "1-gap" or "2-gap" of K,, according to whether it is a gap of S1 or S2 . Notice that not all original gaps contained in K,, are 1-gaps or 2-gaps, only those that have been given a label G in the construction of K,,. When we refer henceforth to left-to-right ordering or adjacency among the 1-gaps and 2-gaps of a given K,,, it is with respect to the ordering ... , G- 2 ,G-1,Go,G 1,G 2,. (Thus, for instance, 1-gaps can only be adjacent to 2-gaps and vice-versa.) The following lemma will be used in bounding both the numerator and denominator of the left side of (3.1). It establishes for all m > 0 a bound on how far K,, can extend to the right of Gm in terms of how far Gm+, extends past G, and similarly for m < 0 on the left. 12 I Lemma 4 Assume t < (-r, 1)/(TI + 2- I T-2 + 2). Let 0 1 (T T - t)(r 2 +1) 2 (Ti -t)(T 2 - t) - (1+ t) = and ( = - t)( + 1) t)( _t)_(I+ t)2 Let C be a 1-gap of K, which is at least as large as all 1-gaps of K" to its right. Let H be the next 2-gap of K,. to the right of G. Then r(K,.) - r(G) _ u2(r(H)- r(G)). l* The same statement with "1" and "2" interchanged holds, as do the corresponding results for left endpoints. Proof Let I be the next 1-gap of K,. to the right of G. Then since IlIII III <-IGI, < d(G. I) <_d(H, I) + r(H) - r(G) _ till + r(H) - r(G). which, because t < T1 , implies that III is bounded above by (r(H) -r(G))/(Ti - t). Hence 1 r(I) - r(H) (1 + t) 111 11 + d(H,I) _< 1+ t (r(H) (t - r(G)). Likewise the next rightward 2-gap of g, extends at most ((1 + t)/(r beyond I, and by induction I 2 (3.2) - t))(r(I) - r(H)) r(K,.) - r(G) = r(H) - r(G) + r(I) - r(H) + ... +-... ) (r(H) - r(G)) < 1+ -- +_±t_ I= u 2 (r(H) - r(G)). The geometric series converges, and the denominator of I 02 is positive, because of our as- sumption that t < (7rr 2 - 1)/(r1 + r 2 + 2). I The next lemma builds on Lemma 4 to obtain a positive lower bound on the distance between a given Km and K,,, provided we can find a 1-gap of Km and a 2-gap of K which are respectively larger than all 1-gaps and 2-gaps between them. The proof is difficult and will be handled later. Lemma 5 There exists a function 0,(r 1 , 72) that is positive whenever (r1 , 72) is in region C and t is sufficiently small, and for which the following statement holds. For m 0 n, let G be a 1-gap of K, and H be a 2-gap of K,,. If all 1-gaps of K,, or K, with at least one endpoint between the closures of G and H are bounded and at most as large as G, and all similarly situated 2-gaps are bounded and at most as large as H, then d(K,,Km) v/t,(r, 72)d(G,H). 13 Case I Case2 Case 3{ S2: = HI- SI: H S --i G S:H I --..-... -H lqH ---iH H ----------.. H J H F---------=_ i Figure 8: Cses in the proof of (3.1) Recall that to construct Ko and K1 , we chose I. and J. to satisfy the above hypotheses. Thus we now know that K0 and I1 are disjoint and separated by a positive dista,,ce (which is at least Ot(ri, -. ) times the distance between I. and J..) Now suppose 0 < n < m and m > 2; we will prove (3.1) by finding a G and H which satisfy the hypotheses of Lemma 5. Assume without loss of generality that K, lies to the left of K,,. Let I be the largest original gap in KIn; say I is a 1-gap. If all 1-gaps of K are smaller than I (Case 1 of Figure 8), let H be the largest original gap in K,,. Since m > n, K,, was constructed before K,, so H must be at least as large as I, and thus is a 2-gap. Let G = I; then G and H satisfy the hypotheses of Lemma 5. Also, d(G, H) > tIG1, since otherwise G would have been included in the construction of K,.. If on the other hand there are 1-gaps of K,, which are at least as large as I (Cases 2 and 3 of Figure 8), let J be the closest such gap to I. Consider all 2-gaps of Km or K,, to the left of J; let K be the largest such 2-gap (any one will do in case of a tie.) Notice that K must be adjacent to I or J. If K is in K,, (Case 2), let G = I and H = K; then G and H satisfy the hypotheses of Lemma 5, and d(G, H) > tIGJ because G was not included in K.. Otherwise (Case 3), let G = K and H = J, and reverse the indices "1" and "2". Once again, G and H satisfy the hypotheses of Lemma 5 and d(G, H) > tIGI. Notice also that in all cases, G is the largest 1-gap of K", and H is at least as large as all 2-gaps of Kn. We now estimate how large K, can be. Let I and J be the 2-gaps of K, adjacent to G on its left and right, respectively. Since I is at most as large as H, T211 <_d(I, H) :_d(I, G) + IGI + d(G,H) _ tIlI + IGI + d(G,H), 14 or in other words III < - 1 -- (IGI + d(G, H)). (3.3) t The same bound holds also for J, so by Lemma 4, SIK.I = IGI + l(G) - (1g.) + r(Km) - r(G) < IGI + o2(l(G) - l(I)) + a 2(r(J) - r(G)) IGI + a2(1 + t)(1I1 + IJI) 1+t <_IGI+ 2U(-r t)(IGI + d(G, 1I)) (+ 2 02 1+t (1 +1+d(G,H) (,r2 - t ) t t 1 2+ 1)d(GH)3 (r,_t)(r2 - t) + (1 + t)(1 2 (2r + t) ) t((rl - t)(r 2 - t) - If on the other hand G is a 2-gap and H is a 1-gap, we 'btain the same bound as (3.4), but with the indices "l" and "2" interchanged. Then in either case, IKm I_< (rI - t)(r 2 - t) + (1 + t) 2 (2 max(ri, r2 ) + 1) d(G, t((T1 - t)( 2 - t) H). + t)2 ) -: Finally, by Lemma 5, d(Km,K) IKI ( (1 + t) 2 )t(r,r)2 ) t((r - t)(r 2 - t) - t)(r 2 - t) + (1 + t) 2 (2max(rl,r2) + 1) (3.5) The right side of (3.5) is positive as long as t is between 0 and (7r12 - 1)/(r + r2 + 2), and 72) > 0, and goes to zero when t approaches any of these borderline values. Therefore the right side of (3.5) attains a maximum value, call it ;(71, 72), at some allowable value of t, say t.. We thus carry out the construction of S with t = t.; then (3.1) holds, and the 3)t(r, proof is complete. I Let (71,r2) = t. (r1 ,r 2 ); then t.((r t.)( t.) (I + t.)2)0(rl, r2) 1 2 (r - t.)('r 2 - t.) + (1 + t.) 2 (2 max(ri, r) + 1) - - - Remark We will see in the proof of Lemma 5 that 0'(r1 ,r 2 ), and hence ;(-r,72), must be very small when (7l,r2) is near the boundary of region C. flowever, if both T1 and r2 are large and t is small compared with the two thicknesse5, it is not hard to check that O.t(rl,r2) is close to one. Then if 71,72 > 1, one finds that t. is of order N/min(Tl, r), whence o(ri, r 2 ) is of order V/min(ri, 72) also. Thus when the thicknesses of S, and S2 are large, the lower bound we obtain on the thickness of S is reasonably large. 15 --- I S2-:.- HH [----- Ho 2 Figure 9: The gaps G'i and Hi We now prove our main technical lemma. Proof of Lemma 5 Let G be a 1-gap of Km and H be a 2-gap of I, satisfying the hypothesis. We assume without loss of generality that rl _ r2 ; then by (1.1) the condition (r1 ,r 2 ) E C implies + 1 2 2+ 2 1 (3.6) and 72 > g(r 1 ) = (2ri + (3.7) 1)2 If d(G, H) = 0, the inequality to be proven is trivial. Otherwise, let us normalize d(G, H) to be one, and assume G lies to the left of H. Let Go = G and H0 = H. Let G1 be the 1-gap of K, adjacent to Ho on its left, and let H1 be the 2-gap of Km adjacent t, Go on its right. Let G2 be the adjacent 1-gap of Km rightward from H1 , and likewise define H2, G 3 ,1I3 , .. (see Figure 9.) For i > 0 let Xi - l(Gi+1 ) { I(Hi) i even ={ r(H'+) - r(G,) i even i odd. r(Gi+) - r(Hi) i odd and il(Gi) - l(Hi+1 ) Let Ri = d(Gi, H,); then R0 = 1 and R,+1 = max(R, - xi - yi, 0) for i > 0. Let R, be the limit as i goes to infinity of Ri. Then d(K,,,K,) = R,, so we wish to show that there is a positive lower bound on R, which depends only on r1, r7 2, and t. In the same way as we obtained (3.2) it follows that for all i, and a dy y Xi+i < 1-+l-tt i (3.8) i+1< 1 + Iti (3.9) T2- t Furthermore, by Lemma 4 we have that Yi + Xi+1 + i+2+" 16 O2Yi and Xi + Yi+i + Xi+2 +... crlX. Thus, for each i, 3R. 3 R xi-,yi - x+ 1 -Yi+i - - > Ri - oixi- 2yi. (3.10) We will show that for some i, the right side of (3.10) is positive. Next let us obtain upper bounds on xo and Yo. We know that Xo = l(Ho) - l(Gi) < jGII + d(G 1 , Ho) :_ (1 + t)1G 11, (3.11) and by hypothesis 1G11 _ IGoI, so xo = I(Ho) - r(Go) - (l(Ga) - r(Go))= 1 - d(Go, G) <_ 1 - ril GI. (3.12) Eliminating 1G1I from these inequalities yields Xo < 1+ t (3.13) Yo : 1+ t (3.14) Similarly, We can obtain similar bounds on x; and y, for i > 1, but the bounds are complicated by the fact that we do not know in general that 1G 1+j1 _<GI (or 114+ 1 _ IH I). The analogues of (3.11) and (3.12) are thus x,. < (1+ t)lG,+l I and xi _ Ri - ri min(IGil, IGj+j 1). If IjG+ 1I < (3.15) eGvi, then as in (3.13) it follows that 3i < 'r, 1++1t+ t Ri. (3.16) If IG,+jI > lGjj, then by (3.15), ?-TIGl ij xi < Rix - -rjlGd R- y l7 i.(3.17) (3.17) If (3.16) fails, then using (3.17) together with the negation of (3.16), one finds that x,- 1 is bounded above by the right side of (3.16). Thus regardless of the relative lengths of G, and Gi+i", min(xi, xi-1) _< 1 + t Ri. 17 (3.18) for i > 1. Likewise, regardless of the relative lengths of Hi and Hi+,, we have for all i > 1 that 1 + t R.