Showing papers on "Coupled map lattice published in 2009"
TL;DR: The horizontal visibility algorithm as mentioned in this paper is 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.
Abstract: networks. This procedure allows us 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 with random series, namely, the degree distribution, the clustering coefficient, and the 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 Pk=1 /32 /3 k2 , independent of the probability distribution from which the series was generated. Accordingly, visibility graphs with other Pk are related to nonrandom 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 independent and identically distributed 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 that are conjectured to be random are analyzed.
547 citations
Book•
01 Sep 2009
TL;DR: The Language of Dynamical Systems Examples of Chaotic Behaviors Probabilistic Approach to Chaos Characterization of Chaotics in Dynamical systems From Order to Chaos in Dissipative Systems Chaos in Hamiltonian Systems Chaos and Information Theory Coarse-Grained Information and Large Scale Predictability Chaos, Numerical Computations and Experiments Chaos in Few-Degrees of Freedom Systems Spatiotemporal Chaos Turbulence as a Dynamical System Problem Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study as mentioned in this paper
Abstract: First Encounter with Chaos The Language of Dynamical Systems Examples of Chaotic Behaviors Probabilistic Approach to Chaos Characterization of Chaotic Dynamical Systems From Order to Chaos in Dissipative Systems Chaos in Hamiltonian Systems Chaos and Information Theory Coarse-Grained Information and Large Scale Predictability Chaos, Numerical Computations and Experiments Chaos in Few-Degrees of Freedom Systems Spatiotemporal Chaos Turbulence as a Dynamical Systems Problem Chaos and Statistical Mechanics: Fermi-Pasta-Ulam a Case Study.
187 citations
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TL;DR: Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems as discussed by the authors, and it owes its name to an underlying irregular and yet linearly stable dynamics.
Abstract: Stable chaos is a generalization of the chaotic behaviour exhibited by cellular automata to continuous-variable systems and it owes its name to an underlying irregular and yet linearly stable dynamics. In this review we discuss analogies and differences with the usual deterministic chaos and introduce several tools for its characterization. Some examples of transitions from ordered behavior to stable chaos are also analyzed to further clarify the underlying dynamical properties. Finally, two models are specifically discussed: the diatomic hard-point gas chain and a network of globally coupled neurons.
52 citations
TL;DR: It is shown, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory.
Abstract: We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally coupled maps, we show, moreover, a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.
41 citations
TL;DR: An adaptive method to maintain synchronization between coupled nonlinear chaotic oscillators, when the coupling between the systems is unknown and time-varying (e.g., due to environmental parameter drift), is experimentally demonstrated and numerically simulated.
Abstract: We experimentally demonstrate and numerically simulate an adaptive method to maintain synchronization between coupled nonlinear chaotic oscillators, when the coupling between the systems is unknown and time-varying (e.g., due to environmental parameter drift). The technique is applied to optoelectronic feedback loops exhibiting high-dimensional chaotic dynamics. In addition to keeping the two systems isochronally synchronized in the presence of a priori unknown time-varying coupling strength, the technique provides an estimate of the time-varying coupling.
40 citations
TL;DR: In this article, the authors derived formulae to calculate wave speed and rates of spread for coupled map lattices, which are dynamical models where space and time are discrete.
25 citations
TL;DR: In this article, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented, and the model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well.
Abstract: Chaotic dynamics have been observed in a wide range of population models. In this study, the complex dynamics in a discrete-time ecological model of parasitoid-host-parasitoid are presented. The model shows that the superiority coefficient not only stabilizes the dynamics, but may strongly destabilize them as well. Many forms of complex dynamics were observed, including pitchfork bifurcation with quasi-periodicity, period-doubling cascade, chaotic crisis, chaotic bands with narrow or wide periodic window, intermittent chaos, and supertransient behavior. Furthermore, computation of the largest Lyapunov exponent demonstrated the chaotic dynamic behavior of the model.
