Coupled Map Lattice

We present a probabilistic approach for the study of systems with exclusions, in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss-network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime, the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with \duplicated-variables expansion", used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.

Currently used public-key cryptosystems are based on difficulties in solving certain numeric theoretic problems, in which the way to predict the private key from the knowledge of the public key is computationally infeasible. Here we propose a method of constructing public-key cryptosystems by generalized synchronization of coupled map lattices, in which the difficulty in predicting the synchronous function is used as the trap-door function to deduce the private key from the public key. In specific, we implement this idea on the method of "Merkle's puzzles," and find that, incorporated with the chaotic dynamics, this traditional method is equipped with some new features and can be practical in certain situations.

Using elementary cellular automata (CA) as an example, we show how to coarse-grain CA in all classes of Wolfram's classification. We find that computationally irreducible (CIR) physical processes can be predictable and even computationally reducible at a coarse-grained level of description. The resulting coarse-grained CA which we construct emulate the large-scale behavior of the original systems without accounting for small-scale details. At least one of the CA that can be coarsegrained is irreducible and known to be a universal Turing machine. PACS numbers: 05.45.Ra, 05.10.Cc, 47.54.+r

Studies of the phase diagram of the coupled sine circle map lattice have identified the presence of two distinct universality classes of spatiotemporal intermittency viz. spatiotemporal intermittency of the directed percolation class with a complete set of directed percolation exponents, and spatial intermittency which does not belong to this class. We show that these two types of behavior are special

By choosing a dynamical system with d different couplings, one can rearrange a system based on the graph with a given vertex dependent on the dynamical system elements. The relation between the dynamical elements (coupling) is replaced by a relation between the vertexes. Based on the E0 transverse projection operator, we addressed synchronization problem of an array of the linearly coupled map lattices of identical discrete time systems. The synchronization rate is determined by the second largest eigenvalue of the transition probability matrix. Algebraic properties of the Bose-Mesner algebra with an associated scheme with definite spectrum has been used in order to study the stability of the coupled map lattice. Associated schemes play a key role and may lead to analytical methods in studying the stability of the dynamical systems. The relation between the coupling parameters and the chaotic region is presented. It is shown that the feasible region is analytically determined by the number of couplings (i.e. by increasing the number of coupled maps, the feasible region is restricted). It is very easy to apply our criteria to the system being studied and they encompass a wide range of coupling schemes including most of the popularly used ones in the literature.

This paper investigates a globally nonlocal coupled map lattice. A rigorous proof to the existence of chaos in the scene of Li–Yorke in that system is presented in terms of the Marotto theorem. Analytical sufficient conditions under which the system is chaotic, and has synchronous behaviors are determined, respectively. The wider regions associated with chaos and synchronous behaviors are shown by simulations. Spatiotemporal chaos, synchronous chaos and some other synchronous behaviors such as fixed points, 2-cycles and 22-cycles are also shown by simulations for some values of the parameters.► A globally nonlocal coupled map lattice is introduced. ► A sufficient condition for the existence of Li–Yorke chaos is determined. ► A sufficient condition for synchronous behaviors is obtained.

We consider weakly coupled analytic expanding circle maps on the lattice Z d (for d ≥ 1), with small coupling strength and coupling between two sites decaying exponentially with the distance. We study the spectrum of the associated (Perron-Frobenius) transfer operators. We give a Fréchet space on which the operator associated to the full system has a simple eigenvalue at 1 (corresponding to the SRB measure µ previously obtained by Bricmont-Kupiainen [BK1]) and the rest of the spectrum, except maybe for continuous spectrum, is inside a disc of radius smaller than one. For d = 1 we also construct Banach spaces of densities with respect to µ on which perturbation theory, applied to the difference of fixed high iterates of the normalised coupled and uncoupled transfer operators, yields localisation of the full spectrum of the coupled operator (i.e., the first spectral gap and beyond). As a side-effect, we show that the whole spectra of the truncated coupled transfer operators (on bounded analytic functions) are O( )-close to the truncated uncoupled spectra, uniformly in the spatial size. Our method uses polymer expansions and also gives the exponential decay of time-correlations for a larger class of observables than those considered in [BK1].

