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Discrete Dynamics in Nature and Society
Volume 2011 (2011), Article ID 174376, 16 pages
http://dx.doi.org/10.1155/2011/174376
Research Article

A Proof for the Existence of Chaos in Diffusively Coupled Map Lattices with Open Boundary Conditions

1School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, China
2Department of Applied Mathematics, South China Agricultural University, Guangzhou 510640, China

Received 2 July 2011; Accepted 13 September 2011

Academic Editor: Recai Kilic

Copyright © 2011 Li-Guo Yuan and Qi-Gui Yang. 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

We first study how to make use of the Marotto theory to prove rigorously the existence of the Li-Yorke chaos in diffusively coupled map lattices with open boundary conditions (i.e., a high-dimensional discrete dynamical system). Then, the recent 0-1 test for chaos is applied to confirm our theoretical claim. In addition, we control the chaotic motions to a fixed point with delay feedback method. Numerical results support the theoretical analysis.

1. Introduction

Extensive research has been carried out to discover complex behaviors of various discrete dynamical systems in the past several decades. However, limited rigorous analysis concerning existence of chaos in high-dimensional discrete dynamical systems has been seen in the literature. Since the 1980s, coupled map lattices (CMLs) as high-dimensional discrete system have caused widespread concern [1]. CMLs as chaotic dynamical system models for spatiotemporal complexity are usually adopted. Spatiotemporal complexity is common in nature, such as biological systems, networks of DNA, economic activities, and neural networks [1]. The complex behaviors of CMLs have been studied extensively [116]. These mainly include bifurcation [2], chaos [6, 7], chaotic synchronization [4, 810], and controlling chaos [5, 11, 12]. However, being able to rigorously prove the existence of chaos in CMLs is an important and open question. A rigorous verification of chaos will provide a theoretical foundation for the researchers to discover the complex behaviors in CMLs. Recently, Li et al. [13, 14] theoretically analyzed the chaos in one-way coupled logistic lattice with periodic boundary conditions and presented a chaotification method for creating spatiotemporal systems strongly chaotic. Tian and Chen [15] discussed the chaos in CMLs with the new chaos definition in the sense of Li-Yorke. These CMLs with the periodic boundary conditions have been most extensively investigated [1, 2, 415]. But, in all of the research so far published, only a few studies have attempted to explore the case of open boundary conditions [16, 17]. In this case, it is almost impossible to obtain all eigenvalues of Jacobian matrix of the CMLs. This partially hindered early research in the CMLs with open boundary conditions.

Until now, the rigorous proof of chaos has not yet been studied in diffusively coupled map lattices (DCMLs) with open boundary conditions, which is one important case of CMLs. Inspired by the ideas of [13, 14, 18, 19], we have tried to answer this question. The DCML is as follows [1, 16, 17]:𝑥𝑛+1𝑥(𝑖)=(1𝜖)𝑓𝑛+𝜖(𝑖)2𝑓𝑥𝑛𝑥(𝑖1)+𝑓𝑛,(𝑖+1)(1.1) where 𝑛 is discrete time step and 𝑖 is lattice point (𝑖=1,2,,𝑁; 𝑁 is the number of the sites in the DCML). 𝜖(0,1) is the coupling strength. 𝑥𝑛(𝑖) represents the state variable for the 𝑖th site at time 𝑛. Throughout this paper, we adopt open boundary conditions [16, 17]: 𝑥𝑛+1𝑥(1)=(1𝜖)𝑓𝑛𝑥(1)+𝜖𝑓𝑛,𝑥(2)𝑛+1(𝑥𝑁)=𝜖𝑓𝑛(𝑥𝑁1)+(1𝜖)𝑓𝑛(.𝑁)(1.2) Here each of the lattice points in (1.1) and (1.2) is chosen to be the logistic map 𝑓(𝑥𝑛(𝑖))=1𝑎𝑥2𝑛(𝑖), where 𝑎(0,2] and 𝑥𝑛(𝑖)(1,1). The logistic function 𝑓(𝑥)=1𝑎𝑥2 is equivalent to the well-known form 𝑔(𝑧)=𝑟𝑧(1𝑧) [20] when the transformations 𝑎=𝑟(𝑟2)/4 and 𝑥=2(2𝑧1)/(𝑟2)are taken. This simple quadratic iteration was only completely understood in the late 1990s [21]. When the lattice points are logistic functions, the CMLs generate more rich and complex dynamic behaviours. What is more is that the dynamical behaviors of CMLs may be different from each other when the lattice points are chosen from 𝑓(𝑥) and 𝑔(𝑧), respectively [1, 2].