(3.19) min(yi, yi-1.) < 7 2 + 1+t Let ai = xi/Ri and bi = yi/Ri provided Ri > 0; then Ri1= inax(1 ai - bi, 0)Ri. - Thus ai+l and bi+l are defined as long as 1 - ai- bi > 0. For j = 1, 2 let (3.20) 1+= Ai 1~+1t' Yj-7j- t The conditions (3.13), (3.14), (3.18), and (3.19) can then be written a0 < A,, bo :5 A2, and(39)beom Alsomi (3.8 codtin 1~+-1 - ab - and 1 1 Weso observeon that8) and 3. - oa2b- 0,(323 aedfie9tlas)sln because a ,<A thn T1 T - 1 r 1 a(-biA (1 + 1)( 0, -aa 2 + +3.1) -A 2 -)(+) r+ t +1)( > 0 t) 2 + t+1) adb 2 (since t < (rlr 2 - 1)/(r 1 + -r2 + 2).) Also, as long as a, < A1 , by (3.22) we have Yz2AI 1-A, - bi* Let Sh(b) = 1- L2 A1 - b The equation h(b) = b has two solutions, -4P2A1 _A (1- 1 - A, 2 and if the roots are real, then h(b) < b for b- < b < b+ (this can be verified by checking the value b = (1 - A1)/2.) We claim that for t sufficiently small, b. are real, with b+ > A2 and (3.24) 1- auAl - a2b-> 0. (3,25) Let us delay the verification of this claim until the end of the proof. Choose b. > b... with 1 - a1 ,1 - o 2b. > 0. Now b0 _<A2 < b+, and as long as ai _< . continues to hold, bi+, :_h(bi) < bi for bi E (b., b+). Then eventually bi _<b., and furthermore since b - h(b) must have a positive minimum value on [b., A2] (if b.> A2 then bo < b. already) there is a maximum number N (depending only on rl, r 2 , and t) of iterations it can take before bi :5 b.. We therefore have shown that if ai < A1 for i < N, then bi < b. for some i < N, and hence (3.26) 1- alai - a 2 bi > !-iA - a2 b. > 0. If on the other hand a,+, > A1 for some i < V, then let i be the smallest index for which this occurs. We claim that then (3.23) holds for i. By the results of the previous paragraph. bi < bi-, < ... < bo _<A2. Also, by (3.21), ai _<Al(1 - ai - bi), or in other words ai :_i-1T1 1-bi). Then 1- 1a - a 2 bi >_1 - l bi. 0.2 Now when t = 0, C7 1 Al I1+A, (71 + 1)r2 r-7 2 -1I (T Tr(7 2 +10 +2)('r1T2-l1) rl(rlr 2 - 1) + 2r-r2 + 2r2 (,r 1 + 2)(7-1r 2 > 0, 19 - 1) and thus for t sufficiently small it remains positive. Then since bi -,la, - AIx ( rA A Tb I_ 012ba I'bT iT (.-, 1# 1 + A,) A 2)L7 a22 When t A2 , (3.27) l 0, by (3.6) i- o 2A 2 = ( 1+1)7 + 2 (r2 + 1)(ri 7 - 1) 7 1 122 - 2r 2 - 1 (r 2 + )(r1 >2 2- (r 2 .)(rT T2 - 7yrT while alAl (A2)i- + r2 = - 1) + 3r 2 + 1 - 2r2 - 1 2 r 1 (r2 2 1)- 1 + 1)_rr2 TA, (r 1 + 2)(rT, 2 -r -) 71 + 2 rr - 1 so .'.e right side of (3.27) is positive for t = 0. It therefore remains positive for t sufficient: small. To summarize, we have shown that if t is sufficiently small, then for some i < N, either (3.26) or (3.27) holds. The right side of each of these equations is positive and depends only on rl,'r2 , and t. Furthermore, ai < AI and by _<A2 for j _<i, so by (3.20), Rj+1 > (1 - Al - A2 )tj, and hence Ri _ (1 - \ 1 - A2 ) . Then by (3.10), Roo > Ri(1 - olai - a bi) >_(1 - A1 - A2 )(1 where 1 -ala, - a2bi), - a2b, is in turn bounded below by the lesser of the right sides of (3.26) and (3.27). We have therefore shown for t sufficiently small how to obtain a positive lower bound on RO, which depends only on rl, r2 , and t; we let 4t(r., r2 ) be this lower bound. It remains for us to verify (3.24) and (3.25). We again show they are true for t = 0, whence they hold for t sufficiently small by continuity. When t = 0, , 1 /(T, 1 + 1) ± Iri/(r '-,2 ± r1 2 + 1)2 - 4/((T 2 b± 2 2 - 4(r, + )7-2 2(rl + 1)r2 Now by (3.6), 712722 - 4(rl + 1)r2 = 71(7722 - 20 4r2 ) - 4r2 + 1)-r2 ) (3.28) > -+I) - 472 l(22r2 + 1)(r22 - (r22 +37-2 = 3 (722 72 - - 2+ 1) - 4r23 (3.29) 1)2 722 Thus bd. are real md distinct (and the same must then hold for t sufficiently small.) Next by (3.28), 2 + 722 - 4T25(1/'r + 1/7i2) 22(I + 1/ri) from which we see that b+ is increasing as a function of ri. Thus b+is greater than the value it would take on if (3.6) were an equality, which owing to (3.29) means (7.22 + 372 + 1)/72 + 22 -2 r2 11/r2 2+ 2(2732 + 3 22 + 1)/r2 U> 722+32+1-(721 - ) 1 2(,2 + 1)(272 + 1) ,r2 + I A2.0 - Hence (3.24) holds for t = 0, and consequently for t sufficiently small. When t = 0, (3.25) can be written b- < 1)2,r2(71 -+ 2r 01 Or 2,r2 (3.30) 1 The right side of (3.30) is an increasing function of 72, and since b. - V1 4714(+1)/r2 2(71 + 1) b- is a decreasing function of 2. Then by (3.7), b ~ b7-1 - Vri2 - 47i,3 (Til + 1)/(2r, + 1)2 2(r, + 1) Ti((2ri + 1) - V(27ri + 1)2 - 4(r,2 + 7r) 2(71 + 1)(2r, + 1) (1r + 1)(2( + I m white w2h72 - 2r - 1 (-r, + 1)27- i) + (2 " + 1)2//1 -(2r1 (T-1 + 21 + 1) 1)2(2--1 + 1)2/713 +1(ri + 1)2 (2,r + 1) 2 7T (+ )(2r, + 1) Thus (3.25) holds for t = 0, and for t sufficiently small. The proof of Lemma 5 is now complete. 4 I Intersecting Three or More Cantor Sets In proving Theorem 1, we chose a subset S of S1 nS 2 in order to guarantee positive thickness. In this section we demonstrate that positive thickness sets are in some sense generic in S, nS 2 . We also explain how Theorem 1 is useful in finding conditions under which three or more Cantor sets must have a nonempty intersection. The set S we constructed in Section 3 need not be dense in S fn S2 nor even in the non-isolated points of S n S2. However, there are subsets of S, n S 2 with thickness at least v('i, r2) near i.ay .ccumulation point. To see this, let {q,} be a sequence of distinct points in S, n S2 which converge to a point q. It is not hard to s: aw that within any neighborhood N of q there are compact subsets T, C S, and T2 C 52, each of which contains all but finitely many q,, with r(T) 2! r(SI) and r(T 2 ) ? -r(S2). Notice that any two compact sets which intersect in three or moe points must be interleaved. Thus T, and T2 are interleaved, and by Theorem 1 their intersection contains a set with thickness at least w(ri, -r2). We conclude that arbitrarily near any non-isolated point of S, n S2 there are subsets of Si n S2 which have thickness at least p(r1 , 72). In addition to showing that there are many subsets of S, n S2 with positive thickness, it is possible to obtain a lower bound on the diameter of the positive thickness subset S of Si n S2. If the two sets Si and S 2 are interlea.,d in such a way that neither is contained in the convex hull of the other, then by the discussion following the statement of Lemma 5, the diameter of S is at least O(ri, 7r2) times the length of overlap between the convex hulls of S and S2. Since the thickness of S is at least V(,rl, r2), we immediately have the following result. Corollary 6 Let S, and S2 be two interleaved compact sets whose thicknesses (r1,r2) lie in region C and for which the intersection Q of their convex hulls contains neither S, nor S 2. If S3 is a compact set with largest bounded gap G such that (i) the hull of S3 contains Q, (ii) IGI < O(ri, 72)IQI (iii) T(S 3 )p(7 1 ,7 2 ) > 1 • • m 99 I then s1fn 2 fn s3 is nonempty. 3 3 3 1 3 35] 3 We note that if instead of condition (iii) we required the pair r(S 3) and V(, rj2) to lie in C, then S, n '2n Ss would contain a set of thickness at least v(r(S3),w(rl, r 2)). Thus one can inductively find thickness conditions guaranteeing the nonempty intersection of any finite (or even countably infinite) number of compact sets, although the analogue of the interleaving condition gets more complicated. sufficiently far from the boundary of region C, then as discussed in the remark If ('ri, r2 ) is preceding the proof of Lemma 5 it is not hard to obtain explicit lower bounds on w(ri,1 r 2 ) and V)(r(, r2). In particular, for rT and r2 large we found that V(r ,r 2) is at least of order /min(rw,r), and 01(ri, r2) is approximately one. We thank the referee for a thorough reading of our paper and many helpful comments. References [1] K. J. Falconer, The Geometry of Fractal Sets, (Cambridge University Press, 1985). [2] R. Kraft, "Intersections of thick Cantor sets," Mem. Amer. Math. Soc. (to appear). [3] J. M. Marstrand, "Some fundamental geometrical properties of plane sets of fractional dimensions," Proc. London Math. Soc. (3) 4 (1954), 257-302. [4] P. Mattila, "Hausdorff dimension and capacities of intersections of sets in n-space," Acta Math. 152 (1984), 77-105. S. Newhouse, "Nondensity of axiom A(a) on 14 (1970), 191-202. 2, Proc. A.M.S. Symp. in Pure Math. [6] S. Newhouse, "Diffeomorphisms with infinitely many sinks," Topology 13 (1974), 9-18. [7] S. Newhouse, "The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms," Publ. Math. IHES 50 (1979), 101-151. [8] S. Newhouse, "Lectures on Dynamical Systems," Progress in Math. 8 (Boston: Birkhauser, 19S0), 1-114. 1 3 I Robinson, "Bifurcation to infinitely many sinks," Comm. Math. Phys. 90 (1983), [9] C. 433-459. [10] R. F. Williams, "How big is the intersection of two thick Cantor sets?", to appear in M. Brown, ed., Contemporary Mathematics, Proceedings of the 1989 Joint Summer Research Conference on Continua and Dynamics. 