We consider biological neural networks of pyramidal cells in a quasistatic approximation. We argue that they can be treated as a coupled map lattice of inhibitory and excitatory site maps, where both maps are derived from perturbation response of rat neocortical pyramidal cells. Inhibitory site maps generate chaotic spike patterns on an open parameter set of w x positive measure R. Stoop, K. Schindler, L.A. Bunimovich, submitted , excitatory site maps are nonchaotic. Our network simulations show that local chaos by inhibition may be used to synchronize cortical networks. q 1999 Elsevier Science B.V. All rights reserved. 0375-9601r99r$ -see front matter q 1999 Elsevier Science B.V. All rights reserved.

We present work in progress on the dynamical analysis of a multi-agent model that allows for temporally distributed asymmetric interactions between agents. The model essentially defines a coupled map lattice in which interactions between local variables obey a random Gaussian law and are transmitted through a gamma pattern of delays. The asymptotic dynamics of the model is investigated employing a dynamic mean field approach. The predictions of the mean field equations are checked numerically through simulations of a finite system of interacting units.

Genetic programming is used to evolve coupled map lattices for density classi cation. The most successful evolved rules depending only on nearest neighbors r = 1 show better performance than existing r = 3 cellular automaton rules on this task.

Numerical simulations of coupled map lattices (CMLs) and other complex model systems show an enormous phenomenological variety that is difficult to classify and understand. It is therefore desirable to establish analytical tools for exploring fundamental features of CMLs, such as their stability properties. Since CMLs can be considered as graphs, we apply methods of spectral graph theory to analyze their stability at locally unstable fixed points for different updating rules, different coupling scenarios, and different types of neighborhoods. Numerical studies are found to be in excellent agreement with our theoretical results. 1 1

Coupled map lattices of non-hyperbolic local maps arise naturally in many physical situations described by discretised reaction diffusion equations or discretised scalar field theories. As a prototype for these types of lattice dynamical systems we study diffusively coupled Tchebyscheff maps of N-th order which exhibit strongest possible chaotic behaviour for small coupling constants a. We prove that the expectations of arbitrary observables scale with \sqrt{a} in the low-coupling limit, contrasting the hyperbolic case which is known to scale with a. Moreover we prove that there are log-periodic oscillations of period \log N^2 modulating the \sqrt{a}-dependence of a given expectation value. We develop a general 1st order perturbation theory to analytically calculate the invariant 1-point density, show that the density exhibits log-periodic oscillations in phase space, and obtain excellent agreement with numerical results.

In certain physical situations, extensive interactions arise naturally in systems. We consider one such situation, namely, small-world couplings. We show that, for a fixed fraction of nonlocal couplings, synchronous chaos is always a stable attractor in the thermodynamic limit. We point out that randomness helps synchronization. We also show that there is a size dependent bifurcation in the collective behavior in such systems.

We study spatio-temporal intermittency (STI) in a system of coupled sine circle maps. The phase diagram of the system shows parameter regimes with STI of both the directed percolation (DP) and non-DP class. STI with synchronized laminar behaviour belongs to the DP class. The regimes of non-DP behaviour show spatial intermittency (SI), where the temporal behaviour of both the laminar and burst regions is regular, and the distribution of laminar lengths scales as a power law.

The scaling hypothesis for the coupled chaotic map lattices (CML) is formulated. Scaling properties of the CML in the regime of extensive chaos observed numerically before is justified analytically. The asymptotic Liapunov exponents spectrum for coupled piece-wise linear chaotic maps is found.

We discuss the spatiotemporal intermittency (STI) seen in coupled map lattices (CML-s). We identify the types of intermittency seen in such systems in the context of several specific CML-s. The Chaté-Manneville CML is introduced and the ongoing debate on the connection of the spatiotemporal intermittency seen in this model with the problem of directed percolation is summarised. We also discuss the STI seen in the sine circle map model and its connection with the directed percolation problem, as well as the inhomogenous logistic map lattice which shows the novel phenomenon of spatial intermittency and other types of behaviour not seen in the other models. The connection of the bifurcation behaviour in this model with STI is touched upon. We conclude with a discussion of open problems.