Based on the Marotto theory [22, 23], we prove theoretically the existence of the Li-Yorke chaos in the DCML (1.1). In the process of proving, the most difficult problem is how to find a snap-back repeller. At the same time, we have exploited different measures such as the chaotic phase, bifurcation diagram, and 0-1 test on time series to confirm our claim of the existence of chaos. The 0-1 test is a new method to distinguish chaotic from ordered motion. It is more suitable to handle high-dimensional systems and does not require phase space reconstruction. Finally, we control spatiotemporal chaotic motion in the DCML (1.1) to period-1 orbit (fixed point) by delay feedback and obtain the stability conditions of control.

The paper is organized as follows. In Section 2, the Marotto theorem is introduced. In Section 3.1, a mathematically rigorous proof of the Li-Yorke chaos in the DCML (1.1) is examined. In Section 3.2, we show numerical simulation results. In Section 3.3, 0-1 test method is used to verify the existence of chaos. In Section 4, delay feedback control method is adopted to control chaos. In the last section, conclusions are given.

2. Marotto Theorem

Li and Yorke [24] state that the period-three orbit exhibits chaos in one-dimensional discrete interval map. This is the first precise definition of discrete chaos. This classical criterion for chaos is extended to higher-dimensional discrete systems by Marotto [22]. Marotto considered the following 𝑛-dimensional discrete system: 𝑥𝑘+1𝑥=𝐹𝑘,𝑘=0,1,2,,(2.1) where 𝑥𝑘𝑅𝑛 and 𝐹𝑅𝑛𝑅𝑛 is continuous. Let 𝐵𝑟(𝑥) denote the closed ball in 𝑅𝑛 of radius 𝑟 centered at point 𝑥 and 𝐵0𝑟(𝑥) its interior. Also, let 𝑥 be the usual Euclidean norm of 𝑥 in 𝑅𝑛[22]. Then, if 𝐹 is differentiable in 𝐵𝑟(𝑧), Marotto claimed that in the following, 𝐴𝐵.AAll eigenvalues of the Jacobian 𝐷𝐹(𝑧) of system (2.1) at the fixed point 𝑧 are greater than one in norm.BThere exist some 𝑠>1 and 𝑟>0 such that, for all 𝑥,𝑦𝐵𝑟(𝑧),𝐹(𝑥)𝐹(𝑦)>𝑠𝑥𝑦.

Marotto thought that, if (A) is satisfied, then (B) can be derived, that is, F is expanding in 𝐵𝑟(𝑧) [22]. But, (A) does not always imply (B) with usual Euclidean norm [25]. Chen et al. [26] first pointed out this problem in the Marotto theorem. During the past decade, several papers tried to fix this error ([19, 23, 25, 26] and some references therein).

In 2005, Marotto redefined the definition of snap-back repeller [23]. He pointed out that (A) does imply (B) with some vector norm in 𝑅𝑛 (which depends on 𝐹 and 𝑧). See, for example, the discussion by Hirsch and Smale in [27]. However, we still do not know what the vector norm is in specific issues. In the application of the Marotto theorem, we need to find a suitable vector norm. With this special vector norm, (A) implies (B). The correct Marotto theorem is given as follows.