23 N I I U BORDER-COLLISION BIFURCATIONS INCLUDING I FOR ''PERIOD TWO TO PERIOD THREE' PIECEWISE SMOOTH SYSTEMS by Helena E. Nussealb James A. Yorkeac I 3 I December 1990 I I U Research in part supported by the Department of Energy (Scientific Computing Staff Office of Energy Research), and by DARPA/ONR. I a. University of Maryland, Institute for Physical Science and Technology, College Park, MD 20742, U.S.A. b. Rijksuniversiteit Groningen, F.E.W., Vakgroep Econometrie, 3 Postbus 800, NL-9700 AV Groningen, The Netherlands c. University of Marylanc., Department of Mathematics, College Park, MD 20742, U.S.A. ABSTRACT. We examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter p. In the simplest case there is a surface F in phase space along which the map has no derivative (or has two one-sided derivatives). r is the border of two regions in which the map is smooth. As the parameter p is varied, a fixed point E may collide with the border F, and we may assume that this collision occurs at p = 0. A variety of bifurcations occur frequently in such situations, but never or almost never occur in smooth systems. In particular E may cross the border and so will exists for p < 0 and for p > 0 but may be a saddle in one case, say p < 0, and may be a repellor for p , Q, For p < 0 there can be a stable period two orbit which shrinks to the point E0 as p - 0, and for p > 0 there may be a stable period 3 orbit which similarly shrinks to E0 as M -+ 0. Hence one observes the following stable periodic orbits: a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit. '.e also see analogously "stable period 2 to stable period p orhtr bifurcations", with p = 5, 11, 52, or period 2 to quasi-perlo,:u or even to a chaotic attractor. We believe this phenomenon ili: bp seen in many applications. 1. INTRODUCTION Certain bifurcation phenomena have been reported repeatedly in numerous studies of low dimensional dynamical systems, that depend on one parameter. The rather familiar bifurcation phenomena describing the evolution of attractors as a parameter is varied include the saddle node bifurcation, the period doubling (or halving) bifurcation, and the Hopf bifurcation. In the literature 1 dealing with bifurcation theory, it is frequently assumed that the map corresponding to the dynamical System is differentiable; see I for example [GH], [K], [RI, or (S]. To remind the reader so that we may draw contrasts, the well known bifurcation diagram of the quadratic map Q (x)= p- x is given in Figure 1 (1 < p < 1.5). All the computer assisted pictures were made by using the DYNAMICS program [Y]. FIGURE 1 We say a map is smooth if the map has a continuous derivative. A region is a closed, connected subset in phase space. We examine I continuous maps which are piecewise smooth. We restrict attention to those which are smooth on two regions of the plane with the border between these regions being a smooth curve. From now on we assume that there is a smooth curve r which separates the plane into two regions denoted by R A and R phase space R . We say, a map F from th- to itself is piecewise-smooth if (1) F is continuous, and (2) F is smooth on both the regions R\ and R.. Note that on the border F between the regions, the mappings musr is assumed to be continuous. A special case tnat be equal since F we shall refer to frequently is the following prototype example, piecewise linear map into which other generic piecewise linear maps in the plane can be transformed by changes in coordinates. Let u and w be vectors in the plane. Let x and y be the ph~tse space coordinates and p is a scalar parameter. Let P be the map defined by 3 P (x,y) xu + JxJw + (y + p)(l,O) and we investigate trajectories (Xn+lYn+I : P (xnYn) ' The regions R and R are the left and right half plane separated by 3 A B 2 t Figure 1. Bifurcation diagram of the quadratic map Q (x) - x. The parameter p (plotted horizontally) varies from 1 to 1.5, and x is plotted vertically, -1 x . 2. F, the Y-axis. To illustrate the "period two to period three" border-collision bifurcation phenomenon, consider tle one-parameter family of maps fP (-m < P < I ) from the plane to itself, defined by f (x,y) = (-l.4x + y, -O.1x) + M(1,0) if x & 0 (-3x + y, -4x) + p(1,0) if x z 0 Notice that the map is smooth in each of the half planes x - 0 and x ; 0, and the Y-axis is the border which is a smooth curve. Note that to write f in the form of P, let u = (-2.2,-2.05), and w (-0.8,-1.951. The bifurcation diagram exhibiting the 'period two to period three" bifurcation, is presented in Figure 2 (-0.1 < P < 0.2). All the bifurcation diagrams in this paper show a projection of the attractor, projecting (x,y) onto the X-axis, which is II plotted vertically; the horizontal coordinate is p. FIGURE 2 The purpose of this paper is to study the occirrence of suc.h a I 3 new bifurcation phenomenon for continuous, piecewise smooth maps. These systems include, for example, two-dimensional continuous, piecewise-linear maps. In (HNS] the dynamics of a simple economic model was studied, and a "period three to period two" bifurcation was observed numerically, and was established rigorously in (HNJ for a degenerate piecewise-linear situation. The "border-collision bifurcation" phenomena is a much richer class of bifurcation phenomena than just a "period two to period three" bifurcation and occur for generic piecewise smooth maps. We present phenomena that occurs when the nature of an unstable fixed point of a piecewise smooth map is changed while the fixed point collides with the 1 3 Figure 2. Bifurcation diagram exhibiting the "period two to period three" bifurcation of the map f P(x,y) = (-1.4x + y + p, -O.Ix) if x :50, and = (-3x + y + p, -4x) if x > 0. The parameter p (plotted horizontally) varies from -0.1 to 0.2, and the coordinate x is plotted vertically, -1 4 x -, 1. 3 border between two regions in which the map is smooth. Since the fixed point is unstable before and after collision, it is not 5 5 shown in the bifurcation diagram in Figure 2. While we consider maps in the plane, higher dimensional analogues exist. We know of no phenomena that can occur only in higher dimensional cases. There is also no difficulty in changing the notation to that there are more than 2 regions on which the map is smooth. We could also 3 allow f to depend on p, but coordinates could be chosen so that it remains fixed, so our case in practice includes moving boundaries. With moving boundaries the map would be piecewise smooth in p. We say, a fixed point E I I 3 is a border crossing fixed)Point if it crosses the border r between two regions in which the map is smooth. We will assume that the crossing occurs at p = 0. The fixed point E called a flip saddle if tne eigenvalues X and P of the is Jacobian matrix DF (E) if X < -1 < u < 1. Assume that there exists a one-parameter family of piecewise smooth maps and assume that there is a border crossing fixed point (or periodic point) I 3 3 Ep the case when E , we emphasize crosses the border r it changes from being a flip saddle to a repellor with complex eigenvalues. The above example has this behavior. In Section 2, we discuss why the border-collision bifurcation phenomenon occurs when the nature of an unstable equilibrium I I 3 3 changes when it crosses the border of two regions. To be somewhat more specific, assume that a border crossing fixed point (or periodic point) E of a one-parameter family of piecewise smooth maps changes from being a flip saddle to a repellor with complex eigenvalues when it crosses the border r. Then at p = 0, a bordercollision-bifurcation occurs at t 'is fixed point E 4 3 on the border. In Section 3, we mainly deal with two piecewise smooth systems of the plane, one piecewise linear and one piecewise nonlinear. The first system is the map P (derived in Section 2) that correspon-ls with a generic piecewise smooth nonlinear map, and the other system is based on the Henon map. For the piecewise linear map P we present several examples including "period 2 to period p" (p = 5, 11, and 52), "period 2 to quasi-periodic" and "period 2 to chaotic" bifurcation. We also present an example of a bordercollision bifurcation for the map P in which no attractors but chaotic saddles are involved. The system of the plane involving the Henon map at the left side and a linear map at the right side of the border, different border-collision bifurcations are observed. We present a variety of examples. Although we we do not have an exhaustive list of types of border-collision bifurcation of one-parameter families of maps under consideration, we point out that several other types of bifurcation occur. We believe this phenomenon will be seen in many applications. In Section 4 we prove that for certain one-parameter fami.l.