# Dynamic Properties of Coupled Maps

Authors: Chunrui Zhang; Huifeng Zheng
Journal: Discrete Dynamics in Nature and Society
 (2010)

---

## 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.


---

## Body

## 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 [2–7]. 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(1.1)xn+1=axn+αf(xn)+β[xn-k-yn-k]+β[xn-k-zn-k],yn+1=ayn+αf(yn)+β[yn-k-zn-k]+β[yn-k-xn-k],zn+1=azn+αf(zn)+β[zn-k-xn-k]+β[zn-k-yn-k],
where α,β,anda 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 Sections2 and 3, we show that the structure of system (1.1) can be represented by a dihedral group D3. 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.D3-Equivariant and Linear Stability of Coupled Maps

Assume thatXn=(xn,yn,zn)T. Equation (1.1) can be rewritten as(2.1)Xn+1=(a000a000a)Xn+(α000α000α)(f((Xn)1)f((Xn)2)f((Xn)3))+(2β-β-β-β2β-β-β-β2β)((Xn-k)1(Xn-k)2(Xn-k)3).
Throughout this section, to establish the main results for (1.1), we make the following hypothesis on the activation functions in (2.1):(H1)
:f:R→R is a C1-smooth function with f(0)=0,f′(0)=1.Under the assumptions(H1), 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:(2.2)Yn+1=(a+α000a+α000a+α)Yn+(2β-β-β-β2β-β-β-β2β)Yn-k.Lemma 2.1.
Both systems (2.1) and (2.2) are D3-equivariant where D3 is the dihedral group of order 6.Proof.
D3 can be generated by matrices
(2.3)P=(010001100),Q=(-1000-1000-1).
Hence,
(2.4)D3={I3,P,P2,Q,QP,QP2}.ForallG∈D3, we have
(2.5)G((a000a000a)Xn+(2β-β-β-β2β-β-β-β2β)Xn-k+(α000α000α)(f((Xn)1)f((Xn)2)f((Xn)3)))=(a000a000a)GXn+(2β-β-β-β2β-β-β-β2β)GXn-k+(α000α000α)(f((GXn)1)f((GXn)2)f((GXn)3)),G((a+α000a+α000a+α)Yn+(2β-β-β-β2β-β-β-β2β)Yn-k)=(a+α000a+α000a+α)GYn+(2β-β-β-β2β-β-β-β2β)GYn-k.
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 Mn+1=AMn, where(2)Mn=(xn,xn-1,xn-2,…,xn-k,yn,yn-1,yn-2,…,yn-k,zn,zn-1,zn-2,…,zn-k)T,A=(R1R2R2R2R1R2R2R2R1)(3k+3)×(3k+3),R1=(a+α0⋯⋯02β10⋯⋯⋯0010⋯⋯00010⋯0⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯)(k+1)×(k+1),R2=(00⋯⋯0-β00⋯⋯⋯0000⋯⋯00000⋯0⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯)(k+1)×(k+1).
The characteristic  equation for (2.2) is given by
(2.7)|λE-A|=[λk+1-(a+α)λk][λk+1-(a+α)λk-3β]2=Δ1Δ22=0,
where
(2.8)Δ1=λk+1-(a+α)λk,Δ2=λk+1-(a+α)λk-3β.
The equationΔ1=0 has a k-fold root λ=0 and a simple root λ=a+α. Assuming that a+α<1, we only need to analyze the distribution of the roots of Δ2=0.
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 a,α,andβ are varied.Theorem 2.2.
Leta+α<1.(1)
The zero solution of (2.1) is local asymptotically stable if (a,α)∈E={(a,α)∣[(a+α-1)2-9β2]≥2|a+α|,a+α<1,a,α∈R}.(2)
Equation (2.1) has equivariant bifurcation at the origin:
(1∘)
ifβj=[1+(a+α)2-2(a+α)cosω]1/2/3(j=0,1,…,k-1), then D3-equivariant Hopf bifurcation occurs;(2∘)
ifa+α+3β=1, then the D3-equivariant pitchfork bifurcation occurs;(3∘)
if1+a+α=(-1)k+13β, then the D3-equivariant doubling bifurcation occurs.Proof.
ConsiderΔ2=0.(1)
It is clears thatΔ2=0 has roots λ=1 if a+α+3β=1 and λ=-1 if 1+a+α=(-1)k+13β.(2)
It is observed that, ifβ=0, then Δ2=0 has simple root λ1=a+α with |λ1|<1 and λ2=0 with multiplies k.(3)
Letλ=eiω be a root of Δ2=0, then we have
(2.9)βj=[1+(a+α)2-2(a+α)cosω]1/23(j=0,1,…,k-1),sin(k+1)ω-e-asinkω=0,ω∈(jπk,(j+1)πk+1)forj=0,1,…,k-1.(4)