Definition 2.1 (see [23]). Suppose that 𝑧 is a fixed point of (2.1) with all eigenvalues of 𝐷𝐹(𝑧) exceeding 1 in magnitude, and suppose that there exists a point 𝑥0𝑧 in a repelling neighborhood of 𝑧, such that 𝑥𝑀=𝑧 and det(𝐷𝐹(𝑥𝑘))0 for 1𝑘𝑀, where 𝑥𝑘=𝐹𝑘(𝑥0). Then, 𝑧 is called a snap-back repeller of 𝐹.

Lemma 2.2 (see [23], the Marotto theorem). If 𝐹 has a snap-back repeller, then 𝐹 is chaotic.

At the same time, Shi and Chen [19] presented a modified Marotto theorem as follows.

Lemma 2.3 (see [19]). Consider the n-dimensional discrete system 𝑥𝑘+1𝑥=𝐹𝑘,𝑥𝑘𝑅𝑛,𝑘=0,1,2,,(2.2) where 𝐹 is a map from 𝑅𝑛 to itself. Assume that 𝐹 has a fixed point 𝑥 satisfying 𝑥=𝐹(𝑥).
Assume, moreover, that(1)𝐹(𝑥) is continuously differentiable in a neighborhood of 𝑥, and all eigenvalues of 𝐷𝐹(𝑥) have absolute values large than 1, where 𝐷𝐹(𝑥) is the Jacobian of F evaluated at 𝑥, which implies that there exist an 𝑟>0 and a norm in 𝑅𝑛 such that 𝐹 is expanding in 𝐵𝑟(𝑥), the closed ball of radius 𝑟 centered at 𝑥 in (𝑅𝑛,),(2)𝑥 is a snap-back repeller of F with 𝐹𝑚(𝑥0)=𝑥,𝑥0𝑥, for some 𝑥0𝐵𝑟(𝑥) and some positive integer 𝑚, where 𝐵𝑟(𝑥) is the open ball of radius 𝑟 centered at 𝑥 in (𝑅𝑛,). Furthermore, 𝐹 is continuously differentiable in some neighborhoods of 𝑥0,𝑥1,,𝑥𝑚1, respectively, and det[𝐷𝐹(𝑥𝑗)]0, where 𝑥𝑗=𝐹(𝑥𝑗1) for 𝑗=1,2,,𝑚.
Then, the system (2.2) is chaotic in the sense of Li-Yorke. Moreover, the system (2.2) has positive topological entropy. Here the topological entropy of 𝐹 is defined to be the supremum of topological entropies of 𝐹 restricted to compact invariant sets.

Remark 2.4. The Marotto theorem is a sufficient condition for the Li-Yorke chaos. Lemmas 2.2 and 2.3 have the same effect. But, direct application of the Marotto theorem is not always easy. In most cases, the verification must be carried out with the aid of a computer [28].

3. Proving Chaos and Simulation Verifications

3.1. Proving Chaos

In this subsection, we prove the existence of the Li-Yorke chaos in the DCML (1.1). Lemmas 3.1 and 3.2 will be useful throughout the proof.

Lemma 3.1 (see [29, 30]). For a matrix 𝐴𝑁×𝑁 with eigenvalues 𝜆1,𝜆2,,𝜆𝑁, the determinant of A is equal to 𝑁𝑖=1𝜆𝑖. Denote det(𝐴)=𝑁𝑖=1𝜆𝑖.

Lemma 3.2 (see [29, 30], the Gershgorin circle theorem). Let 𝐴 be an 𝑛×𝑛 matrix, and let 𝑅𝑖 denote the circle in the complex plane with center 𝑎𝑖𝑖 and radius 𝑛𝑗=1,𝑗i|𝑎𝑖𝑗|; that is, 𝑅𝑖=||𝑧𝒞𝑧𝑎𝑖𝑖||𝑛𝑗=1,𝑗𝑖||𝑎𝑖𝑗||,(3.1) where 𝒞 denotes the complex plane. The eigenvalues of 𝐴 are contained within 𝑅=𝑛𝑖=1𝑅𝑖. Moreover, the union of any 𝑘 of these circles that do not intersect the remaining (𝑛𝑘)contains precisely 𝑘 (counting multiplicities) of the eigenvalues.