22 citations
TL;DR: This paper introduces a chaotic cryptosystem based on the symbolic dynamics of random maps with position dependent weighting probabilities, and demonstrates that the proposed algorithm using symbolic dynamics achieves the optimal security criteria.
Abstract: Chaotic cryptology has been widely investigated recently A common feature in the most recent developments of chaotic cryptosystems is the use of a single dynamical rule in the encoding–decoding process The main objective of this paper is to provide a set of chaotic systems instead of a single one for cryptography In this paper, we introduce a chaotic cryptosystem based on the symbolic dynamics of random maps with position dependent weighting probabilities The random maps model is a deterministic dynamical system in a finite phase space with n points The maps that establish the dynamics of the system are chosen randomly for every point The essential idea of this paper is that, given two dynamical systems that behave in a certain way, it is possible to combine them (by composing) into a new dynamical system This dynamically composed system behaves in a completely different way compared to the constituent systems The proposed scheme exploits the symbolic dynamics of a set of chaotic maps in order to encode the binary information The performance of the new cryptosystem based on chaotic dynamical systems properties is examined Both theoretical and experimental results demonstrate that the proposed algorithm using symbolic dynamics achieves the optimal security criteria
15 citations
TL;DR: In this article, the anti-synchronization problem for different chaotic dynamical systems with fully unknown parameters in a response system is analyzed, where the vanishing of the sum of relevant variables is defined.
Abstract: We have observed anti-synchronization phenomena in different chaotic dynamical systems. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. Anti-synchronization problem for different chaotic dynamical systems with fully unknown parameters in response system is analyzed. This technique is applied to achieve anti-synchronization between Lorenz system, Lu system and Four-scroll system. Numerical simulations are provided to verify the effectiveness of the proposed methods.
15 citations
TL;DR: The results show that depending on many features such as position, size, and fragmentation of a refuge, as well as the dispersal parameters of hosts and parasitoids, together with the parasitoid attack rate, the inclusion of refuges may as well stabilize as destabilize the host-parasitoid dynamics.
TL;DR: Dąbrowski et al. as discussed by the authors used chaos to reduce the amplitude of the oscillations in the neighborhood of the resonance in a regular dynamical system and showed that chaotic dynamic systems are usually controlled in a way that allows the replacement of chaotic behavior by the desired periodic motion.
Abstract: Chaotic dynamic systems are usually controlled in a way, which allows the replacement of chaotic behavior by the desired periodic motion. We give the example in which an originally regular (periodic) system is controlled in such a way as to make it chaotic. This approach based on the idea of dynamical absorber allows the significant reduction of the amplitude of the oscillations in the neighborhood of the resonance. We present experimental results, which confirm our previous numerical studies [Dąbrowski A, Kapitaniak T. Using chaos to reduce oscillations. Nonlinear Phenomen Complex Syst 2001;4(2):206–11].
TL;DR: Feedback methods for controlling spatio-temporal on–off intermittency which is an aperiodic switching between an “off” state and an ‘on’ state are proposed and an important advantage of the methods is that the feedback signal is vanished when control is realized.
Abstract: In this paper, we propose feedback methods for controlling spatio-temporal on–off intermittency which is an aperiodic switching between an “off” state and an “on” state. Diffusively coupled map lattice with spatially non-uniform random driving is used for showing spatio-temporal on–off intermittency. For this purpose, we apply three different feedbacks. First, we use a linear feedback which is a simple method but has a long transient time. To overcome this problem, two nonlinear feedbacks based on prediction strategy are proposed. An important advantage of the methods is that the feedback signal is vanished when control is realized. Simulation results show that all methods have suppressed the chaotic behavior.
TL;DR: In this article, a coupled map lattice model was introduced in which the weak interaction takes place via rare "collisions" and the uniqueness of the SRB measure and exponential space-time decay of correlations were proved.