-s of piecewise smooth maps exhibit a "period 2 to period 3" bortercollision bifurcation. This phenomenon persists under small perturbations of the involved maps. In Section 5, we discuss the state of the art, and pose several questions which remain unresolved. This paper does not give a complete theory, but can be considered as initiating a bifurcation theory of piecewise smooth maps. 5 2 THE BORDER-COLLISION BIFURCATION PHENOMENON In the bifurcation theory for maps, attention is focused on differentiable maps when one or more eigenvalues of a fixed point (or periodic point) cross the unit circle. When this occurs, the 3I nature of the fixed point changes. For example, a fixed point attractor becomes a saddle (possibly a flip saddle) or a repellor. For border crossing fixed points, the Jacobian matrix of the fixed point generally changes discontinuously, and the fixed point can for example change from being a repellor to a saddle as p crosses zero. Let F(.,P) F be a one-parameter family of piecewise smooth maps from the phase space R2 to itself, depending smoothly on the I parameter p, and where p varies in a certain interval on the real line. Let E 3 denote a fixed point of F defined on -c < p < c, for some e > 0. For a general approach (which is given below) we need the concept of the "orbit index" of a periodic orbit [MYI. The orbit index is a number associated with a periodic orbit, arvi 'his number is useful in understanding the patterns of bifurcations the orbit undergoes. We say an orbit of period p is typical if its 3 Jacobian matrix exists (that is, the Jacobian matrix of the p-th iterate of the map at a point of the orbit) and neither +1 nor -l is an eigenvalue (of the Jacobian matrix). For typical orbits, the I orbit index is -1, 0, or +1. The orbit index is a bifurcation invariant in the sense that if one examines the periodic orbits 3 that collapse to the fixed point E as p -. 0, and adds the orbit indexes of the periodic orbits that exist just before a bifurcation, then that sum equals the corresponding sum just after I that bifurcation. Suppose a typical periodic orbit PO of a map F 1 6 has (minimum) period p. The orbit index of that orbit depends on the eigenvalues of the Jacobian matrix A of the map Fp at one of the points in PO. Now we Oefine the orbit index I of PO. Let m be the number of real eigenvalues of Ap smaller than -1, and let n be the number of real eigenvalues of A p greater than +1. The orbit index IPO of PO is defined by IPO = 0 if m is odd; IPO = -1 if m is even and n is odd; IPO = +1 if both m and n are even. If the orbit index = -1, then the orbit is called a regular saddle. The typical orbits with orbit index +1 in the plane are repellors and attractors and fixed points with non-real eigenvalues. The def'nition of orbit index is technical when a p;.int of the orbit lies on the boundary and so does not have a Jacobian matrix, and the definition is unnecessary since we consider orbits for p # 0. For a moment, assume that E R is in the interior of the region (or the region RB), and write X and u for the eigenvalues of DF (E). If neither of the two eigenvalues X and v is on the unit circle, then the fixed point E index 0) if X < -1 < v < 1; -1) if -1 < L < I < X; E lXi HXJ < 1 and Hol is a regular saddle (and has index is a repellor (and has index +1) if both > I and juj > 1; and E both E is called a flip saddle (and has is an attractor (and has index +1) if IAI < 1. Note that E has orbit index +1 if the eigenvalues are not real. Hence, a typical fixed point is a flip saddle, a regular saddle, a repellor or an attractor. Similarly, the nature of periodic points is defined. Now we are able to provide a definition of the notion "border- 7 J collision bifurcation". Let the regions RA and RB, the map F the fixed point (periodic point) E and be as above. Assume there exists a number c > 0 such that (1) E0 iz cn the border of the two regions RA and RB, (2) for -E < p < 0 the fixed point E is in the region RA , and its index is IA , and (3) for 0 < p < e the fixed point E is in the region RB , and its index is B- If A and IB are different, then (as stated below) some bifurcation must occur at E0 , since the orbit index of E is changing from IA to IB 1 while the parameter p is increasing from -e to +c. We say a periodic orbit PO is an isolated border crossing orbit if (1) PO includes a point that is a border crossing fixed point under some iterate of the map, and (2) the orbit PO is isolated in phase space when p = 0, that is, in the plane there exist neighborhood U of the orbit PO such that PO is the only periodic orbit in U when p = 0. From the topological degree theory as described in (MY] (see also [AYYI for the two dimensional case), the following "Border-Collision Bifurcation" result follows after some minor modifications. BORDER-COLLISION BIFURCATION THEOREM. For each two-dimensional piecewise smooth map and depending smoothly on a parameter p, if the index of an isolated border crossing orbit changes as M crosses 0, then at p = 0 a bifurcation occurs at this point, a bifurcation involving at least one additional 'periodic orbit. This result says that additional fixed points or periodic points must bifurcate from E0 at g = 0. These bifurcating orbit6 need not to be stable. An example of the preservation of orbit 8 index occurs with a period doubling bifurcation. If for p < 0 there is an attracting fixed point (and no other entering orbits), the total index is +1. Then for p > 0 we can have a flip saddle (orbit index 0) and a period 2 attractor (orbit index +1). Hence, the sum of the orbit indices before and after P = 0 is + 1. Note that the two points of the period 2 orbit are collectively assigned +1, not individually, since that orbit has index +I. Since this bifurcation occurs while the fixed point (or periodic point) collides with the border of the regions RA and RB1 we call it a border-collision bifurcation. In other words, a bordercollision bifurcation is a bifurcation at a fixed point (or periodic point) on the border of two regions when the orbit index of the fixed point (or periodic point) Lefore the collision with the border is different from the orbit index of the fixed point after the collision. We derive the map P that was introduced in Section 1, froin nonlinear piecewise smooth maps. We assume coordinates are chosen so that the curve r is a straight line. Let z denote any vector in the plane, and write F (z) = F(z;u), and write z0 = E04 From the is piecewise smooth, we have that on each of the assumption F regions RA and RB F(z;p) = F(zo;0) + DZF(z 0 ,O)(z-z0 ) + D F(z0,O)p + H.O.T. where H.O.T. stands for Higher Order Terms. Hence, there exist matrices MA and MB and vectors vA and v B such that if z is in the region RA then F(z;p) = F(z0 ;0) + MA(z-z 0 ) + vA M + H.O.T. and if z is in the region RB then = F(z 0 ;0) + MB(Z-z 0 ) + VBP + H.O.T. F(z;i) Let e I be the unit vector tangent to F at z0 . The assumption F is piecewise smooth and depends smoothly on M implies MAeI = MBe 1 = e 2 and vA = VB = v. Choose coordinates so that z 0 = 0, so F(z0 ,0) = 0. Assume that e2 is independent of el, so we may use e l and e2 as basis vectors. We let e1 and e2 be the basis vectors of the plane. We assume that e2 is independent of v and that v is not parallel with e I-e2 We claim that by change of variables and by . rescaling M we may assume that v = e . Write v = (VxVy). We now assume that v * 0. We can make vy = 0 after a change of vaIls andv x variables, an y. rescaling of p. If v is not 0 then *e 1 =Iby can change variables, setting y = y - VyM (where x is unchanged), and the new vector v for the (x,y) system will have its second coordinate 0. By rescaling p, that is, by introducing p can change the system so that the new vector v is (1,0), , we may write MA . the parameter. Therefore, pv x we when o is [c and v = (1,0). Since all these assumptions are generic, we sa" tnh prototype piecewise linear form of F F(z;p) F(z;p) =+ U [ b 0 + p(1,0) for p small is defined ov if z is in the region RAs p(1,0) if z is in the region RB To write the prototype piecewise linear form of F II. of the map P , let u 2 b+d and w in the form a c b-d We observe the following fact. Assume that the fixed point E a flip saddle (orbit index 0) in region RA and a repellor with complex eigenvalues (orbit index +1) in region RB. If there ex.ists 10 s a stable periodic orbit with period 2 in RA that converges to E0 when u approaches 0, then the total degree in RA is +1. Hence, if there exists a stable periodic orbit in RB that converges tco E 0 when P goes to 0, then there must exist a regular saddle periodic orbit of the same period (orbit index -1) in RB that converges to E0 when p goes to 0, since the total orbit index is a bifurcation invariant. Consequently, for the family of maps f in the Section 1 exhibiting a "period two to period three" bifurcation in figure 2, there must also exist a regular saddle periodic orbit with period 3. PERIOD TWO TO PERIOD THREE BORDER-COLLISION BIFURCATION THEOREM. Let F be a one-parameter family of liecewise smooth ma-i which has a prototype piecewise linear form at M = 0, and assume that (1) a < -1, c < -1, d < -1; (2) c 2 + 4d < 0; and (3) 0 < a(ac + d) < 1. Then, there exists c > 0 such that if IbI < e, then the family F has a "period two to period three" border-collision bifurca~ton at (0,0). We point out that the border-collision bifurcations persist under small perturbations. The proof follows of the Theorem from the result obtained in Section 4. The geometrical proof given in Section 4, might give insight why other bifurcations (for example, period 5 to period 2 bifurcation) may occur in piecewise smooth systems. Presumably, the method of proof only works if one of the two maps involved has a small Jacobian. Hence, when the piecewise smooth map consists of maps that all have Jacobian bounded (far) away from zero, new techniques have to be developed to obtain 11 I I rigorous border-collision bifurcation results. 