This paper studies the stability of the slab reactor with respect to the enrichment. For this purpose, the coupled map lattice theory is applied to the multi-group diffusion equations. Applying mean Lyapunov exponent theory introduced by Shibata [H. Shibata, Physica A 264 (1999) 226] on the model shows that, two successive phases: subcritical and supercritical. In order to compare the performance of the selected method by using the MCNP and ANISN codes the obtained results controlled. The model, in spite of its simplicity in form, shows a greater efficiency in prediction of critical enrichment.

This paper studies the stability of the slab reactor with respect to the enrichment. For this purpose, the coupled map lattice theory is applied to the multi-group diffusion equations. Applying mean Lyapunov exponent theory introduced by Shibata [H. Shibata, Physica A 264 (1999) 226] on the model shows that, two successive phases: subcritical and supercritical. In order to compare the performance of the selected method by using the MCNP and ANISN codes the obtained results controlled. The model, in spite of its simplicity in form, shows a greater efficiency in prediction of critical enrichment.

We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

The dynamics of two coupled piece-wise linear one-dimensional monostable maps is investigated. The single map is associated with Poincaré section of the FitzHugh-Nagumo neuron model. It is found that a diffusive coupling leads to the appearance of chaotic attractor. The attractor exists in an invariant region of phase space bounded by the manifolds of the saddle fixed point and the saddle periodic point. The oscillations from the chaotic attractor have a spike-burst shape with anti-phase synchronized spiking.

We investigate a system of coupled phase oscillators with nearest neighbors coupling in a chain with fixed ends. We find that the system synchronizes to a common value of the time-averaged frequency, which depends on the initial phases of the oscillators at the ends of the chain. This time-averaged frequency decays as the coupling strength increases. Near the transition to the frozen state, the time-averaged frequency has a power law behavior as a function of the coupling strength, with synchronized time-averaged frequency equal to zero. Associated with this power law, there is an increase in phases of each oscillator with 2 jumps with a scaling law of the elapsed time between jumps. During the interval between the full frequency synchronization and the transition to the frozen state, the maximum Lyapunov exponent indicates quasiperiodicity. Time series analysis of the oscillators frequency shows this quasiperiodicity, as the coupling strength increases.

Chaos synchronization in coupled systems is often characterized by a map between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set-by which we mean graph͑͒-can be extremely complicated. We identify, describe, and give examples of several different complications that can arise, and we link each to inherent properties of the underlying dynamics. In brief, synchronization sets can in general become nondifferentiable, and in the more severe case of noninvertible dynamics, they might even be multivalued. We suggest two different ways to quantify these features, and we discuss possible failures in detecting chaos synchrony using standard continuity-based methods when these features are present.

We generalize the exact solution to the Bernoulli shift map. Under certain conditions, the generalized functions can produce unpredictable dynamics. We use the properties of the generalized functions to show that certain dynamical systems can generate random dynamics. For instance, the chaotic Chua's circuit coupled to a circuit with a non-invertible I-V characteristic can generate unpredictable dynamics. In general, a nonperiodic time-series with truncated exponential behavior can be converted into unpredictable dynamics using non-invertible transformations. Using a new theoretical framework for chaos and randomness, we investigate some classes of coupled map lattices. We show that, in some cases, these systems can produce completely unpredictable dynamics. In a similar fashion, we explain why some wellknown spatiotemporal systems have been found to produce very complex dynamics in numerical simulations. We discuss real physical systems that can generate random dynamics.

We study the synchronization properties of a lattice of chaotic piecewise linear maps. The coupling strength decreases with the lattice distance in a power-law fashion. We obtain the Lyapunov spectrum of the coupled map lattice and investigate the relation between spatiotemporal chaos and synchronization of amplitudes and phases, using suitable numerical diagnostics.