Abstract: We introduce a new coupled map lattice model in which the weak interaction takes place via rare “collisions”. By “collision” we mean a strong (possibly discontinuous) change in the system. For such models we prove uniqueness of the SRB measure and exponential space-time decay of correlations.
TL;DR: In this paper, it was shown that the fine structure of 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.
Abstract: Coupled map lattices are a paradigm of higher-dimensional dynamical systems exhibiting spatio-temporal 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.
TL;DR: In this article, the authors studied the dynamics of cold atoms inside a well of a red detuned lattice, with the aim to understand the dynamical mechanisms leading to the disappearance of chaos.
Abstract: The dynamics of cold atoms in conservative optical lattices obviously depends on the geometry of the lattice. But very similar lattices may lead to deeply different dynamics. For example, in a 2D optical lattice with a square mesh, the sign of the detuning plays a crucial role: in the blue detuned case, trajectories of an atom inside a well are chaotic for high enough energies. On the contrary, in the red detuned case, chaos is completely inhibited inside the wells. Here, we study in details the dynamical regimes of atoms inside a well of a red detuned lattice, with the aim to understand the dynamical mechanisms leading to the disappearance of chaos. We show that the motions in the two directions of space are frequency locked in most of the phase space, for most of the parameters of the lattice and atoms. This synchronization, not as strict as that of a dissipative system, is nevertheless a mechanism powerful enough to explain that chaos cannot appear in red detuned lattices.
TL;DR: In this article, the authors studied the transition to intermittent spatiotemporal chaos in the regularized long-wave equation, a nonlinear model of shallow water waves, and explored a mechanism for the onset of on-off spatio-temporal intermittency.
Abstract: Transition to intermittent spatiotemporal chaos is studied in the regularized long-wave equation, a nonlinear model of shallow water waves. A mechanism for the onset of on-off spatiotemporal intermittency is explored. In this mechanism, the coupling of two chaotic saddles triggers random switching between phases of laminar and bursty behaviors. The average time between bursts as a function of the control parameter follows a power law typical of crisis transitions in chaotic systems. The degree of spatiotemporal disorder in the observed fluid patterns is quantified by means of the time-averaged spectral entropy for both chaotic attractors and chaotic saddles. The implications of these results to other fluid systems are discussed.
TL;DR: In this article, a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics, which is relevant for the question at which scale in complex dynamics regularities and patterns emerge.
Abstract: We show how a simple scheme of symbolic dynamics distinguishes a chaotic from a random time series and how it can be used to detect structural relationships in coupled dynamics. This is relevant for the question at which scale in complex dynamics regularities and patterns emerge. © 2009 Wiley Periodicals, Inc. Complexity 2009
TL;DR: Chaotic neuronal maps are studied with threshold activated coupling at selected pinning sites with increasing pinning density to observe a transition from spatiotemporal chaos to a fixed spatial profile with synchronized temporal cycles.
Abstract: Chaotic neuronal maps are studied with threshold activated coupling at selected pinning sites with increasing pinning density. A transition from spatiotemporal chaos to a fixed spatial profile with synchronized temporal cycles is observed. There is an optimal fraction of sites where it is necessary to apply the control algorithm in order to effectively suppress chaotic dynamics.
TL;DR: In this article, the analysis of one-, two-, and three-dimensional coupled map lattices is developed under a statistical and dynamical perspective, which leads to an integrated understanding of the most important properties of these universal models of spatiotemporal chaos.
Abstract: The analysis of one-, two-, and three-dimensional coupled map lattices is here developed under a statistical and dynamical perspective. We show that the three-dimensional CML exhibits low dimensional behavior with long range correlation and the power spectrum follows 1 / f noise. This approach leads to an integrated understanding of the most important properties of these universal models of spatiotemporal chaos. We perform a complete time series analysis of the model and investigate the dependence of the signal properties by change of dimension.
