Showing papers on "Coupled map lattice published in 2008"

Chaotic behaviour of deterministic dissipative systems

Abstract: Introduction 1 Differential equations, maps and asymptotic behaviour 2 Transition from order to chaos 3 Numerical methods for studies of parametric dependences, bifurcations and chaos 4 Chaotic dynamics in experiments 5 Forced and coupled chemical oscillators: a case study of chaos 6 Chaos in distributed systems Appendices Bibliography Index

Increasing average period lengths by switching of robust chaos maps in finite precision

Abstract: Grebogi, Ott and Yorke (Phys. Rev. A 38, 1988) have investigated the effect of finite precision on average period length of chaotic maps. They showed that the average length of periodic orbits (T) of a dynamical system scales as a function of computer precision (e) and the correlation dimension (d) of the chaotic attractor: T ∼e-d/2. In this work, we are concerned with increasing the average period length which is desirable for chaotic cryptography applications. Our experiments reveal that random and chaotic switching of deterministic chaotic dynamical systems yield higher average length of periodic orbits as compared to simple sequential switching or absence of switching. To illustrate the application of switching, a novel generalization of the Logistic map that exhibits Robust Chaos (absence of attracting periodic orbits) is first introduced. We then propose a pseudo-random number generator based on chaotic switching between Robust Chaos maps which is found to successfully pass stringent statistical tests of randomness.

Rapidly switched random links enhance spatiotemporal regularity.

Abstract: We investigate the spatiotemporal properties of a lattice of chaotic maps whose coupling connections are rewired to random sites with probability p . Keeping p constant, we change the random links at different frequencies in order to discern the effect (if any) of the time dependence of the links. We observe two different regimes in this network: (i) when the network is rewired slowly, namely, when the random connections are quite static, the dynamics of the network is spatiotemporally chaotic and (ii) when these random links are switched around fast, namely, the network is rewired frequently, one obtains a spatiotemporal fixed point over a large range of coupling strengths. We provide evidence of a sharp transition from a globally attracting spatiotemporal fixed point to spatiotemporal chaos as the rewiring frequency is decreased. Thus, in addition to geometrical properties such as the fraction of random links in the network, dynamical information on the time dependence of these links is crucial in determining the spatiotemporal properties of complex dynamical networks.

Analysis of Chaotic Dynamics Using Measures of the Complex Network Theory

Abstract: Complex phenomena are observed in various situations. These complex phenomena are produced from deterministic dynamical systems or stochastic systems. Then, it is an important issue to clarify what is a source of the complex phenomena and to analyze what kind of response will emerge. Then, in this paper, we analyze deterministic chaos from a new aspect. The analysis method is based on the idea that attractors of nonlinear dynamical systems and networks are characterized by a two-dimensional matrix: a recurrence plot and an adjacent matrix. Then, we transformed the attractors to the networks, and evaluated the clustering coefficients and the characteristic path length to the networks. As a result, the networks constructed from the chaotic systems show a small world property.

Chaos synchronization in networks of coupled maps with time-varying topologies

Abstract: Complexity of dynamical networks can arise not only from the complexity of the topological structure but also from the time evolution of the topology. In this paper, we study the synchronous motion of coupled maps in time-varying complex networks both analytically and numerically. The temporal variation is rather general and formalized as being driven by a metric dynamical system. Four network models are discussed in detail in which the interconnections between vertices vary through time randomly. These models are: 1) i.i.d. sequences of random graphs with fixed wiring probability, 2) groups of graphs with random switches between the individual graphs, 3) graphs with temporary random failures of nodes, and 4) the meet-for-dinner model where the vertices are randomly grouped. We show that the temporal variation and randomness of the connection topology can enhance synchronizability in many cases; however, there are also instances where they reduce synchronizability. In analytical terms, the Hajnal diameter of the coupling matrix sequence is presented as a measure for the synchronizability of the graph topology. In topological terms, the decisive criterion for synchronization of coupled chaotic maps is that the union of the time-varying graphs contains a spanning tree.

Improving Image Encryption Using Multi-Chaotic Map

Abstract: Chen G et al. presented a 3D chaotic cat map based symmetric image encryption method and Kai W et al. broke it only with the knowledge of symbolic dynamics and some specially designed plain-images. In this letter we point out why the original scheme was vulnerable to the proposed attacks. Some essential weaknesses in the original scheme are discussed here. Based on the analysis, an improved encryption scheme with multi-chaotic map is presented to achieve higher security. In our proposed method, the 3D cat map with random scan processing is developed to increase the confusion process; the combinations of coupled map lattice (CML) model, tent map and logistic chaotic map are employed in the diffusion process to improve the initial sensitivity and security of the encryption system. In addition, entropy is used to evaluate the performance of the proposed encryption system. Experimental results show that the presented method can resist various kinds of attacks and provide an effective way for high security image encryption.