We find that the global symbolic dynamics of a diffusively coupled map lattice (CML) is wellapproximated by a very small local model for weak to moderate coupling strengths. A local symbolic model is a truncation of the full symbolic model to one that considers only a single element and a few neighbors. Using interval analysis we give rigorous results for a range of coupling strengths and different local model widths. Examples are presented of extracting a local symbolic model from data and of controlling spatiotemporal chaos.

In this work we address the statistical periodicity phenomenon on a coupled map lattice. The study was done based on the asymptotic binary patterns. The pattern multiplicity gives us the lattice information capacity, while the entropy rate allows us to calculate the lockingtime. Our results suggest that the lattice has low locking-time and high capacity information when the coupling is weak. This is the condition for the system to reproduce a kind of behavior observed in neural networks.

We investigate quasiperiodic travelling waves (QTWs) in lattices of diffusively coupled logistic maps. Starting from the assumption that any spatial structure can be broken down into simpler elementary structures, a classification scheme for QTWs is introduced. Within this framework, the phenomenon of discrete velocities is reviewed and further investigated. In addition, a new technique is proposed for predicting whether QTWs can occur for given parameter values and which they might be.

We consider networks of coupled maps where the connections between units involve time delays. We show that, similar to the undelayed case, the synchronization of the network depends on the connection topology, characterized by the spectrum of the graph Laplacian. Consequently, scale-free and random networks are capable of synchronizing despite the delayed flow of information, whereas regular networks with nearest-neighbor

We describe methods of estimating the entire Lyapunov spectrum of a spatially extended system from multivariate time-series observations. Provided that the coupling in the system is short range, the Jacobian has a banded structure and can be estimated using spatially localized reconstructions in low embedding dimensions. This circumvents the "curse of dimensionality" that prevents the accurate reconstruction of high-dimensional dynamics from observed time series. The technique is illustrated using coupled map lattices as prototype models for spatiotemporal chaos and is found to work even when the coupling is not strictly local but only exponentially decaying.

A coupled map lattice of generalized Lotka–Volterra equations in the presence of colored multiplicative noise is used to analyze the spatiotemporal evolution of three interacting species: one predator and two preys symmetrically competing each other. The correlation of the species concentration over the grid as a function of time and of the noise intensity is investigated. The presence of noise induces pattern formation, whose dimensions show a nonmonotonic behavior as a function of the noise intensity. The colored noise induces a greater dimension of the patterns with respect to the white noise case and a shift of the maximum of its area towards higher values of the noise intensity.

Coexistence of various ordered chaotic states in a Hamiltonian system is studied with the use of a symplectic coupled map lattice. Besides the clustered states for the attractive interaction, a novel chaotic ordered state is found for a system with repulsive interaction, characterized by a dispersed state of particles. The dispersed and clustered states form an onion-like structure in phase space. The degree of order increases towards the center of the onion, while chaos is enhanced at the edge between ordered and random chaotic states. For a longer time scale, orbits itinerate over ordered and random states. The existence of these ordered states leads to anomalous long-time correlation for many quantifiers such as the global diffusion.

Coupled map lattices are a paradigm of higher-dimensional dynamical systems exhibiting spatiotemporal chaos. A special case of non-hyperbolic maps are one-dimensional map lattices of coupled Chebyshev maps with periodic boundary conditions, called chaotic strings. In this short note we show that the fine structure of the self energy of this chaotic string in the scaling region (i.e. for very small coupling) is retained if we reduce the length of the string to three lattice points.

We study 1-dimensional coupled map lattices consisting of diffusively coupled Chebyshev maps of N -th order. For small coupling constants a we determine the invariant 1-point and 2-point densities of these nonhyperbolic systems in a perturbative way. For arbitrarily small couplings a > 0 the densities exhibit a selfsimilar cascade of patterns, which we analyse in detail. We prove that there are log-periodic oscillations of the density both in phase space as well as in parameter space. We show that expectations of arbitrary observables scale with √ a in the low-coupling limit, contrasting the case of hyperbolic maps where one has scaling with a. Moreover we prove that there are log-periodic oscillations of period log N 2 modulating the √ a-dependence of the expectation value of any given observable.

