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Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 905102, 10 pages
http://dx.doi.org/10.1155/2010/905102
Research Article

Dynamic Properties of Coupled Maps

1Department of Mathematics, Northeast Forestry University, Harbin 150040, China
2Department of Electrical and Computer Engineering, Pratt School of Engineering, Duke University, Durham, NC 27708, USA

Received 24 March 2010; Revised 16 July 2010; Accepted 8 August 2010

Academic Editor: Elena Braverman

Copyright © 2010 Chunrui Zhang and Huifeng Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Dynamic properties are investigated in the coupled system of three maps with symmetric nearest neighbor coupling and periodic boundary conditions. The dynamics of the system is controlled by certain coupling parameters. We show that, for some values of the parameters, the system exhibits nontrivial collective behavior, such as multiple bifurcations, and chaos. We give computer simulations to support the theoretical predictions.

1. Introduction

Coupled maps is one of the most interesting topics on spatial extended systems. Bifurcation and chaos of coupled maps generated by iterated maps of nonlinear difference equations have attracted considerable attention in both theoretical studies and practical applications [1]. When many identical maps are coupled to a larger system (coupled map lattices or CML), the system can exhibit nontrivial collective behavior, such as stability, multiple bifurcations, and chaos [27]. In this paper, we extend this work by considering three-dimensional discrete dynamical systems which arise from three coupled one-dimensional maps with delay. In general, delayed coupled maps of three maps with symmetric nearest neighbor coupling and periodic boundary conditions can be described as where are parameters.

Symmetry breaking Hopf bifurcations from steady states to periodic orbits can occur in systems with some symmetry. The coexistence of more chaotic attractors in phase space is phenomenon that has been proven to exist in many fields of science. There are many researches of equivariant bifurcations of ODEs. We refer the readers to the study by Golubitsky et al. in [8]. But, until now, there are fewer papers to discuss equivariant bifurcation problems in maps, which motivates us to write this paper. The goal of this paper is to investigate how parameters affect coupled maps with delay (1.1) by using the symmetric groups theory of Chossat and Golubitsky [9].

Accordingly, the paper is organized as follows. In Sections 2 and 3, we show that the structure of system (1.1) can be represented by a dihedral group . The generalized center subspace is invariant under the action of the symmetry group, and the center manifold reduction can be performed in such a way that the reduced equations commute with the restricted action of the symmetry group. We obtain some important results about the stability and chaos and spontaneous bifurcations of multiple branches of periodic solutions and their spatiotemporal patterns, which describe the oscillatory mode of each oscillator. Finally, some numerical simulations are carried out to support the analysis results, and the existence of chaotic attractors is also exhibited numerically.

2. -Equivariant and Linear Stability of Coupled Maps

Assume that Equation (1.1) can be rewritten as Throughout this section, to establish the main results for (1.1), we make the following hypothesis on the activation functions in (2.1): is a -smooth function with

Under the assumptions , the origin (0,0,0) is an equilibrium of (1.1). Linearizing system (2.1) about origin (0,0,0) gives the following linear system:

Lemma 2.1. Both systems (2.1) and (2.2) are -equivariant where is the dihedral group of order .

Proof. can be generated by matrices Hence, , we have
For discussing the linear stability and Hopf bifurcations of (2.1), we need to consider the characteristic equation for (2.2). We can induce the method of Zhang and Zheng [10] to mark , where The characteristic equation for (2.2) is given by where
The equation has a -fold root and a simple root . Assuming that , we only need to analyze the distribution of the roots of .
In what follows, the analysis on the distribution of the roots to (2.7) is based on the conclusion given in [10]: the sum of the order of the zeros of (2.7) can change only if a zero appears or accesses the unit circle as parameters are varied.

Theorem 2.2. Let .The zero solution of (2.1) is local asymptotically stable if .Equation (2.1) has equivariant bifurcation at the origin:if , then -equivariant Hopf bifurcation occurs; if , then the -equivariant pitchfork bifurcation occurs;if , then the -equivariant doubling bifurcation occurs.