A minimal 2-d quadratic map with quasi-periodic route to chaos

Abstract: The aim of this paper is to present and analyze a minimal chaotic map of the plane, then we describe in detail the dynamical behavior of this map, along with some other dynamical phenomena.

Pareto and Boltzmann Gibbs behaviors in a deterministic multi-agent system

Abstract: a b s t r a c t A deterministic system of interacting agents is considered as a model for economic dynamics. The dynamics of the system is described by a coupled map lattice with nearest neighbor interactions. The evolution of each agent results from the competition between two factors: the agent’s own tendency to grow and the environmental influence that moderates this growth. Depending on the values of the parameters that control these factors, the system can display Pareto or Boltzmann‐Gibbs statistical behaviors in its asymptotic dynamical regime. The regions where these behaviors appear are calculated on the space of parameters of the system. Other statistical properties, such as the mean wealth, the standard deviation, and the Gini coefficient characterizing the degree of equity in the wealth distribution are also calculated.

Stabilizing unstable fixed points of chaotic maps via minimum entropy control

Abstract: In this paper the problem of chaos control in nonlinear maps using minimization of entropy function is investigated. Invariant probability measure of a chaotic dynamics can be used to produce an entropy function in the sense of Shannon. In this paper it is shown that how the entropy control technique is utilized for chaos elimination. Using only the measured states of a chaotic map the probability measure of the system is numerically estimated and this estimated measure is used to obtain an estimation for the entropy of the chaotic map. The control variable of the chaotic system is determined in such a way that the entropy function descends until the chaotic trajectory of the map is replaced with a regular one. The proposed idea is applied for stabilizing the fixed points of the logistic and the Henon maps as some cases of study. Simulation results show the effectiveness of the method in chaos rejection when only the statistical information is available from the under-study systems.

Transition to chaos in small-world dynamical network

Abstract: The transition from a non-chaotic state to a chaotic state is an important issue in the study of coupled dynamical networks. In this paper, by using the theoretical analysis and numerical simulation, we study the dynamical behaviors of the NW small-world dynamical network consisting of nodes that are in non-chaotic states before they are coupled together. It is found that, for any given coupling strength and a sufficiently large number of nodes, the small-world dynamical network can be chaotic, even if the nearest-neighbor coupled network cannot be chaotic under the same condition. More interesting, the numerical results show that the measurement 1 R of the transition ability from non-chaos to chaos approximately obeys power-law forms as 1 R ∼ p - r 1 and 1 R ∼ N - r 2 . Furthermore, based on dissipative system criteria, we obtain the relationship between the network topology parameters and the coupling strength when the network is stable in the sense of Lyapunov (i. s. L.).

Novel tracking function of moving target using chaotic dynamics in a recurrent neural network model

Abstract: Chaotic dynamics introduced in a recurrent neural network model is applied to controlling an object to track a moving target in two-dimensional space, which is set as an ill-posed problem. The motion increments of the object are determined by a group of motion functions calculated in real time with firing states of the neurons in the network. Several cyclic memory attractors that correspond to several simple motions of the object in two-dimensional space are embedded. Chaotic dynamics introduced in the network causes corresponding complex motions of the object in two-dimensional space. Adaptively real-time switching of control parameter results in constrained chaos (chaotic itinerancy) in the state space of the network and enables the object to track a moving target along a certain trajectory successfully. The performance of tracking is evaluated by calculating the success rate over 100 trials with respect to nine kinds of trajectories along which the target moves respectively. Computer experiments show that chaotic dynamics is useful to track a moving target. To understand the relations between these cases and chaotic dynamics, dynamical structure of chaotic dynamics is investigated from dynamical viewpoint.

Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors

Abstract: The reaction-diffusion equation for the Brusselator model produces a coupled map lattice (CML) by discretization. The two-dimensional nonlinear local map of this lattice has rich and interesting dynamics. In [7] we studied the dynamics of the local map, focusing on trajectories escaping to infinity, and the Julia set. In this paper we build a correspondence between CML and its local map via traveling waves, and then using this correspondence we study asymptotic properties of this CML. We show the existence of a bounded region in which every trajectory in the Julia set is eventually trapped. We also find a region where every bounded trajectory visits. Finally, we present some strange attractors that are numerically observed in the Julia set.