The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows to apply methods of complex network theory for characterizing time series. In this work we present the horizontal visibility algorithm, a geometrically simpler and analytically solvable version of our former algorithm, focusing on the mapping of random series (series of independent identically distributed random variables). After presenting some properties of the algorithm, we present exact results on the topological properties of graphs associated to random series, namely the degree distribution, clustering coefficient, and mean path length. We show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series, since any random series maps to a graph with an exponential degree distribution of the shape P (k) = (1/3)(2/3) k−2 , independently of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P (k) are related to non-random series. Numerical simulations confirm the accuracy of the theorems for finite series. In a second part, we show that the method is able to distinguish chaotic series from i.i.d. theory, studying the following situations: (i) noise-free low-dimensional chaotic series, (ii) low-dimensional noisy chaotic series, even in the presence of large amounts of noise, and (iii) high-dimensional chaotic series (coupled map lattice), without needs for additional techniques such as surrogate data or noise reduction methods. Finally, heuristic arguments are given to explain the topological properties of chaotic series and several sequences which are conjectured to be random are analyzed.

In this paper, we consider the spatiotemporal dynamics in a ring of N mutually coupled self-sustained oscillators in the regular state. When there are no parameter mismatches, the good coupling parameters leading to full, partial, and no synchronization are derived using the properties of the variational equations of stability. The effects of the spatial dimension of the ring on the stability boundaries of the synchronized states are performed. Numerical simulations validate and complement the results of analytical investigations. The influences of coupling parameter mismatch on the forecasted stability boundaries are also highlighted.

1. Models that represent space as a lattice have a critical function in theoretical and applied ecology. Despite their significance, there is a dearth of appropriate theoretical developments for the description of dispersal across such lattices. 2. We present a series of methods for approximating continuous dispersal in discrete landscapes (denoted as centroid-to-centroid, centroid-to-area, area-to-centroid and area-to-area dispersal). We describe how these methods can be extended to incorporate different conditions at the boundary of the simulation arena and a framework for approximating continuous dispersal between irregularly shaped patches. 3. Each approximation method was tested against a baseline of continuous Gaussian dispersal in a periodic simulation arena. The residence probabilities for an individual dispersing in each time step according to a Gaussian kernel across grids of three differing resolutions were calculated over a number of dispersal steps. In addition, the steady-state asymptotic properties for the transition matrices for each approximation method and cell resolution were calculated and compared against the uniform expectation under continuous dispersal. 4. All four methods described in this article provide a reasonable approximation to the continuous baseline (<0AE03 absolute error in probability calculations) on landscapes with grid cells of length equal to the expected dispersal distance or finer, but error increases as grid cells become progressively larger than the expected dispersal distance. 5. Each approximation method exhibits a different spatial pattern of approximation error. Centroid-to-centroid dispersal overestimates residence probabilities near the origin, resulting in decreased invasion rates relative to the baseline diffusion process. All other approximation methods underestimate residence probabilities near the origin and overestimate such probabilities in the peripheries, leading to an overestimation of invasion rates. 6. The asymptotic properties of centroid-to-centroid and area-to-centroid dispersal approximation methods deviate from that which is expected under continuous dispersal. This characteristic renders these methods unsuitable for use in long-term simulation studies where the equilibrium properties of the system are of interest. 7. Centroid-to-area and area-to-area approximation methods exhibit both low approximation error and desirable asymptotic properties. These methods provide a viable mechanism for linking individual-level dispersal to larger-scale characteristics such as metapopulation connectivity.

