Observability is a very useful concept for determining whether the dynamics of complicated systems can be correctly reconstructed from a single (univariate or multivariate) time series. When the governing equations of dynamical systems...
moreObservability is a very useful concept for determining whether the dynamics of complicated systems can be correctly reconstructed from a single (univariate or multivariate) time series. When the governing equations of dynamical systems are high-dimensional and/or rational, analytical computations of observability coefficients produce large polynomial functions with a number of terms that become exponentially large with the dimension and the nature of the system. In order to overcome this difficulty, we introduce here a symbolic observability coefficient based on a symbolic computation of the determinant of the observability matrix. The computation of such coefficients is straightforward and can be easily analytically carried out, as demonstrated in this paper for a five-dimensional rational system.
Heart rate variability analysis using 24-h Holter monitoring is frequently performed to assess the cardiovascular status of a patient. The present retrospective study is based on the beat-to-beat interval variations or ΔRR, which offer a...
moreHeart rate variability analysis using 24-h Holter monitoring is frequently performed to assess the cardiovascular status of a patient. The present retrospective study is based on the beat-to-beat interval variations or ΔRR, which offer a better view of the underlying structures governing the cardiodynamics than the common RR-intervals. By investigating data for three groups of adults (with normal sinus rhythm, congestive heart failure, and atrial fibrillation, respectively), we showed that the first-return maps built on ΔRR can be classified according to three structures: (i) a moderate central disk, (ii) a reduced central disk with well-defined segments, and (iii) a large triangular shape. These three very different structures can be distinguished by computing a Shannon entropy based on a symbolic dynamics and an asymmetry coefficient, here introduced to quantify the balance between accelerations and decelerations in the cardiac rhythm. The probability P111111 of successive heart beats without large beat-to-beat fluctuations allows to assess the regularity of the cardiodynamics. A characteristic time scale, corresponding to the partition inducing the largest Shannon entropy, was also introduced to quantify the ability of the heart to modulate its rhythm: it was significantly different for the three structures of first-return maps. A blind validation was performed to validate the technique.
The transformation of a nonlinear dynamical system into a standard form by using one of its variables and its successive derivatives can be used to identify the relationships that may exist between the parameters of the original system...
moreThe transformation of a nonlinear dynamical system into a standard form by using one of its variables and its successive derivatives can be used to identify the relationships that may exist between the parameters of the original system such as the subset of the parameter space over which the dynamics is left invariant. We show how the size of the attractor or the time scale (the pseudo-period) can be varied without affecting the underlying dynamics. This is demonstrated for the Rössler and the Lorenz systems. We also consider the case when two Rössler systems are unidirectionally coupled and when a Lorenz system is driven by a Rössler system. In both cases, the dynamics of the coupled system is affected.
We analyze chemical experimental time series by means of recent tools from nonlinear dynamics. More specifically, experiments on the Belousov-Zhabotinskii reaction [1] in a continuous flow reactor reveal a spiralling strange attractor...
moreWe analyze chemical experimental time series by means of recent tools from nonlinear dynamics. More specifically, experiments on the Belousov-Zhabotinskii reaction [1] in a continuous flow reactor reveal a spiralling strange attractor which arises from a (global) homoclinic bifurcation. By using a global vector field reconstruction method, a set of ordinary differential equations is obtained from the measurements of the
Resume. Lesecoulements a†eurants presentent generalement des regimes d'oscillations entretenues resultant d'un processus de retro-action complexe, en particulier pour des nombres de Reynolds moderes. Des travaux anterieurs ont...
moreResume. Lesecoulements a†eurants presentent generalement des regimes d'oscillations entretenues resultant d'un processus de retro-action complexe, en particulier pour des nombres de Reynolds moderes. Des travaux anterieurs ont revele un phenomµene d'¶echange de modes qui restait µa caracteriser d'un point de vue dynamique. L'intermittence entre les deux modes dominants est ici caracterisee par dynamique symbolique. La premiµereetape consiste µa reconstruire, µa
When a dynamical system is studied, one of the most interesting goals is to obtain a set of differential equations which provides a model of its behavior. Indeed, it has been pointed out that the knowledge of a global model which captures...
moreWhen a dynamical system is studied, one of the most interesting goals is to obtain a set of differential equations which provides a model of its behavior. Indeed, it has been pointed out that the knowledge of a global model which captures all the underlying dynamics provides a significant step in the understanding of physical processes. In the case of
Global vector field reconstruction is a well established technique to provide phenomenological models from nonlinear data, in particular when all information is contained in a so-called standard function. In the case when the standard...
moreGlobal vector field reconstruction is a well established technique to provide phenomenological models from nonlinear data, in particular when all information is contained in a so-called standard function. In the case when the standard function is taken as a ratio of polynomials, we establish that information about the fixed points of the system can be automatically retrieved from the data, allowing one to build a better model by selecting an appropriate structure. The method is exemplified in the case of the variable z of the Rössler system, which constitutes a rather acid test case.