TL;DR: In this paper, a new discrete adaptive method is proposed in order to control spatiotemporal chaos in coupled map lattices, which is an adaptive control which is based on the quasi sliding mode control design.
Abstract: In this paper, a new discrete adaptive method is proposed in order to control spatiotemporal chaos in coupled map lattices. The proposed method is an adaptive control which is based on the quasi sliding mode control design. It is a general method for tracking the control of spatiotemporal chaotic systems modeled by using coupled map lattices, and is robust with respect to unmodeled dynamic, unknown parameters, and bounded disturbance. The robustness of stability of the controlled system based on the Lyapunov direct method is proved even though bounds of disturbance are unknown. The unknown parameters of the model are updated with an adaptive algorithm, which is an extension of the gradient algorithm with a deadzone function. Diffusively coupled map lattice is used as an example to show the suitability of the method. Simulation results reveal the robustness and the applicability of the method in controlling spatiotemporal chaos in coupled map lattice models.
TL;DR: A two-species spatially extended system of hosts and parasitoids is studied and several metrics are introduced to characterize the patterns and onset thereof and build a consistent sequence of corrections to the mean-field equations using a posteriori knowledge from simulations.
Abstract: A two-species spatially extended system of hosts and parasitoids is studied. There are two distinct kinds of coexistence; one with populations distributed homogeneously in space and another one with spatiotemporal patterns. In the latter case, there are noise-sustained oscillations in the population densities, whereas in the former one the densities are essentially constants in time with small fluctuations. We introduce several metrics to characterize the patterns and onset thereof. We also build a consistent sequence of corrections to the mean-field equations using a posteriori knowledge from simulations. These corrections both lead to a better description of the dynamics and connect the patterns to it. The analysis is readily applicable to realistic systems, which we demonstrate by an example using an empirical metapopulation landscape.
TL;DR: In this article, the anti-synchronization of two nonlinear-coupled spatiotemporal chaotic systems is discussed, and a special nonlinearcouple term is constructed through suitably separating the spatio-temporal chaotic system to linear and nonlinear terms.
Abstract: The anti-synchronization of two nonlinear-coupled spatiotemporal chaotic systems is discussed. A special nonlinear-coupled term is constructed through suitably separating the spatiotemporal chaotic systems to linear and nonlinear terms, and anti-synchronization of two two-dimensional coupled map lattices is realized. The nonlinear-coupled method is further generalized to anti-synchronize the complex networks composed of two-dimensional coupled map lattices. The artificial simulation results show that this method is still effective.
TL;DR: In this paper, a hierarchy of solvable chaotic maps with dynamical parameter as a control parameter is defined and a method for dynamical control of chaos is described based on the hierarchy of chaotic maps.
Abstract: Techniques for stabilizing unstable state in nonlinear dynamical systems using small perturbations fall into three general categories: feedback, non-feedback schemes, and a combination of feedback and non-feedback. However, the general problem of finding conditions for creation or suppression of chaos still remains open. We describe a method for dynamical control of chaos. This method is based on a definition of the hierarchy of solvable chaotic maps with dynamical parameter as a control parameter. In order to study the new mechanism of control of chaotic process, Kolmogorov–Sinai entropy of the chaotic map with dynamical parameter based on discussion the properties of invariant measure have been calculated and confirmed by calculation of Lyapunov exponents. The introduced chaotic maps can be used as dynamical control.
TL;DR: In this paper, recurrence relations are found that rule weakly CML evolution, with both global and diffusive coupling, and solutions obtained from these relations are very general because they do not hold restrictions about boundary conditions, initial conditions and number of oscilators in the CML.
TL;DR: In this paper, the synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state.