The visibility algorithm has been recently introduced as a mapping between time series and complex networks. This procedure allows to apply methods of complex network theory for characterizing time series. In this work we present the horizontal visibility algorithm, a geometrically simpler and analytically solvable version of our former algorithm, focusing on the mapping of random series (series of independent identically distributed random variables). After presenting some properties of the algorithm, we present exact results on the topological properties of graphs associated to random series, namely the degree distribution, clustering coefficient, and mean path length. We show that the horizontal visibility algorithm stands as a simple method to discriminate randomness in time series, since any random series maps to a graph with an exponential degree distribution of the shape P (k) = (1/3)(2/3) k−2 , independently of the probability distribution from which the series was generated. Accordingly, visibility graphs with other P (k) are related to non-random series. Numerical simulations confirm the accuracy of the theorems for finite series. In a second part, we show that the method is able to distinguish chaotic series from i.i.d. theory, studying the following situations: (i) noise-free low-dimensional chaotic series, (ii) low-dimensional noisy chaotic series, even in the presence of large amounts of noise, and (iii) high-dimensional chaotic series (coupled map lattice), without needs for additional techniques such as surrogate data or noise reduction methods. Finally, heuristic arguments are given to explain the topological properties of chaotic series and several sequences which are conjectured to be random are analyzed.

By considering a symmetric N-dimensional map which possesses invariant measure in its diagonal and anti-diagonal invariant sub-manifolds, we have been able to propose an N-coupled map which possesses invariant measure in synchronized or anti-synchronized states. Then chaotic synchronization and anti-synchronization are investigated in the introduced model. We have calculated Kolmogrov-Sinai entropy and Lyapunov exponent as another tool to study the stability of N-coupled map in synchronized and anti-synchronization states.

Hyperchaos occurs in a dynamical system with more than one positive Lyapunov exponent. When the equations governing the time evolution of the dynamical system are known, the transition from chaos to hyperchaos can be readily obtained when the second largest Lyapunov exponent crosses zero. If the only information available on the system is a time series, however, such method is difficult to apply. We propose the use of recurrence quantification analysis of a time series to characterize the chaos-hyperchaos transition. We present results obtained from recurrence plots of coupled chaotic piecewise-linear maps and Chua-Matsumoto circuits, but the method can be applied as well to other systems, even when one does not know their dynamical equations.

This paper describes interactive digital painting technique where brush stroke is controlled via coupled map lattice model. Coupled map lattice model is an extension of cellular automata model where discrete (integer) state values of CA cells are replaced with continuous real values. Using this technique, various brush styles and strokes can be achieved.

Conventional approaches to modeling any system try to incorporate increasingly realistic features into the model, thereby making it more and more complex. An opposite approach seeks to build simpler and simpler conceptual models capable of capturing some observed features of a system. This trend began with Lorenz, who simplified models of the atmosphere to obtain the Lorenz model consisting of a system of only three equations. Despite the simplicity of these equations, this system displayed surprisingly rich properties, and has been used as a conceptual model in diverse disciplines. Poincare maps help study ordinary differential equations from a qualitative perspective. Several investigators like Henon and Feigenbaum followed this simplification approach. Instead of investigating Poincare maps of realistic systems, they, along with several others, investigated simple maps for their own sake. Despite lack of realism, this approach proved to be very fruitful. A map, as simple as the Logistic map, became an important conceptual modeling paradigm. It provided a tool for understanding bifurcation routes to chaos, which were verified experimentally through various experiments in diverse fields. Coupled map lattices (CML) help explore partial differential equations (PDE). Further simplification led to the introduction of Cellular Automata (CA). These fields continue to be explored with vigor and have given rise to a rich body of knowledge, conceptually useful over a wide spectrum of disciplines. In this paper, we follow the simplification approach for modeling the N-body problem. N-body simulations, say in Gravitation, give rise to filamentary structures. Such structures are observed in the actual observed Galactic distribution. The mechanism for creation of such structures is not well understood. We present a simple iterative dynamical model, motivated by the N-body problem, which, though unrealistic, produces such filamentary structures. This model also exhibits a variety of intriguing structures. Attempts to understand these structures may lead to useful insights similar to those provided by investigations in maps, CML, CA etc.

Chaos synchronization in coupled systems is often characterized by a map ϕ between the states of the components. In noninvertible systems, or in systems without inherent symmetries, the synchronization set—by which we mean graph(ϕ)—can be extremely complicated. We identify, describe, ...