Page 1. PHYSICAL REVIEW E VOLUME 51, NUMBER 5 MAY 1995 Global vector field reconstruction from a chaotic experimental signal in copper electrodissolution C. Letellier, L. Le Sceller, E. Marechal, P. Dutertre, B.Maheu ...
An algebraic expression for evaluation of linking numbers of unstable periodic orbits in chaotic attractors is demonstrated. An illustrating example (horseshoe dynamics) is provided.
A multivariate polynomial L2 approximation on nets is designed for global vector-field reconstructions of time continuous dynamical systems. The technique is tested by investigating standard forms of the Rössler band, the Lorenz mask, and...
moreA multivariate polynomial L2 approximation on nets is designed for global vector-field reconstructions of time continuous dynamical systems. The technique is tested by investigating standard forms of the Rössler band, the Lorenz mask, and a chaotic attractor produced by a simple model of thermal lens oscillations.
AND MOTIVATION Many examples exist in nature where the output of a dynamic system is a spiky
Rational functions are not very useful for obtaining global differential models because they involve poles that may eject the trajectory to infinity. In contrast, it is here shown that they allow one to significantly improve the quality...
moreRational functions are not very useful for obtaining global differential models because they involve poles that may eject the trajectory to infinity. In contrast, it is here shown that they allow one to significantly improve the quality of models for maps. In such a case, the presence of poles does not involve any numerical difficulty when the models are iterated. The models then take advantage of the ability of rational functions to capture complicated structures that may be generated by maps. The method is applied to experimental data from copper electrodissolution.
We study the relation between a dynamical system, which is unchanged (equivariant) under a discrete symmetry group G and another locally identical dynamical system with no residual symmetry. We also study the converse mapping: lifting a...
moreWe study the relation between a dynamical system, which is unchanged (equivariant) under a discrete symmetry group G and another locally identical dynamical system with no residual symmetry. We also study the converse mapping: lifting a dynamical system without symmetry to a multiple cover, which is equivariant under G. This is done in R3 for the two element rotation and inversion groups. Comparisons are done for the equations of motion, the strange attractors that they generate, and the branched manifolds that classify these strange attractors. A dynamical system can have many inequivalent multiple covers, all equivariant under the same symmetry group G. These are distinguished by the value of a certain topological index. Many examples are presented. A new global bifurcation, the "peeling bifurcation," is described.
Type-I intermittencies are common phenomena that are often observed in the neighborhood of periodic windows when a control parameter is varied. These intermittencies usually have a single reinjection channel, that is, a single type of...
moreType-I intermittencies are common phenomena that are often observed in the neighborhood of periodic windows when a control parameter is varied. These intermittencies usually have a single reinjection channel, that is, a single type of laminar phase was observed. Recently, type-I intermittencies with two reinjection channels were reported in several systems. In this paper, it will be shown that type-I intermittencies with n channels of reinjection are associated with the coexistence of n stable periodic orbits that are mapped into each other under a symmetry. A procedure to build type-I intermittency with n reinjection channels using the n-fold cover of an image system is presented. Cases up to n=3 are explicitly given with the covers of the centered Rössler system.
Stretching and squeezing mechanisms act together in phase space to build up strange attractors of low dimension (d_L<3). Through the development of a branched manifold, the stretching and squeezing mechanisms acting on a strange...
moreStretching and squeezing mechanisms act together in phase space to build up strange attractors of low dimension (d_L<3). Through the development of a branched manifold, the stretching and squeezing mechanisms acting on a strange attractor can be revealed. The branched manifold also indicates the presence, location, and symmetry properties of fixed points. A second but less obvious way to get
Ineffective inspiratory triggering efforts are a major cause of poor patient-ventilator interactions during mechanical ventilation, but their routine identification requires the insertion of an esophageal catheter. We developed a...