One, moreover, has(2.10)dλdβ|β=βj=2(k+1)sinω-2sin(k+1)ωcosωsinkω|(k+1)eiωk-(a+α)keiω(k+1)|2>0.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).Letβj=([1+(a+α)2-2(a+α)cosω]1/2/3)(j=0,1,…,k-1). We will consider the periodic solutions of (2.1).Assume that(3.1)g:R2n×R→R2n
and that (Dg)(0,0) has eigenvalues e±2πiθ, each with multiplicity n, where θ≠0,1/2.Denote bySPT the subspace of PT consisting of all T-periodic solution of (2.1) under condition (2.4). Let Σ be a subgroup and let (Σ,SPT) be fixed-point subspace of SPT.Lemma 3.1 (see [8]).
LetΣ be a subgroup such that dimFix(Σ,SPT)=2. Assume that the eigenvalues cross the unit circle with nonzero speed. Then generically there exists a unique branch of g-invariant circles emanating from the trivial fixed point x=0, and this branch is tangent to Fix(Σ)⊂R2n×R at x=0.Consider the subgroup ofD3: (3.2)Σ1={I9,P,P2},Σ1=(P,θ(t)){P,θ(ωt)=ωt-2π3},Σ2={I9,Q},Σ2=(Q,θ(t)){Q,θ(ωt)=ωt},Σ3={I9,QP},Σ3=(QP,θ(t)){QP,θ(ωt)=ωt-π}.
It is clear that (3.3)(1)dimFix(Σ1x,SPT)=2,(2)dimFix(Σ2x,SPT)=2,(3)dimFix(Σ3x,SPT)=2.System (2.1) is equivariant with respect to the D3-action where the subgroups Σ1, Σ2, and Σ3 of D3 act by permutation (sending xi to xi+1) and by interchanging (sending xi to x3-i).Applying Lemma3.1 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 thatβj is defined as in (2.9). When near β=βj, there exist eight branches of asynchronous periodic solutions of period PT near 2π/ω, bifurcated simultaneously from the zero solution of system (2.1), and these are(1)
two phase-locked oscillations,xi(n)=xi-1(n±PT/3)=xi-1(n±2π/3ω);(2)
three mirror-reflecting waves,xi(n)=xj(n)≠xk(n);(3)
three standing,xj(n)=xi(n+PT/2)=xi(n+π/ω).To illustrate analytical results found, let us consider the following example:(3.4)xn+1=axn+αtanh(xn)+β(xn-2-yn-2)+β(xn-2-zn-2),yn+1=ayn+αtanh(yn)+β(yn-2-zn-2)+β(yn-2-xn-2),zn+1=azn+αtanh(zn)+β(zn-2-xn-2)+β(zn-2-yn-2).Firstly, we choose parameters such that (2.9) met. When a=0.5,andα=0.48,β=-0.18, we have the result that the origin of (3.4) is local asymptotically stable. See Figure 1.The origin is local asymptotically stable for parametersa=0.5,α=0.48,andβ=-0.18.
(a)(b)(c)Next, one possible choice of the parameters isa=0.5,α=0.48,andβ=-0.21. In this case, multiple branches of asynchronous periodic solutions appear. These solutions are shown in Figures 2-3.Completely symmetrical periodic solution of(xn,yn); antisymmetrical periodic solutions (xn,zn),(yn,zn).
(a)(b)(c)Three waveforms have2π/3ω out of phase.
(a)(b)(c)It is shown that, in Figures1, 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, whena=0.6,α=0.66,andβ=-0.21, system (3.4) has coexistence of chaotic attractors. See Figures 4, 5, 6, 7.Time series for(n,xn);(n,yn);(n,zn).
(a)(b)(c)Figure 5
Phase portrait of the chaotic attractor(xn,yn).Figure 6
Phase portrait of the chaotic attractor(xn,zn).Figure 7
Phase portrait of the chaotic attractor(yn,zn).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 fora=0.6,α=0.66,andβ=-0.21 are calculated by means of Matlab and are illustrated in Figure 8.Figure 8
Lyapunov exponents of system (3.4) when a=0.6,α=0.66,β=-0.21.


## 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.



---

*Source: 905102-2010-09-02.xml*