Theorem 3.3. If 0<𝜖<1/2 and 𝜖 is small enough, 𝑎{𝑎𝑎>(1+2)/21.2071}{𝑎𝑎>(1𝜖)2/(12𝜖)2(1/4)}, and 𝑐=1/(3𝑁+2)𝜖24𝑁𝜖+2𝑁<0.0613/2, then the DCML (1.1) is chaotic in the sense of Li-Yorke.

Proof. We will prove that the DCML (1.1) has a snap-back repeller 𝐱. Rewrite the DCML (1.1) in the vector form as follows: 𝐱𝑘+1𝐱=𝐹𝑘,(3.2) where 𝐱𝑘=[𝑥𝑘(1),𝑥𝑘(2),,𝑥𝑘(𝑁)]𝑇 and 𝑇 denotes the vector (or matrix) transpose. Using Definition 2.1 and Lemma 2.3, we have to verify the following three conditions.(a)𝐱 is a fixed point of 𝐹 and all the eigenvalues of 𝐷𝐹(𝐱) have absolute values larger than 1. Moreover, there exist 𝑟>0 and a norm in 𝑅𝑛 such that 𝐹 is expanding in 𝐵𝑟(𝐱).(b)There exist a 𝐱0𝐵(𝐱,𝑟) and 𝐱0𝐱 such that 𝐹𝑚(𝐱0)=𝐱 for some 𝑚 and 𝑚2.(c)det[𝐷𝐹𝑚(𝐱0)]0.
The proof consists of four steps. The ideas are motivated chiefly by [13, 18, 19].
Step 1. Let 𝐱=[(4𝑎+11)/2𝑎,,(4𝑎+11)/2𝑎]𝑇=𝑧𝟏𝑅𝑁, where 𝑧=(4𝑎+11)/2𝑎, 𝟏=[1,,1]𝑇. Then 𝐱 is a fixed point of the DCML (3.2), that is, 𝐱=𝐹(𝐱). 𝐹(𝐱) is continuously differentiable in 𝐵𝑟(𝐱) for some 𝑟>0. Its Jacobian matrix at 𝐱 is 𝐱𝐷𝐹=(1𝜖)𝑓𝑧𝜖𝑓𝑧𝜖0002𝑓𝑧(1𝜖)𝑓𝑧𝜖2𝑓𝑧0𝜖002𝑓𝑧(1𝜖)𝑓𝑧𝜖2𝑓𝑧𝜖0002𝑓𝑧(1𝜖)𝑓𝑧𝜖2𝑓𝑧000𝜖𝑓𝑧(1𝜖)𝑓𝑧,(3.3) where 𝑓(𝑧)=14𝑎+1<0. We denote 𝐷𝐹(𝐱) by (14𝑎+1)𝑀, where 𝑀=𝜖1𝜖𝜖0002𝜖1𝜖20𝜖002𝜖1𝜖2𝜖0002𝜖1𝜖2000𝜖1𝜖.(3.4) Obviously, 𝑀 is not a circulant matrix. When 𝑁 is large, it will be difficult to calculate all the eigenvalues of the matrix 𝐷𝐹(𝐱). With the Marotto theorem (Lemmas 2.2 and 2.3), we do not need to know the size of eigenvalues and only need to know that the absolute value of eigenvalues is greater than one. According to the Gershgorin circle theorem (Lemma 3.2), all the eigenvalues of 𝐷𝐹(𝐱), 𝜆𝑗(𝑗=1,2,,𝑁), are given by 14𝑎+1𝜆𝑗(14𝑎+1)(12𝜖). Under the conditions of Theorem 3.3, that is, 0<𝜖<0.5 and 𝑎>(1𝜖)2/(12𝜖)2(1/4), the following results are obtained: 1<||𝜆4𝑎+11(12𝜖)𝑗||4𝑎+11,𝑗=1,2,,𝑁,(3.5) that is, all the eigenvalues of 𝐷𝐹(𝐱) are greater (in absolute value) than one. 