3. A VARIETY OF BORDER-COLLISION BIFURCATIONS. In this Section we present a variety of numerical examples exhibiting a border-collision bifurcation. The first series of examples is from the piecewise linear map and the second series is based on the H~non map. We will present examples showing I that in a border-collision bifurcation not only attracting periodic orbits are involved, but also chaotic saddles may play a role. Therefore, in order to describe the qualitatively different border-collision bifurcations in a consistent manner, we refer to the invariant sets that are involved in the border-collision I bifurcation. A chaotic saddle is a compact, invariant set that is not an attractor which contains a chaotic trajectory. If an attractor A of a map F is an attracting periodic orbit with period p, then we call A a period D attractor, and we say instead of "period two to period three" bifurcation a bifurcation from a I * period 2 attractor to a period 3 attractor. The bifurcation diagrams below show the long term behavior of the coordinate x for p between -0.1 and 0.2. The diagrams have been constructed as follows. For the minimum value -0.1 of p, and initial value (0,0), calculate the first 200 points (transient I I time 200) of the orbit and plot the next 1000 points of the orbit. Increase M slightly, say by 0.001, take for the initial value the last point which was plotted, calculate 200 points of this orbit and plot the next 1000 points. Increase p again, and continue increasing until p achieves the maximum value 0.2. Hence, once the I I12 orbit is close to an attractor, as the parameter is increased, this attractor is "followed" as long as it exists. In the diagrams, the x-coordinate is plotted vertically, and the parameter p is plotted horizontally. Define the map GL T-d2 from the plane to itself to be the prototype piecewise linear form of F , that is, GL (x,y) = (ax + y, bx) + p(1,0) if x S 0 GL (x,y) = (cx + y, dx) + p(1,0) if x 2 0 Rezall that the map GLP is equivalent to the map P , since to a+c Ib+d , -u = write the map GL in the form of the map P , let P and w . We present a few numerical examples for this a-c Ib-d-- map GL exhibiting a border-collision bifurcation. In all these examples, the fixed point is a flip saddle for p < 0 and a repellor with complex eigenvalues for p > 0. EXAMPLE 1. The presumably simplest border-collision bifurcation is from a period 2 attractor to a period 3 attractor presented in Figure 1. We present parameter values for which the map GL shows a bifurcation from a period 2 attractor to a period p attractor for a variety of period p. For a = -1.25, b = -.035, c = -2, d = -1.75, the bifurcation diagram in Figure 3a exhibits a bifurcation from a period 2 attra:tor to a period 5 attractor. For a = -1.25, b = -0.0435, c = -2, d = -2.175, the bifurcation diagram in Figure 3b exhibits a bifurcation from a period 2 attractor to a period 11 attractor. For a = -1.25, b = -0.03943, c =-2, d = -1.9715, the bifurcation diagram in Figure 3c exhibits a bifurcation from a period 2 attractor to a period 52 attractor. 13 II 'I• • • • • • • • •w• • m••m Figur"e 3a. II Bifurcation diagram of GL Ij(x,y) =(-1.25x + y + Mj,-0.035x) if x Ii I = (-2x exhibits at p. Ill + y + Ml, -1.75x) = if 0, and x a 0 0 a border-collision bifurcation from a period 2 attractor to a period 5 attractor. The parameter M (plotted horizontally) varies from -0.1 to 0.2, and the coordinate x is ploted vertcly -0. O5x_03 N/ N/ N/ N/ /N N Figure 3b. Bifurcation diagram of GL 11(x,y) =(-1.25x + y + p1,-0.0435x) if x .50, and =(-2x + y + 11, -2.175x) if x a 0 exhibits at p0 0 a border-collision bifurcat'ion from a period 2 attractor to a period 11 attractor. The parameter p (plotted horizontally) varies from -0.1 to 0.2, and the coordinate x is plotted vertically, -0.3 _ x ; 0.3. Figure 3c. GLBifurcation diagram of ,y)(~.2~+ y + It -0,03943x) (-x+ y + M, -1-9715x) eXhibits at if p0 =0 a if x ,and a 0 attactr t aPerod border..lli. 2 atrcor n bifurcation 1 a tra tot y o a pr from a Period i s fo 52 a t c o The Parame ter hPlo td verti a ries fr g (Plotted _ 0.1 to 0.2) and the coordinate s P ed l tv r ic l y -0.3 'S x Is.3 0.3 For other choices for a, b, c, and d we have found bifurcations from a period 2 attractor to a period p attractor, where p = 6, 7, 8, 9, 10, 11, 13, 19, 21, 23, 29, 31, 37, 41, etc. EXAMPLE 2. The simplest border-collision bifurcation in which chaotic attractors are involved is presumably the bifurcation from a period 2 attractor to a (1-piece) chaotic attractor. Frequently, the border-collision bifurcation from a period 2 attractor to a p-piece chaotic attractor is observed. For a = -1.25, b = -0.042, c = -2, and d = -2.1, the bifurcation diagram in Figure 4a exhibits a bifurcation from a period 2 attractor to a 1-piece chaotic attractor. For a = -1.36, b = -0.12, c = -2, and d = -2, the kifurcation diagram in Figure 4b seems to exhibit a bifurcation from a period 2 attractor to a 12-piece chaotic attractor, but using the phase space it turns out that the bifurcation is from a period 2 attractor to a 18-piece chaotic attractor. We have observed many other values of p, the map GL shows a bifurcation from period 2 attractor to p-piece chaotic attractor. For the selection a = -1.25, b = -0.03865, c = -2, and d = -1.9325, we obtain a bifurcation diagram similar to figure 4a, but in this case the border-collision bifurcation is from a period 2 attractor to a what appears to be quasi-periodic attractor. EXAMPLE 3. A border-collision bifurcation in which chaotic saddles (rather than attractors) are involved, will not be exhibited by bifurcation diagrams. Therefore, some other numerical method is needed to detect these sets. We use the Saddle Straddle I I I I I I 4a. I I a t I c o t o° a - i.°c h o i a t a c o . T h a r m t r p l t e Figure 4a. I I I r m Bifurcation diagram of GL (x~y) =(,-l.25x + y + p, -0.042x) if x 0, and (-2x + y + p, -2.1x) if x a 0 exhibits at 0:0 a border-collision bifurcation from a period 2 attractor to a 1-piece chaotic attractor. The parameter Ns (plotted horizontally) varies from -0.1 to 0.2; the coordinate x is plotted vertically, -0.3 < x < 0.3. .Iwmm w ww mmm~ mw~ wm nm m .~m w, w um mwmwm ~m m Figure 4b. Bifurcation diagram of GL 11(x,y) (-1.36x + y + g, -0.12x) if' x s,0, and y + M, -2x) if x a 0 exhibits at go=0 a border-collision b-ifurcation from a period2 attractor to a 18-piece chaotic attractor. The parameter u (plotted horizontally) varies from -0.1 to 0.2; the coordinate -(-2x + is plotted vertically, -0.3 -e X :5 0.3. Trajectory (SST) method introduced in [NY] to detect such sets. We select a = -1.25, b = 0.18, c = 2, and d = -3. For p = -0.05 the invariant set (obtained by the SST method) is presented in Figure 5a, and the invariant set for p = 0.05 is in Figure 5b. Presumably, it is correct to say that the border-collision bifurcation is a bifurcation from a chaotic saddle to another chaotic saddle. Now we present a few examples based on the Henon map. In fact, in these examples we have a moving border. Define the map H from the plane to itself by H(x,y) and define the map L (-vs (A - x + By, x) < p< m) from the plane to itself by L (x,y) = (A + Cx + By - (p+C)p, Dx + (1-D)p) The regions R A and R B are the half planes to the left and the right of the straight line x = p. The map we are investigatin-, defined being the Henon map on RA and the "linear" map L Define the one-parameter family of maps F on from the plane to itself by if x FH(x,y) (<yP I L P(x,y) if x a P Notice that the map is smooth in each of the half planes x a p and x a p, and the line x = p is the border which is a smooth curve. Since the map F is continuous, it is a piecewise smooth map. Nol, that for this family F border-collision bifurcations occur presumably for values p0 different from zero. 15 rho -0.0500000000 I~ 'I 'I' Figure 5a. Chaotic saddle of GL (x,y) (-1.25x + y + p, O.18x) if x s 0, and -0.05. (2x + y + p, -3x) if x a 0 when p and the The coordinate x (-0.2 . x < 0.1) is plotted horizontally, coordinate y ( -0.25 < y - 0.02) is plotted vertically. I rho 0.0500000000 I •~A ,4 ,.. / I I AV 7! - I I I 3 I 3 Figure 5b. Chaotic saddle of GL (x,y) (-1.25x + y + 11, 0.18x) if x 0, and = 0.05. (2x + y + M, -3x) if x a 0 when P horizontally, and the The coordinate x (-1 5 x - 0.6) is plotted coordinate y ( -1.8 . y : 0.2) is plotted vertically. EXAMPLE 4. Simple border-collision bifurcations are bifurcations from a period p attractor to a period q attractor. For A = 1.4, B = 0.3, C = 0.9, and D = -5, the 5ifurcation diagram in Figure 6a exhibits a bifurcation from a period 3 attractor to a period 4 attractor, where p (plotted horizontally) varies from 0.89 to 0.87. In the region RA the fixed point is a flip saddle and in the region RB the fixed point is a repellor. The border-collision bifurcation occurs at g = go s 0.884. For p > go (the side of the period 3 attractor which has orbit index +1) the fixed point is a flip saddle (orbit index 0) and we find no other periodic orbits on this side of the bifurcation. For g ' 0 (the side of the period 4 attractor which has orbit index +1) the fixed point is a repellor (orbit index +1); there also exists a period 4 regular saddle (orbit index -1). The regular saddle also shrinks to a point (the fixed point) as g -+ 0o. Hence, the orbit index is +1 on both sides of g0. For A = 1.4, B = 0.3, C = 1, and D = -5, the bifurcation diagram in Figure 6b exhibits a bifurcation from a period 6 attractor to a period 4 attractor, where g (plotted horizontally) varies from 1.05 to 0.8. In the figure one might first notice a bifurcation from a 6-piece chaotic attractor to a period 4 attractor, but closer examination gives the above mentioned bifurcation from a period 6 attractor to a period 4 attractor. Similarly as above, the periodic orbits involved in the border-collision bifurcation that occurs at g = g 0 0.884 are the following. For M > g0 there is period 6 attractor and the fixed point is a flip saddle, and for p < g0 the fixed point is a repellor and there is a period 4 attractor a period 4 regular 16 I.