Generalized synchronization of chaos in autonomous systems.

Abstract: We extend the concept of generalized synchronization of chaos, a phenomenon that occurs in driven dynamical systems, to the context of autonomous spatiotemporal systems. It means a situation where the chaotic state variables in an autonomous system can be synchronized to each other, but not to a coupling function defined from them. The form of the coupling function is not crucial; it may not depend on all the state variables. Nor does it need to be active for all times for achieving generalized synchronization. The procedure is based on an analogy between a response map subject to an external drive acting with a probability p and an autonomous system of coupled maps where a global interaction between the maps takes place with this same probability. It is shown that, under some circumstances, the conditions for stability of generalized synchronized states are equivalent in both types of systems. Our results reveal the existence of similar minimal conditions for the emergence of generalized synchronization of chaos in driven and in autonomous spatiotemporal systems.

Chaotic Dynamics in Hybrid Systems

Abstract: In this paper we give an overview of some aspects of chaotic dynamics in hybrid systems, which comprise different types of behaviour Hybrid systems may exhibit discontinuous dependence on initial conditions leading to new dynamical phenomena We indicate how methods from topological dynamics and ergodic theory may be used to study hybrid systems, and review existing bifurcation theory for one-dimensional non-smooth maps, including the spontaneous formation of robust chaotic attractors We present case studies of chaotic dynamics in a switched arrival system and in a system with periodic forcing

Cascades with coupled map lattices in preferential attachment community networks

Abstract: In this paper, cascading failure is studied by coupled map lattice (CML) methods in preferential attachment community networks. It is found that external perturbation R is increasing with modularity Q growing by simulation. In particular, the large modularity Q can hold off the cascading failure dynamic process in community networks. Furthermore, different attack strategies also greatly affect the cascading failure dynamic process. It is particularly significant to control cascading failure process in real community networks.

Chaotic control of the coupled Logistic map

Abstract: Based on the stability criterion of discrete systems, state feedback is used to stabilize unstable low-periodic orbits of the coupled Logistic map, and a new scheme is proposed to change the parameter value of the first bifurcation point of this dynamic system optionally. Numerical simulations show the effectiveness of our methods.

Map Lattices coupled by collisions

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.

The identification of complex spatiotemporal patterns using coupled map lattice models

Abstract: Many complex and interesting spatiotemporal patterns have been observed in a wide range of scientific areas. In this paper, two kinds of spatiotemporal patterns including spot replication and Turing systems are investigated and new identification methods are proposed to obtain Coupled Map Lattice (CML) models for this class of systems. Initially, a new correlation analysis method is introduced to determine an appropriate temporal and spatial data sampling procedure for the identification of spatiotemporal systems. A new combined Orthogonal Forward Regression and Bayesian Learning algorithm with Laplace priors is introduced to identify sparse and robust CML models for complex spatiotemporal patterns. The final identified CML models are validated using correlation-based model validation tests for spatiotemporal systems. Numerical results illustrate the identification procedure and demonstrate the validity of the identified models.

Neighborhood Detection for the Identification of Spatiotemporal Systems

Abstract: Neighborhood detection and local state vector construction for the identification of spatiotemporal systems is considered in this paper Determining the neighborhood size both in the space and time domain can considerably reduce the complexity of the set of candidate model terms for the identification of coupled map lattice models The computation requirements of the model identification algorithm can also be greatly reduced instead of the more direct identification approach of searching over the entire spatiotemporal neighborhood in the original space In this paper, a new neighborhood detection method is introduced based on embedding theory for nonlinear dynamical systems to produce an initial spatiotemporal neighborhood for the identification of spatiotemporal systems Numerical examples are provided to demonstrate the feasibility and applicability of the new neighborhood detection method

Generalized N-coupled maps with invariant measure in Bose-Mesner algebra perspective

Abstract: 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 E 0 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.