𝐱 is an expanding fixed point of 𝐹. Therefore, there exist some 𝑟>0 and a special vector norm such that 𝐹 is expanding in 𝐵𝑟(𝐱). That is, for any two distinct points 𝐱, 𝐲𝐵𝑟(𝐱), we have 𝐹(𝐲)𝐹(𝐱)>𝑠𝐲𝐱,(3.6) where 𝑠>1 and 𝐱,𝐲 are sufficiently close to 𝐱. Since 𝐹(𝐲)𝐹(𝐱)=𝐷𝐹(𝐱)(𝐲𝐱)+𝛼, where 𝛼/𝐲𝐱0 as 𝐲𝐱0 [19], specially, 𝐹(𝐱)𝐹(𝐱)=𝐷𝐹(𝐱)(𝐱𝐱)+𝛼. When 𝜀 is small enough, we can prove that the operator 𝐷𝐹(𝐱) is expanding with Frobenius matrix norm 𝐹, where 𝐷𝐹(𝐱)𝐹=(𝑁𝑗=1𝑁𝑖=1𝑎2𝑖𝑗)1/2. With the conditions of Theorem 3.3, we get |(14𝑎+1)(1𝜖)|>1. For any point 𝐱𝐵𝑟(𝐱) and 𝜖 small enough, there exists some 𝑠>1 such that 𝐷𝐹(𝐱)𝐱𝐹=1𝑀𝑥4𝑎+11,𝑥2,𝑥3,,𝑥𝑁2,𝑥𝑁1,𝑥𝑁𝑇𝐹=14𝑎+1(1𝜖)𝑥1+𝜖𝑥2𝜖2𝑥1+(1𝜖)𝑥2+𝜖2𝑥3𝜖2𝑥𝑁2+(1𝜖)𝑥𝑁1+𝜖2𝑥𝑁𝜖𝑥𝑁1+(1𝜖)𝑥𝑁𝐹𝑥𝑠1𝑥2𝑥𝑁1𝑥𝑁𝐹.(3.7) Since 𝐹(𝐱) is continuously differentiable, 𝐷𝐹(𝐱) is also expanding for 𝐱𝐵𝑟(𝐱). Let the bound of the maximal open expanding ball 𝐵𝑟(𝐱) be denoted by 𝜌𝟏, where 𝜌 satisfies the following inequality [13]: 𝐷𝐹(𝜌𝟏)=4𝑎2(1𝜖)2𝑁𝜌2+2𝜖2𝑎2(𝑁2)𝜌2+8𝜖2𝑎2𝜌2>1.(3.8) Moreover, the equation 2𝑎2𝜌2(3𝑁+2)𝜖24𝑁𝜖+2𝑁=1(3.9) has two solutions 𝜌1,21=2𝑎×(3𝑁+2)𝜖2𝑐4𝑁𝜖+2𝑁=,2𝑎(3.10) where 𝑐=1/(3𝑁+2)𝜖24𝑁𝜖+2𝑁. One has 𝑐(0,1) (because 𝑓(𝜖)=(3𝑁+2)𝜖24𝑁𝜖+2𝑁 is a quadratic function, the discriminant Δ=8𝑁216𝑁<0, when 𝜖=4𝑁/2(3𝑁+2), min𝑓(𝜖)=(4(3𝑁+2)2𝑁16𝑁2)/4(3𝑁+2)>1). In fact, 𝑐(1/2𝑁,1/(3/4)𝑁+(1/2)). Since 𝑎>(1+2)/21.2071 and 0<𝜌2<𝑧<1, we take 𝜌=𝜌2=𝑐/2𝑎. Then, 𝑧𝜌<1𝑧, and we denote 𝑟=𝑧𝜌=4𝑎+112𝑐2𝑎>0.(3.11) Thus, condition (a) of Definition 2.1 and Lemma 2.2 is satisfied.Step 2. For all 𝐳=𝑧𝟏𝐵𝑟(𝐱), we have |𝑧𝑧|<𝑟, that is, 𝜎1<𝑧<𝜎2, where 𝜎1=𝑧𝑟=𝑐/2𝑎,𝜎2=𝑧+𝑟=(4𝑎+11(2/2)𝑐)/𝑎. Now let 𝐱=(𝑥1,𝑥2,,𝑥𝑁)𝑇 and 𝐹(𝐱)=𝐱, that is, (1𝜖)1𝑎𝑥21+𝜖1𝑎𝑥22=𝑧,(1𝜖)1𝑎𝑥2𝑖+1+𝜖21𝑎𝑥2𝑖+𝜖21𝑎𝑥2𝑖+2=𝑧,𝜖1𝑎𝑥2𝑁1+(1𝜖)1𝑎𝑥2𝑁=𝑧,(3.12) where 𝑖=1,2,,𝑁2. Summing all the above equations, we obtain 𝜖121𝑎𝑥21+𝜖1+21𝑎𝑥22+𝑁2𝑘=31𝑎𝑥2𝑘+𝜖1+21𝑎𝑥2𝑁1+𝜖121𝑎𝑥2𝑁=𝑁𝑧.