moreIneffective inspiratory triggering efforts are a major cause of poor patient-ventilator interactions during mechanical ventilation, but their routine identification requires the insertion of an esophageal catheter. We developed a mathematical analysis of ventilatory tracings recorded under noninvasive pressure ventilation in order to identify ineffective triggering efforts and their consequences without recording esophageal pressure. We assessed 2,183 cycles from 44 pressure support tracings in 14 children with cystic fibrosis treated by noninvasive home ventilation. Airway pressure, flow and esophageal pressure time series were visually analyzed and manually counted. Airway pressure versus time and flow versus time were then analyzed using a dedicated algorithm written by us. Esophageal pressure was only used for validation. A mathematical treatment of flow time series allowed us to draw phase portraits that had specific periodic trajectories for triggered ventilatory cycles and ineffective triggering efforts. From flow and pressure tracings, our algorithm correctly identified 100% of triggered cycles and 53/56 (94.6%) of ineffective triggering efforts. Ineffective triggering was associated with a significant reduction in minute ventilation, inspiratory flows and a significant increase in inspiratory efforts. A noninvasive analysis of flow and airway pressure can reliably identify ineffective triggering efforts during noninvasive pressure support ventilation. This approach may be a valuable tool for evaluating patient-ventilator interactions and their consequences during long-term recordings.
Strange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism....
moreStrange attractors can exhibit bifurcations just as periodic orbits in these attractors can exhibit bifurcations. We describe two classes of large-scale bifurcations that strange attractors can undergo. For each we provide a mechanism. These bifurcations are illustrated in a simple class of three-dimensional dynamical systems that contains the Lorenz system.
The van der Pol attractor exhibits a wide variety of behavior depending on the control parameter values: limit cycles, quasiperiodic motion on a torus, mode locking, period doubling, banded chaos, boundary crises, torus wrinkling, breakup...
moreThe van der Pol attractor exhibits a wide variety of behavior depending on the control parameter values: limit cycles, quasiperiodic motion on a torus, mode locking, period doubling, banded chaos, boundary crises, torus wrinkling, breakup of a torus, and toroidal chaos. The organization of these phenomena with respect to each other is well described by studying a partition of the control parameter plane of the Curry-Yorke map.
In practical problems, the observability of a system not only depends on the choice of observable(s) but also on the space which is reconstructed. In fact starting from a given set of observables, the reconstructed space is not unique,...
moreIn practical problems, the observability of a system not only depends on the choice of observable(s) but also on the space which is reconstructed. In fact starting from a given set of observables, the reconstructed space is not unique, since the dimension can be varied and, in the case of multivariate measurement functions, there are various ways to combine the measured observables. Using a graphical approach recently introduced, we analytically compute symbolic observability coefficients which allow to choose from the system equations the best observable, in the case of scalar reconstructions, and the best way to combine the observables in the case of multivariate reconstructions. It is shown how the proposed coefficients are also helpful for analysis in higher dimension.
Simple jerk systems are very useful for combining analytical computations and dynamical analysis in phase space. This is particularly relevant since there is still no direct link between the algebraic structure of ordinary differential...
moreSimple jerk systems are very useful for combining analytical computations and dynamical analysis in phase space. This is particularly relevant since there is still no direct link between the algebraic structure of ordinary differential equations and the topology of the chaotic attractors which they generate. In this paper, particular analytical solutions are identified for three simple chaotic flows. It is
In order to investigate a possible correspondence between differential and difference equations, it is important to possess discretization of ordinary differential equations. It is well known that when differential equations are...
moreIn order to investigate a possible correspondence between differential and difference equations, it is important to possess discretization of ordinary differential equations. It is well known that when differential equations are discretized, the solution thus obtained ...
Proceedings from the Montebello Round Table Discussion. Second annual conference on Complexity and Variability discusses research that brings innovation to the bedside. ... Seely AJ, Kauffman SA, Bates JH, Macklem PT, Suki B, Marshall JC,...
moreProceedings from the Montebello Round Table Discussion. Second annual conference on Complexity and Variability discusses research that brings innovation to the bedside. ... Seely AJ, Kauffman SA, Bates JH, Macklem PT, Suki B, Marshall JC, Batchinsky AI, Perez-Velazquez JL, ...
ABSTRACT A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which...
moreABSTRACT A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which is the product of the phase space ℝm of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to define a Poincaré section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is ℝm+2 instead of the usual extended phase space ℝm × S1. It is also demonstrated that a global model may then be obtained by using m + 2 dynamical variables with two variables associated with the driving force. The obtained model characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input.