% Figure 6a. Bifurcation diagram of F (x,y) = (1.4 - x 2 + 0.3y, x) if x & , and = (1.4 + 0.9x + 0.3y - (g+0.9)p, -5x + 6p) if x a p, 0.884 a border-collision bifurcation from a period 3 attractor to a period 4 attractor. The parameter m (plotted horizontally) varies from 0.89 to 0.87; the coordinate x exhibits at 0 is plotted vertically, 0.6 x : 1.2. / d Figure 6b. Bifurcation diagram of (1.4 - x 2 + 0.3y, F (x,y) (1.4 + x + 0.3y - x) if x s P, and (p+l)p, -5x + 6g) if x a g, bifurcation from a exhibits at g0 A 0.884 a border-collision The parameter g period 6 attractor to a period 4 attractor. to 0.8; the coordinate "z (plotted horizontally) varies from 1.05 is plotted vertically, -0.5 S x s 2. saddle. Hence, the orbit index is +1 on both sides of M0 . EXAMPLE 5. In this example we present two cases of a border-collision bifurcation from a period p attractor to a q-piece chaotic attractor. For A = 1.4, B = 0.3, C = 1.1, and D = -5, the bifurcation diagram in Figure 7a exhibits a bifurcation from a 1-piece chaotic attractor to a period 4 attractor, where p (plotted horizontally) varies from 1.05 to 0.8. The border-collision bifurcation occurs at g= go A 0.885. For g > g0 (the side with the chaotic attractor) we do not know the (total) orbit index since the chaotic attractor contains a lot of periodic orbits. For g > M0 the fixed point is a flip saddle (orbit index 0). For p < g0 (the side of the period 4 attractor which has orbit index +1) the fixed point is a repellor (orbit index +1) there also exists a period 4 regular saddle (orbit index -1). point as p -+ p. The regular saddle also shrinks to the fi.ei Hence, presumably we have a border-collisiou bifurcation from a period 4 attractor to a 1-piece chaotic attractor. For A = 1.4, B = 0.3, C = 1.5, and D = -4, the bifurcation diagram in Figure 7b exhibits a bifurcation from a 8-piece chatLc attractor to a period 5 attractor, where g (plotted horizontally) varies from 0.91 to 0.86. The border-collision bifurcation occurs at p = g ss 0.884. For g > go (the side of the 8-piece chaotic attractor) we do not know the (total) orbit index since the chaotic attractor contains a lot of periodic orbits, and the fixed point is a flip saddle (orbit index 0). For g < g0 (the side of the period 5 attractor which has orbit index +1) the fixed point 17 Figure 7a. Bifurcation diagram of F (x,y) = (1.4 - x 2 + 0.3y, x) = (1.4 + 1.Ix + 0. 3 y - if x j, and (M+I.1)M, -5x + 6p) if ; p, exhibits at g0 s 0.885 a border-collision bifurcation from a 1-piece chaotic attractor to a period 4 attractor. The parameter g (plotted horizontally) varies from 1.05 to 0.8; the coordinate x is plotted vertically, -0.5 x . 2. I ...... ... ............. ........ Fiue7. daga fIdi Iiucto ,2"()3,-info =ki 4 I .c4 3 Iu F pyk . c rh o i k%~1Otted horizO I ~ otted p ee cta odr cl i i nt ra tr h aIj t va ie nta , 2Y 2-i il ve2. l from ~~ ~ J I I I I F ( - +,I I I I I Figure 8.1 Bifurcation diagram of + 0.3y, IF(x,y) = (1.4 - x x) if x H: and , + 1.2x + 0.3y - (p+1.2)p, -4x + 5p) if x bifurcation from a exhibits at go % 0.884 a border-collision chaotic attractor. The 1-piece chaotic attractor to a 1-piece varies from 0.95 to 0.85; the parameter g (plotted horizontally) 1.6. 0.4 s x coordinate x is plotted vertically, (1.4 u n~ um nm nn um 'm m unw ~ mnm i~mnmmnnunum uwm m um wl~ I is a repellor (orbit index +1); regular saddle (orbit index -1). 3 Ito there also exists a period 5 The regular saddle also shrinks the fixed point as u -+go In the figure one might first notice a bifurcation from a 5-piece chaotic attractor to a period 5 attractor, but closer examination in the phase space gives the above mentioned bifurcation from a 8-piece chaotic attractor to a period 5 attractor. Hence, presumably we have a border-collision 3 bifurcation from a period 5 attractor to a 8-piece chaotic attractor. IEXAMPLE 6. Border-collision bifurcation from a p-plece chaoti attractor to a q-piece chaotic attractor. We present just one 3 example, namely p = a = 1. For A = 1.4, B = 0.3, C = 1.2, and D = -4, the bifurcation diagram in Figure 8 exhibits a bifurcation from a 1-piece chaotic attractor to a 1-piece chaotic attractor, where pa (plotted horizontally) varies from 0.95 to 0.85. The border-collision bifurcation occurs at y = go s 0.884 and we only can say that rk both sides infinitely many periodic orbits are involved in the border-collision bifurcation, since the attractors are chaotic. Hence, presumably we have a border-collision bifurcation from 1 1-piece chaotic attractor to a 1-piece chaotic attractor. EXAMPLE 7. In this example we show that coexisting attractors of different nature can be involved on the same side of a border-collision bifurcation. For A = 1.4, B = 0.3, C = 1.4, and D = -4, the bifurcation diagram in Figure 9a exhibits a bifurcation from a 5-piece chaot v 18 Figure a Bifurcation diagram of F11(x,y) =(1.4 x2+ 0.3y, x) if x I~ u, and =(1.4 + 1.4x + 0.3y - (p'+l. 4 )p, -4x + 5pj) if x a exhibits at ju 0 s~ 0.884 a border-coll~ision bifurcation from a 5 -piece chaotic attractor to a 1-piece chaotic attractor. The parameter p~ (plotted horizontally) varies from 0.87 to 0.895; the coordinate x is plotted vertically, 0.3 :5 x :5 1.6. - I I I I I I diagra Bifurcatio of I 2 + 0.3y, : 11.4 - ,. x) if x - , and (11. 4 + I.4x + 0. 3y - (1 +. -1)M, -4x + 51j) i f x a exhibits at go ; 0.884 a border-collision bifurcation from a F9(x,y) I 5-piece chaotic attT.,actor to a period 4 attractor. The parameter P (plotted horizontally) varies from 0.874 to 0.895; the coordinate x is plotted vertically, 0.3 -s x e, 1.6. attr-actor to a 1-piece chaotic attractor, where p (plotted horizontally) varies from 0.87 to 0.895. On both sides of the collision-bifurcation. which occurs at p0 s 0.884, there are infinitely many unstable periodic orbits involved, since the attractors are chaotic. Due to the projection of the picture onto one phase space coordinate the bifurcation diagram seems to show a 2-piece chaotic attractor, but again in phase space one has clearly a 5-piece chaotic attractor. For the same parameter values, the bifurcation diagram in Figure 9b exhibits a bifurcation from a 5-piece chaotic attractor to a period 4 attractor, where p (plotted horizontally) varies from 0.874 to 0.895. Hence, we may have a border-collision bifurcation from a 5-piece chaotic attractor to a coexisting 1-piece chaotic attractor and a period 4 attractor. EXAMPLE 8. Now we consider an example in which the rurve fr is the straight line y = -x + p. In this example we have a moving border. Let the map H from the plane to itself be defined as above, that is, H(x,y) = (A - x2 + By, x), and define the map G (- < p< ) from the plane to itself by G (x,y) = (A - PC - x 2 + Cx + (B+C)y, (B+D)x - Dy -pD) The regions RA and RB are the half planes to the left and the right of the curve F . The map we are investigating is defined being the Henon map on R and the "linear" map G the one-parameter family of maps F Define from the plane to itself by ( = H(x,y if x S -y + P G (xy) if x a -y + 19 on R Uf Ii I~mN lI NI INl l l ml Figure 10a. Bifurcation diagram of (114 - x 2 + 0.3y, F l(x,y) = (1.4 x a -y + p, x) if x -y + 0.5p - x 2 + 0.5.x + 0.2y, exhibits at p0 s + p, ana -1.3x + Y + .)if 1.015 a border-collision bifurcation from a period 4 attractor to a strange chaotic attractor. The parameter p (plotted horizontally) varies from 1.2 to 1; the coordinate x is plotted vertically, -2 s x s 2. Figure 10b. F The chaotic strange attractor _ 2 (. Ixy 4 0.3y, x) if -x+ (1.4 + 0. 5 11 - x + 0.5x x S -y + 0. 