(3.13) Assume that (3.12) has a solution, and denote 𝐲1=𝑧1𝟏, that is 𝑁(1𝑎𝑧21)=𝑁𝑧, which has two solutions: 𝑧1=±(1𝑧)/𝑎=±(4𝑎+11)/2𝑎. We choose 𝑧1=(14𝑎+1)/2𝑎 since 𝑧1𝜎1=(14𝑎+12𝑐)/2𝑎<0,thatis,𝑧1<𝜎1 and 𝑧1(𝜎1,𝜎2).Step 3. Now, let 𝐹(𝐱)=𝐲1, that is, (1𝜖)1𝑎𝑥21+𝜖1𝑎𝑥22=𝑧1,(1𝜖)1𝑎𝑥2𝑖+1+𝜖21𝑎𝑥2𝑖+𝜖21𝑎𝑥2𝑖+2=𝑧1,𝜖1𝑎𝑥2𝑁1+(1𝜖)1𝑎𝑥2𝑁=𝑧1,(3.14) where 𝑖=1,2,,𝑁2. Summing the above 𝑁 equations, we get 𝜖121𝑎𝑥21+𝜖1+21𝑎𝑥22+𝑁2𝑘=31𝑎𝑥2𝑘+𝜖1+21𝑎𝑥2𝑁1+𝜖121𝑎𝑥2𝑁=𝑁𝑧1.(3.15) Assume that the system of (3.14) has a solution, and denote 𝐲2=𝑧2𝟏, that is, 𝑁(1𝑎𝑧22)=𝑁𝑧1, that is,1𝑎𝑧22=(14𝑎+1)/2𝑎, which has two solutions: 𝑧2=±(2𝑎+4𝑎+11)/2𝑎2. We take 𝑧2=(2𝑎+4𝑎+11)/2𝑎2. Thus, 𝜎2𝑧2=4𝑎+11𝑐2/2𝑎2𝑎+4𝑎+112𝑎2=8𝑎+22𝑐2𝑎1+4𝑎+1.2𝑎(3.16) Denote 4𝑎+1=𝑡; since 𝑎>(1+2)/2,thatis,3>𝑡>3+222.4142 and 𝑎=(𝑡21)/4, then 8𝑎+22𝑐2𝑎1+4𝑎+1=2𝑡2𝑐𝑡212=+𝑡12𝑡𝑡2+2𝑡322𝑐2.(3.17) Denoting 𝑦(𝑡)=2𝑡𝑡2+2𝑡3, we get 𝑦(𝑡)=2(𝑡+1)/((𝑡+1)24)>0,𝑡(3+22,3). So, 𝑦(𝑡) is monotone increasing continuous function, and min𝑦(𝑡)=23+222(2+3+22)2.0613. We get 2𝑡𝑡2+2𝑡322𝑐0.06132𝑐>0 (since the condition 𝑐<(0.0613/2)). Therefore, 𝑧2<𝜎2. On the other hand, 𝑧2𝜎1=(2𝑎+4𝑎+11)/2𝑎2(𝑐/2𝑎)=(2𝑎+4𝑎+11𝑐)/2𝑎>0,thatis,𝑧2>𝜎1. Thus, 𝜎1<𝑧2<𝜎2,𝐲2𝐵𝑟(𝐱), and 𝑧2𝑧,thatis,𝐲2𝐱. Let 𝐱0=𝐲2, 𝐱1=𝐲1; then, 𝐹2(𝐱0)=𝐱. Steps 2 and 3 complete the proof of condition (b).Step 4. According to 𝐷𝐹(𝐲1)=(4𝑎+11)𝑀𝜔𝑀, where 𝜔=(4𝑎+11)>0, with Lemma 3.2, all eigenvalues of 𝐷𝐹(𝐲1) lie in the interval 0<𝜔(12𝜖)𝜆𝑗𝜔. Thus, with Lemma 3.1, det[𝐷𝐹(𝐲1)]=𝑁𝑗=1𝜆𝑗0. Moreover, according to  𝐷𝐹(𝐲2)=4𝑎+24𝑎+12𝑀Θ𝑀, where Θ=4𝑎+24𝑎+12<0, with Lemma 3.2, all eigenvalues of 𝐷𝐹(𝐲2) lie in the interval Θ𝜆𝑗Θ(12𝜖)<0. Thus, with Lemma 3.1, det[𝐷𝐹(𝐲2)]=𝑁𝑗=1𝜆𝑗0. Then, we have 𝐹𝑚(𝐱0)=𝐱 and det[𝐷𝐹𝑚(𝐱0)]0(𝑚=2). Thus, condition (c) is complete. The system (1.1) has a snap-back repeller 𝐱. Under the conditions of the Theorem 3.3, the DCML (1.1) is chaotic in the sense of Li-Yorke. The proof is completed.