Abstract: The synchronization transition between two coupled replicas of spatio-temporal chaotic systems in 2+1 dimensions is studied as a phase transition into an absorbing state—the synchronized state. Confirming the scenario drawn in (1+1)-dimensional systems, the transition is found to belong to two different universality classes—multiplicative noise (MN) and directed percolation (DP)—depending on the linear or nonlinear character of damage spreading occurring in the coupled systems. By comparing a coupled map lattice with two different stochastic models, accurate numerical estimates for MN in 2+1 dimensions are obtained. Finally, aiming to pave the way for future experimental studies, slightly non-identical replicas have been considered. It is shown that the presence of small differences between the dynamics of the two replicas acts as an external field in the context of absorbing phase transitions and can be characterized in terms of a suitable critical exponent.
TL;DR: In this article, the spatio-temporal dynamics of three interacting species, two preys and one predator, in the presence of two different kinds of noise sources is studied, by using Lotka-Volterra equations.
Abstract: The spatio-temporal dynamics of three interacting species, two preys and one predator, in the presence of two different kinds of noise sources is studied, by using Lotka-Volterra equations. A correlated dichotomous noise acts on the interaction parameter between the two preys, and a multiplicative white noise affects directly the dynamics of the three species. After analyzing the time behaviour of the three species in a single site, we consider a two-dimensional spatial domain, applying a mean field approach and obtaining the time behaviour of the first and second order moments for different multiplicative noise intensities. We find noise-induced oscillations of the three species with an anticorrelated behaviour of the two preys. Finally, we compare our results with those obtained by using a coupled map lattice (CML) model, finding a good qualitative agreement. However, some quantitative discrepancies appear, that can be explained as follows: i) different stationary values occur in the two approaches; ii) in the mean field formalism the interaction between sites is extended to the whole spatial domain, conversely in the CML model the species interaction is restricted to the nearest neighbors; iii) the dynamics of the CML model is faster since an unitary time step is considered.
12 Dec 2009
TL;DR: This paper proposes a new stream cipher Gemstone by discretizing coupled map lattices (CML), which is a nonlinear system of coupled chaotic maps, and shows that there is no high probability difference propagations or high correlations over the IV setup scheme.
Abstract: In this paper, we propose a new stream cipher Gemstone by discretizing coupled map lattices (CML), which is a nonlinear system of coupled chaotic maps. Gemstone uses a 128-bit key and a 64-bit initialization vector (IV). We show that there is no high probability difference propagations or high correlations over the IV setup scheme. Thus the IV setup of Gemstone is very secure. We also verify that the largest linear correlations between consecutive key streams are below the safe bounds. Gemstone is slightly slower than AES-CTR, but its initialization speeds are higher than some finalists of eSTREAM.
TL;DR: In this paper, the authors considered coupled map lattices with long-range interactions to study the spatio-temporal behaviour of spatially extended dynamical systems and derived the Lyapunov dimension and the transversal distance to the synchronization manifold.
Abstract: We considered coupled map lattices with long-range interactions to study the spatiotemporal behaviour of spatially extended dynamical systems. Coupled map lattices have been intensively investigated as models to understand many spatiotemporal phenomena observed in extended system, and consequently spatiotemporal chaos. We used the complex order parameter to quantify chaos synchronization for a one-dimensional chain of coupled logistic maps with a coupling strength which varies with the lattice in a power-law fashion. Depending on the range of the interactions, complete chaos synchronization and chaos suppression may be attained. Furthermore, we also calculated the Lyapunov dimension and the transversal distance to the synchronization manifold.
TL;DR: By using symbolic dynamics and shadowing, this work analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy, providing a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.
Abstract: We study the complexity of stable waves in unidirectional bistable coupled map lattices as a test tube to spatial chaos of traveling patterns in open flows. Numerical calculations reveal that, grouping patterns into sets according to their velocity, at most one set of waves has positive topological entropy for fixed parameters. By using symbolic dynamics and shadowing, we analytically determine velocity-dependent parameter domains of existence of pattern families with positive entropy. These arguments provide a method to exhibit chaotic sets of stable waves with arbitrary velocity in extended systems.