2 + 1p, and y, -1.3x t. y + p) x ;-I -Y + m, where p if =1. The coordinate x (-2 s x s 2) is plotted horizontally, and the coordinate y ( -2 zs y :5 2) is plotted vertically. - 3 Notice that the map F is a piecewise smooth map. We present an example for which the map F 5 5 has a the border-collision bifurcation from a period 4 attractor to a chaotic strange attractor. For A = 1.4, B = -0.3, C = 0.5, and D = -1, the bifurcation diagram in Figure 10a exhibits a bifurcation from a period 4 attractor to a chaotic strange attractor, where p (plotted horizontally) varies from 1.2 to 1. The border-collision I 3 3 1 bifurcation occurs at p = po 0 1.015. The chaotic strange attractor for p = 1 is given in Figure lOb. Hence, we may have a border-collision bifurcation from a period 4 attractor to a chaotic strange attractor. 4. "PERIOD TWO TO PERIOD THREE" BORDER-COLLISION BIFURCATION In this Section we explain why "period two to period three" 3 I border-collision bifurcations occur for two-dimensional piecewise smooth maps. Let a, b, c, and d denote real numbers. Define the one-parameter family GL '4 GL (x,y) = from the plane to itself, by (ax + y, GL (x,y) = (cx + y, + p(1,0) if x ! 0 dx) +P(1,0) if x a 0 bx) where p is in an open interval I including zero. Recall that this family GL 3 Let F is equivalent with the piecewise linear map P . be a one-parameter family of piecewise smooth maps which has a prototype piecewise linear form at p = 0, and assume that (Al) 3 3 * -a > 1, -c > 1, -d > 1; (A2) c 2 + 4d < 0; (A3) 0 <a(ac + d) < 1. We want to show that there exists e > 0 such that if IbI < c, then the family F has a "period two to period three" border-collision 20 bifurcation at (0,0). First, we show that for b = 0, the family GL has a border-collision bifurcation from a period 2 attractor to a period 3 attractor. We write C for the set of all one-parameter families of maps GL defined above such that b = 0. PROPOSITION. At M = 0, every family GL in C has a "period two to period three" border-collision bifurcation at (0,0). PROOF OF THE THEOREM. Assume that the Proposition has been proved. Apply the Proposition and it follows immediately from perturbation results. The geometrical proof of the Proposi.ion (given below) -ight give insight why other bifurcations (for example, period 5 to period 2 bifurcation) may occur in piecewise smooth systems. Presumably, the method of proof only works if one of the two maps involved has a zero Jacobian. We first show that a border-collision bifurcation occurs at p = 0, and we present an example to give an idea of the proof. Let GL 1 be in C. The fixed point E 0) if = ( 0 and 1 of F d -'lc--ii l-c-d. is given by E if .i 0. In the notation of Section 2, define the matrices MA and MB by MA [ 1]' MB = [c 11. The eigenvalues of MA are 0 and a, so if p < 0 then the fixed point E particular, E is unstable since -a > 1. In is a flip saddle if M < 0. The eigenvalues of MB are 0.5c ± 0.5 /c + 4d and are complex, since c 2 + 4d < 0. For p > 0 the fixed point E is unstable (repelling), since the M 21 3 product -d of the eigenvalues of MB exceeds 1. The nature of the fixed point E is changing from being a flip saddle (in region RA which is the left half plane) to a ripellor with complex eigenvalues (in region RB) when the parameter p is varied from say -0.1 to 0.1. We conclude that a border-collision bifurcation occurs at 0 when M is continuously varied from some negative j value to a positive value, since the orbit index of E changes from 0 to +1. For simplicity of the explanation of this bordercollision bifurcation phenomenon, we offer the following example. 3 EXAMPLE. Consider the one-parameter family g from the plane to itself, defined by l,,y) = (-4.x + y, 0) + je.(1,0) if x g(x,y) = (-2x + y, ---x) + 1.(1,0) 0 if x z 0 The bifurcation diagram exhibiting the "period two to period Sthree"bifurcation, family of maps g I 3 is similar to the diagram in figure 1. The is in the class C, so it is an example for %,hich the result above applies. The idea why a "period two tc period three" border-collision bifurcation occurs for the family i, s the following. For P < 0, write W for the interval [-.,-.P,) on the o h f- 3 p a p X-axis. We have (1) the image g (p) of each point p on the X-axis but not in W is in W , and (2) each point p in W is mapped to a point p* on the X-axis after two iterates, so g/ (p) = p*. In figure 11, the graph of the corresponding return map G on W whicn is defined by G(x) = g 2(x,O), is given. To be more specific, G(x) 25 16 fa 1 T p for x - 0 and G(x) 22 1 1 for x a 0. Gtz) PS Pa Figure bYg0 i defined The maP g 9 reunmp A G defined by thde on fied POI~ n + PI 2 . 25 r th. o ino val 2 k%~teirtra 5 G has Pi . h maP ~h u s a l mafia h o~P a stable ie thean tal 3 FIGURE 11 4 ±.M < 0 and P The map G has two fixed points Pu 2 -p . > The fixed point pu is unstable since the slope of G in Pu is -5, and the fixed point p5 is stable since the slope of G at p The properties (1) 4" I4 4.I < 3 I . 4 M < 0, (2) G has slope -25 at x for 1 x < 0, (3) G has slope -for 0, imply that g points P1 (-*. I < Pu is x > 0, and (4) 1 = -1.p G(0) > has a period 2 attractor consisting of the two N , 0) and P2 gg(P 1 ) = (-7-P, the norms of both these points converge to zero as p goes to zero, that is, both lHP 211-* 0 as p -+ 0. In other words, 1 11-o 0 and 11P period 2 attractor shrinks to a point as p goes to zero; the this point to which the period 2 attractor converges is the fixed point of g at p = 0. For p > 0, each point p on the X-axis 3is mapped to a point p* on the X-axis after three iterates, so g1 3 (p) p*. The graph of the corresponding return map H, defined by H(x) 3 3 (x,0), is given in figure 12. In particular, H(x) for x < 0, H(x) I-.p 113 =--.x - 3 1 .p for 0 e x L -'p, - 3 and H(x) 1 '21 for x a FIGURE 12 * The map H has an unstable fixed point Pu fixed points qs I 4 -7 p < 14 4.p > 0 and two stable 0 and ps = -- 'p > 0. Furthermore, for ail x with x <pu we have lim Hn(x) = qs, and for all x with x > p, .. p . The properties (1) H has slope between 1).%nd have lim Hn(x) 5465 S 1 for x < 0, (2) H has slope bigger than I for 0 < x H(0) has slope between 0 and 1 for x > 1.p, and (4) .iP () f - imply g has a period 3 attractor 1 89 1 and 14 I4 0), and S 14(--., ( -. P., 0), S2 consisting of the points S= 1 23 Pig~~~12. the ret.. the X- .x and23 d f)z(2 rap 'R dei e d by 'I +( The map q has ~n4two stable t'jxe (-12-X -2 6 5x if aps 3fl (xoo un fsta l e (x L) fixed points 4Iblethe five S P 0 [ap 0 an p P 4' -- sit 0,n 4t r7*9> S19 19, -- 49. -4-) Notice that the norms of all three points converge to zero as p goes to zero, that is, all three 11S11 -+ 0, 11S 211- 0, and U1S 3111 0 as p -* 0. In other words, the period 3 attractor * shrinks to a point as p goes to zero; this point to which the period 3 attractor converges is the fixed point of g at p = 0. 4 The point (-7.p, 0) is a point of a period 3 orbit which is a reguLar saddle of the map g Conclusion: at p = 0, there is a "period two to period three" END OF THE EXAMPLE. border-coilision bifurcation. ! PROOF OF THE PROPOSITION. Let GL 3 3 be a one-parameter family in the class C, where p is in some interval I. We write pO = (xOyo) for an initial condition and p that is, pn = GL n('), . value (0,0), we write A0 = (Xny) for its n-th iterate, each p. For the particular initial (0,0), A = GL(A 0 A2 = GL(A ), A3 GL (A ), and A 4 = GL (A3 ). For each initial value p0 = (X0 ,y0 ) we observe the following I fact. If x 0 0 then yl = 0, and if x 0 > 0 then yl = dx 0 < 0. Hence, it is sufficient to consider initial values in the lower half plane. Hence, from now on, we assume that yo - 0. ( L.P, Assume first, p < 0. Recall that the fixed point E 0) is unstable, and is a flip saddle, since -a > I. Assume that p 0 I = (x0 ,y0 ) is any initial value with y 0 . 0. Then, Yl = 0, and if x 0 > 0, then x 1 = cx 0 + y 0 + p < if x 0 , 0 then 0 and so Y2 = 0. Therefore, it is sufficient to consider points on the X-axis, and 3 we will do so. £ Consider the initial value p 0 = (0,0) = A 0 . Computation of the 24 first four iterates of A 0 yields A 1 = (M,O), A 2 = ((a+l)p, A 3 = ((c(a+l) + i)p, 0), d(a+i)p), and A 4 = ((a+l)(ac + d + 1)p, 0). The assumptions 0 < a(ac + d) < 1 and -a > I imply -1 % ac + d < 0 yielding 0 < x 4 < x 2 . From -1 < ac + d < 0, and the assumptions, -a > 1, and -c > 1 follows that c(a + 1) > 0 and d < c; therefore Ix3 1 > 1y3 1. Hence, A 1 is on the X-axis to the left of A 0 , A 3 is under and to the left of A1 , and both A 2 and A 4 are on the X-axis to the right of A0 and A 4 is between A0 and A 2 . First we consider the image of the X-axis. Let p 0 = (x0 9Y0 ) be any point on the X-axis. The image of the right half of the X-axis with end point A 0 is the half line through A 3 with end point A[ = GL (A0), since p1 = (cx0 + p, dx 0 ) for x 0 > 0. The image of the left half of the X-axis with end point A 0 is the half line on the X-axis to the right of A = with end point A I, since p1 0) for x 0 - 0. (ax0 + p, 2 Define Q = (--P,O) = (xQ)0) and R a ( 0) d)' (I-a)(ac + d - (XR'O). The point Q is mapped to A 0 iterating GL once, that is, GL (Q) A, and Q is on the X-axis between A and E since A P = 0 (p,O), E 1.a 1 , 0) and -a > 1. The point R is on the X-axis to the right of A 0 , and R is mapped to E iterating GL twice, that is, GL 2(R) = E . Let P 0 = (x0 0) be any point. Straightforward computation gives the following. If x 0 > 0 (that is, P 0 is on the X-axis to the (1+a)p, 0), then p1 = ([ac + d]x 0 + so p 2 is on the X-axis. If x 0 = 0 (that is, p0 = right of A 0 ) then p1 = (cx0 + p, dx0 ) and P 2 = (p, 0) and P 2 = ((l+a)p, 0), 1 A0 ) so p. is on the X-axis to the right of A 0 . If ---. 