3.2. Numerical Simulation of Chaos

When 𝑁=300, 𝑎=1.8, and 𝜖=0.01, the conditions of Theorem 3.3 are satisfied. The DCML (1.1) can be denoted as follows:𝑥𝑛+1(1)=(1𝜖)1𝑎𝑥2𝑛(1)+𝜖1𝑎𝑥2𝑛,𝑥(2)𝑛+1(2)=(1𝜖)1𝑎𝑥2𝑛+𝜖(2)21𝑎𝑥2𝑛(1)+1𝑎𝑥2𝑛,𝑥(3)𝑛+1(299)=(1𝜖)1𝑎𝑥2𝑛+𝜖(299)21𝑎𝑥2𝑛(298)+1𝑎𝑥2𝑛,𝑥(300)𝑛+1(300)=𝜖1𝑎𝑥2𝑛(299)+(1𝜖)1𝑎𝑥2𝑛.(300)(3.18) The corresponding eigenvalues of 𝐷𝐹(𝐱) lie in the interval (14𝑎+1,(14𝑎+1)(12𝜖)), that is, 𝜆𝑖(1.8636,1.8263)(𝑖=1,2,,300). These eigenvalues are strictly larger than one in absolute value. Starting from a random initial state, the number of iterations is 140. Simulation result is shown in Figure 1. When fixed 𝑁=300, 𝑎=1.8, and 𝜖<0.0582; these satisfy the conditions of Theorem 3.3. Thus, the system (3.18) should display chaotic behavior. The bifurcation diagram in Figure 2 also confirms the above statement.