The Complex Spatiotemporal Dynamics of a Shallow Fluid Layer

Abstract: The nonlinear and chaotic dynamics of a shallow fluid layer are investigated numerically using large-scale parallel numerical simulations. Two particular situations are studied in detail. First, a fluid layer is placed between rigid boundaries and heated from below to yield the chaotic dynamics of thermal convection rolls (Rayleigh-Bénard convection). Second is a free-surface fluid layer placed on a shaker to yield nonlinear surface waves (Faraday waves). In both cases the full governing partial differential equations are solved using parallel spectral element methods. Rayleigh-Bénard convection is studied in a cylindrical dish with realistic boundaries. The complete flow field is obtained as well as the spectrum of Lyapunov exponents and Lyapunov vectors. The Lyapunov exponents and their corresponding perturbation fields are used to determine when and where events occur that contribute most to the chaotic dynamics. Roll pinch-off and roll mergers are found to be the largest contributors. Two dimensional and three dimensional Faraday waves are studied with periodic boundary conditions. The full Navier-Stokes equations are solved including the complex dynamics of the free surface waves to gain a better understanding of the interplay between the viscous boundary layers, the nonlinear streaming flow, and the bulk flow. The vortices in the bulk flow are weak compared to the flow in the viscous boundary layers. The surface waves are found to be non-sinusoidal and the time evolution of the waves are explored for both large and small amplitude waves. iii Acknowledgments I would first and foremost like to thank my advisor Dr. Mark Paul for giving me this wonderful research opportunity. It was a great learning experience and I am very privileged to have worked with him. I cherish all of the encouragement, guidance, and thought provoking discussions we have had during my time at Virginia Tech. I want to thank my parents Audrey and Jon. Without them and their support, none of this would have been possible. I also want to thank my significant other and best friend Rachel for all of her inspirations and encouragement. Thanks go to my committee members Dr. Tafti and Dr. Iliescu for their support of this research. I would finally like to acknowledge all of my friends, classmates, and labmates for their help and support.

The characteristics of nonlinear chaotic dynamics in quantum cellular neural networks

Abstract: With the polarization of quantum-dot cell and quantum phase serving as state variables, this paper does both theoretical analysis and simulation for the complex nonlinear dynamical behaviour of a three-cell-coupled Quantum Cellular Neural Network (QCNN), including equilibrium points, bifurcation and chaotic behaviour. Different phenomena, such as quasi-periodic, chaotic and hyper-chaotic states as well as bifurcations are revealed. The system's bifurcation and chaotic behaviour under the influence of the different coupling parameters are analysed. And it finds that the unbalanced cells coupled QCNN is easy to cause chaotic oscillation and the system response enters into chaotic state from quasi-periodic state by quasi-period bifurcation; however, the balanced cells coupled QCNN also can be chaotic when coupling parameters is in some region. Additionally, both the unbalanced and balanced cells coupled QCNNs can possess hyper-chaotic behaviour. It provides valuable information about QCNNs for future application in high-parallel signal processing and novel ultra-small chaotic generators.

Parameter estimation for systems with peak-to-peak dynamics

Abstract: A parameter estimation method for chaotic systems with low-dimensional chaos is proposed. Such systems display peak-to-peak dynamics, namely their chaotic dynamics can approximately (but accurately...

The synchronization of spatiotemporal chaos of unilateral coupled map lattice

Abstract: Coupled map lattices are taken as examples to study synchronization of spatiotemporal chaotic systems. Based on Lyapunov stability theory, global synchronization of two unilateral coupled map lattices is realized through appropriate choice of driving functions. Simulation results show the effectiveness of the method. The effect of control parameters on the rate of synchronization is further discussed. Simulation results also show that synchronization can also be realized when there is a systematic bias and system noise, which shows that the method has certain anti-jamming capability.

Collective behavior in coupled chaotic map lattices with random perturbations

Abstract: Numerical simulations of coupled map lattices with non-local interactions (i.e., the coupling of a given map occurs with all lattice sites) often involve a large computer time if the lattice size is too large. In order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution. This emulates a “thermal bath”, where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude. We demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied. We considered one-dimensional lattices of chaotic logistic maps at outer crisis x → 4 x ( 1 − x ) .

Coexistence of synchronized and desynchronized patterns in coupled chaotic dynamical systems

Abstract: We analyse the synchronous chaos and its stability in coupled map lattices and coupled nonlinear oscillators. The existence of desynchronized patterns such as standing wave like spatiotemporal periodic patterns within the critical size limit is identified for some sets of random initial conditions. The emergence of such patterns is explained using the concept of controlling of chaos. In the case of coupled map lattices, an expression for standing wave like pattern is given in terms of unstable periodic orbits of the isolated map. Finally, the role of initial conditions on the existence of multiple stable states in both coupled map lattices and coupled oscillators is studied.