5 x 0 < 0 (that is, p 0 is on the X-axis between Q and A 0 ) then p1 = (ax 0 + p, 0) and P 2 = (a(ax0 + p) + p, 25 I 1 0), so P2 is on the X-axis. If x 0 < -!-A (that is, P0 is to the left of Q) then p1 = (ax0 + p, 0) and p 2 = (c(ax 0 + g) + p, d(ax0 W) and P 3 + = ([a c + adlx 0 + 'ad + a + d + 1).P, 0), and so P 3 is on the X-axis while p 2 is not. Summarizing, for each point P0 2 on the X-axis to the right of Q we have P 2 = GL is on the (p0) X-axis. Therefore, we have a return map on the interval consisting of the points on the X-axis to the right of Q. Let G denote the return map of GL on [Q,cD), so G(x) = GL 2(x,0) for each x z xQ* The above results imply G(x) = a x + (1+a)g for I. " I x < 0, and G(x) = (ac+d)x + (1+a)p for x a 0. The graph of Q. is similar to 1 - and The map G has two fixed points, namely figure 11. a + 1 Ps 1 - ac - d ' and Pu 1 0 < ps. The fixed > 1, and the point pu is unstable since the slope of G in Pu is a fixed point ps is stable since the slope of G at ps is ac + d for which -1 < ac + d < 0. Furthermore, for all x with Pu < x < xR we have iim Gn(x) = Ps. The properties (1) xQ < slope a2 > i f XQ < x < 0, (3) G has slope -1 > 0, and (4) G(0) > 0, imply that GL consisting of the points P1 (a + I)d c - d + I " ) ' 1 - ac - ac - d (1 -L- < 0, (2) G has 0 for: < ac + d has a period 2 attractor aac 1 d ' 0) and P 2 GL(P Notice that the norms of both these points converge to zero as p goes to zero, that is, both 1 II11 -+ 0 and 11P 2 11- 0 as p -* 0. Hence, the period 2 attractor shrinks to a point as p goes to zero; this point to which the period 2 attractor converges is the fixed point of GL at p = 0. Now assume p = 0. Assume p 0 = (x0 ,y 0 ) is any initial value with YO < 0, then x 0 & 0 implies y 1 = 0, and x 0 > 0 implies x I cx 0 yielding y 2 = 0. Hence, it is sufficient to consider points on the X-axis. Let P0 = (x0 '0) be given. If x 0 < 0 then p, = (ax0 ,0) £26 which is on the positive X-axis. If x 0 = 0 then P, = (ax0 '0) and so p 0 is the fixed point of GL = and P 2 ((ac+d)x 0 , 0). If x 0 > 0, then P1 = (cx0 ' dx 0 ), Consequently the point A 0 = (0,0) is a globally stable fixed point of GL0 , since -1 < ac + d < 0. d 1 Now assume M > 0. The fixed point E P -c-d~ J i 1--~ unstable with complex eigenvalues since it was assumed -d > I and c 2 + 4d < 0. Assume p0 = Then x 0 - 0 implies y = p Y1 0 and so y 2 = dx 0 < -y < (c < 0; 0 (x0 ,y0 ) is any initial value with y0 0, and if x 0 a -. If 0 < x 0. hence, < -p then x then x = if x 1 s 0 then Y2 = 1 + P = + Y0 = cx 0 =cx 0 + y 0 + + p and 2 c x 0 + cy 0 + cp + dx 0 +p < P + cx 0 ), X2 - c(M + cx0 ) + cp + dx 0 + P = dx 0 + M < 0, and so y 3 2 0. 0, else if x, > 0 (and so 0 cX 1 + < = 0. Therefore, it is sufficient to consider points on the X-axis. Let pO then p 1 = = (x0 ,y0 ) = GL (P0 q0 = (w0 O0) = (x0 ,O) be any point on the X-axis. If x 0 e 0 (axO + P, 0) = (xlYl), so x1 > 0. Every point such that w 0 < X0 s 0 satisfies q 1 = GLP(qo) = (aw) + P, 0) = (w1 ,zI), so w I > X1 > 0. The conclusion is that poLflLs on the X-axis to the left of A 0 = (0,0) are mapped monotonically, into the X-axis to the right of (p,O). Let P 0 ((1+c)p, = do), (0,0). A simple computation shows p1 P3 = ((ac + a + d + i)p, 0), = A P., = and p 4 = (ax3 + P, 0). Notice x 3 < 0, hence x 4 > P = x1 . Recall that P0 = A 2, P 3 = A 3 , and P 4 = (p, 0), A 0, p, = A,, p2 The conclusion is that A 0 , A1 , A 3 , and A4 are on the X-axis, and A 3 is to the left of A 0 , and both A, and A 4 are to the right of A 0 with A 1 between A 0 and A 4 . Let p0 = (x0 '0) be any point on the X-axis for which x 0 Then p1 = (cxo + P, dxo). Notice that if x 0 = d1B) B d p. W B2 GL (B0) - . , 0), B 1 Write B 0 = ---. Yl > 0. then x 0 and h(B), and and GL (BI) d (0,-i.m), B 2 B3 =GL (B2 ). Then B1 ([a(1--) + 1I], 0). Notice that B at which the line segment [AI,A d ((I-I)M, 0), and B= denotes the point on the Y-axis 21 intersects the Y-axis, and that B 2 is a point on the X-axis to the left of A . The assumptions -a 0 > 1 and 0 < a(ac + d) < 1 imply ac + d < 0 and we obtain that the a point A 3 = ((ac + a + d + l)M, 0) is on the X-axis to the left of B2. The image of the half line (A1 ,a) through A 2 under the map GL is the kinked half line (A2 ,B2 ] U [B2 ,cD) through A The image of this kinked half line is on the X-axis. In particular, the image of the half line (B2 , ) through A 3 is [B3 D) on the X-axis to the I right of A BB31 [A3 ,B 3 ]. (P,0), and the image of the line segment (A2,B2 1A 31 Let p0 2 (x0 ,0) be any point on the X-axis. Straightforward computation shows the following. If x 0 ; - .p (that is, p 0 the right of B 0 ) then p 1 P 2 n d = (cx0 + p, dx , p 2 = ([ac + d]x dx 0 ), P2 = 2 = 0 c *0 + d~x0 + (c+1)4, cdx 0 + dp), (c+1)P < -cp - il P3 3 is on the X-axis between A 0 and B 0 ) then p1 (Cc2 d + c1 c d + P = (l--)p c is on the X-axis. If x 0 < 0 then p 1 (c+l)p, adx 0 + dp), and p3 = = 0 Hence, + both --.p (that (cx0 + u and since (c 2+d)x + 0 < 0, we have + ad + cd]x 0 + [ac + a + d + 1}p, ((ac is to 0 c0 (a+l)p, 0,, and p 3 = (a[ac+dlx 0 + [a(a+l) + 1I.P, 0). P2 and are on the X-axis for x z -. P. If 0 : x is, p0 is = 0), so the point p 3 (ax 0 + P, 0), (atac + dx 0 P 2 = (acx 0 + + (ac + a + d + i]p, 0), so the point P3 is on the X-axis. The conclusion is that for each point p 0 = (x0 ,0) on the X-axis, the third iterate of p 0 is also on the X-axis, that is, GL 3(p 0 ) = (x3 , 0). of GL Hence, a return map exists on the X-axis. We call this return map H, so H(x) 28 GL 3(x,O). The above results imply H(x) = (a 2c + ad)x + (ac H(x) = (ac H(x) = (a 2 2 + a + d + 1)'p for x < O, + ad + cd)x + (ac + a + d + 1)'p for 0 < x c + ad)x + (a2 + a + 1).p for x 0 z _l.p. 1c 1 -.. p, and The graph of H is similar to figure 12. The map H has three fixed points, namely ac +- a ~c~)* + d + I -(ac + a + + add -+ 1) c-ac+d) I1 p, and p5 S+ a(ac+d) > 0. The fixed point p > 2 ac < 0 is unstable since the slope + ad + cdof H in Pu is bigger than 1, and the two fixed points qs and ps is stable since the slope a2 c + ad of H at both qs and PS is between 0 and 1. Furthermore, for all x with x < Pu we have im Hn (x) = qs, and for all x with x > Pu we have lir Hn (x) = P -0 ns The properties (1) H has slope between 0 and 1 for x < 0, (2) H has slope bigger than I for 0 < x < -. p, (3) H has slope between 0 and 1 for x > -L'p, and (4) H(O) < 0 and H(-!.P) > imply that GL consisting of the points p has a period 3 attractor 2 = ( ac + a + d + 1 - a(ac+d) S Sa = ([c 3 -P, +a+ 1 1 - a(ac+d) + a $ "' I , d , + a+ 0), a a+ +a+ -a(ac+d) 1 and aand Notice that the norms of all three points converge to zero as p goes to zero, that is, all three IISlIl 1 0, IS2 I -+ 0, and IIS3i1 -+ 0 as p -+ 0. Hence, the period 3 attractor shrinks to a point as p goes to zero; this point to which the period 3 attractor converges is the fixed point of GL at M = 0. The point (pu, 0) is a point of a period 3 orbit which is a regular saddle of the map GL . We conclude: at M = 0, there is a "period two to period three" border-collision bifurcation. This completes the proof of the Proposition. 29 I 5. DISCUSSION AND CONCLUDING REMARKS. We have presented bifurcation phenomena, which we call "border-collision bifurcations". These bifurcations occur when the nature of a fixed point (or periodic point) of a piecewise smooth system changes when it collides with the border of two regions. An interesting case occurs when the fixed point changes from being a 5 3 flip saddle to a repellor with complex eigenvalues~at the parameter value where it collides with the border of two regions. We have presented a variety of examples based on the piecewise linear map P and the Henon map. In particular, we have shown the occurrence of a "period two to period three" border-collision I bifurcation for maps in the class C. We point out that the border-collision bifurcation can be 3 expected to occur in many piecewise smooth models. In particular, the "period two to period three" bifurcation phenomenon can be expected to occur in many linear models with constraints. Assume for the piecewise linear map P 5 I 3 5 that the fixed point E is a flip saddle in the left half plane and a repellor with complex eigenvalues in the right half plane. QUESTION 1. Does there exist a classification of the bordercollision bifurcations for P in the case where a period 2 attractor converges to the fixed point (0,0) when p goes to 0? QUESTION 2. More generally, is is possible to give a classification of the border- ollision bifurcations for the piecewise linear map P ? QUESTION 3. 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