Proof. Consider .It is clears that has roots if and if .It is observed that, if , then has simple root with and with multiplies .Let be a root of , then we have One, moreover, has Using the theorem of [10], the conclusions are obtained.

3. Multiple Bifurcations

Recently, there has been an increasing interest in the multistability problem in dynamical systems theory. The purpose of this section is to explore the coexistence of multiple stable patterns such as multiple periodic orbits of maps (2.1).

Let . We will consider the periodic solutions of (2.1).

Assume that and that has eigenvalues each with multiplicity , where

Denote by the subspace of consisting of all -periodic solution of (2.1) under condition (2.4). Let be a subgroup and let be fixed-point subspace of .

Lemma 3.1 (see [8]). Let be a subgroup such that Assume that the eigenvalues cross the unit circle with nonzero speed. Then generically there exists a unique branch of -invariant circles emanating from the trivial fixed point , and this branch is tangent to at .

Consider the subgroup of : It is clear that

System (2.1) is equivariant with respect to the -action where the subgroups , , and of act by permutation (sending to ) and by interchanging (sending to ).

Applying Lemma 3.1 of the symmetric Hopf bifurcation theorem for maps, eight branches of asynchronous periodic solutions are obtained due to their corresponding two-dimensional fixed-point subspace. More precisely, we have the following theorem.

Theorem 3.2. Assume that is defined as in (2.9). When near , there exist eight branches of asynchronous periodic solutions of period near , bifurcated simultaneously from the zero solution of system (2.1), and these aretwo phase-locked oscillations, ;three mirror-reflecting waves, ;three standing,

To illustrate analytical results found, let us consider the following example:

Firstly, we choose parameters such that (2.9) met. When we have the result that the origin of (3.4) is local asymptotically stable. See Figure 1.

fig1
Figure 1: The origin is local asymptotically stable for parameters

Next, one possible choice of the parameters is In this case, multiple branches of asynchronous periodic solutions appear. These solutions are shown in Figures 2-3.

fig2
Figure 2: Completely symmetrical periodic solution of antisymmetrical periodic solutions
fig3
Figure 3: Three waveforms have out of phase.

It is shown that, in Figures 1, 2, and 3 for different values of parameters, system (3.4) exhibits its different dynamics. At first, the trivial solution is local asymptotically stable, then loses its stability, and several different periodic patterns described by the equivariant Hopf bifurcation can be observed, which depend on different values of parameters.

4. Chaos and Lyapunov Exponents

The presence of chaos in discrete dynamic systems is well known. In this section, the numerical method is used to indicate the chaotic behavior of system (3.4). Analytic approach is usually limited to some simple cases in studying nonlinear systems, and numerical simulation by computer is more efficient sometimes. Therefore, we study the chaos of (3.4) by numerical methods.

We notice that, when system (3.4) has coexistence of chaotic attractors. See Figures 4, 5, 6, 7.

fig4
Figure 4: Time series for
905102.fig.005
Figure 5: Phase portrait of the chaotic attractor
905102.fig.006
Figure 6: Phase portrait of the chaotic attractor
905102.fig.007
Figure 7: Phase portrait of the chaotic attractor

Lyapunov exponents are a quantitative measure for distinguishing among the various types of orbits based upon their sensitive dependence on the initial conditions and are used to determine the stability of any steady-state behavior, including chaotic solutions.

The Lyapunov exponents for are calculated by means of Matlab and are illustrated in Figure 8.

905102.fig.008
Figure 8: Lyapunov exponents of system (3.4) when .

5. Conclusions

In this paper we have shown a rich dynamics in a new class coupled maps described by (1.1). In particular, we have demonstrated that system (1.1) with very simple connection matrices can exhibit stability-to Multiple bifurcations-to-chaos when a parameter varies. These properties are similar to those in symmetric delayed differential equations. It is expected that finding more such simple coupled maps would be helpful for studying the role of symmetry in discrete system.

Acknowledgments

The authors thank the referees for their many valuable suggestions that lead to a significant improvement of their paper. C. Zhang author was supported by the National Natural Science Foundations of China (no. 10771045; no. 10871056).

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