174376.fig.001
Figure 1: Spatiotemporal chaos in the DCML (3.18) without any control, with parameters 𝑁=300, 𝑎=1.8, 𝜖=0.01.
174376.fig.002
Figure 2: Bifurcation diagram of the DCML (3.18) versus 𝜖(0.01,0.14) and 𝑥(111), with initial point (0.6,,0.6).
3.3. 0-1 Test for Chaos in the DCML

The 0-1 test for chaos was first reported in [31]. It and its modified version are applied directly to the time series data and do not require phase space reconstruction [3136]. Moreover, the dimension and origin of the dynamical system are irrelevant. The 0-1 test can efficiently distinguish chaotic behavior from regular (periodic or quasiperiodic) behavior in deterministic systems. The test result is 0 or 1, depending on whether the dynamics is regular or chaotic, respectively. This method has been successfully applied to some typical chaotic systems [3744] and experiment data [45]. We apply this method to the DCML. From another point of view, we show the existence of chaos in the DCML using the 0-1 test. Now, we describe the implementation of the 0-1 test. The interested reader can consult [35] for further details. Consider discrete data sets 𝜙(𝑛) sampled at times 𝑛=1,2,3,,𝑁, where 𝑁 is the total number of data points. 𝜙(𝑛) is an observable data from the underlying dynamic system.

Step 1. For a random number 𝑐(𝜋/5,4𝜋/5), define the translation variables 𝑝𝑐(𝑛)=𝑛𝑗=1𝜙(𝑗)cos(𝑗𝑐),𝑞𝑐(𝑛)=𝑛𝑗=1𝜙(𝑗)sin(𝑗𝑐).(3.19)

Step 2. Define the mean square displacement 𝑀𝑐(𝑛) as follows: 𝑀𝑐(𝑛)=lim𝑁1𝑁𝑁𝑗=1𝑝𝑐(𝑗+𝑛)𝑝𝑐(𝑗)2+𝑞𝑐(𝑗+𝑛)𝑞𝑐(𝑗)2.(3.20) Note that this definition requires 𝑛𝑁. In practice, 𝑛𝑁/10 yields good results. Denote 𝑛cut=round(𝑁/10), where the function round(𝑥) rounds the elements of 𝑥 to the nearest integers.

Step 3. Define the modified mean square displacement 𝐷𝑐(𝑛)=𝑀𝑐(𝑛)𝑉osc(𝑐,𝑛),(3.21) where 𝑉osc(𝑐,𝑛)=(𝐸𝜙)2(1cos𝑛𝑐)/(1cos𝑐) and 𝐸𝜙=lim𝑁(1/𝑁)𝑁𝑗=1𝜙(𝑗).

Step 4. Form the vectors 𝜉=(1,2,,𝑛cut) and Δ=(𝐷𝑐(1),𝐷𝑐(2),,𝐷𝑐(𝑛cut)). Then define the correlation coefficient 𝐾𝑐[].=corr(𝜉,Δ)1,1(3.22)

Step 5. Steps 14 are performed for 𝑁𝑐 values of 𝑐 chosen randomly in the interval (𝜋/5,4𝜋/5). In practice, 𝑁𝑐=100 is sufficient. We then compute the median of these 𝑁𝑐 values of 𝐾𝑐 to compute the final result 𝐾=median(𝐾𝑐). 𝐾0 indicates regular dynamics, and 𝐾1 indicates chaotic dynamics.

Note that the (𝑝𝑐(𝑛),𝑞𝑐(𝑛))-trajectories provide a direct visual test of whether the underlying dynamics is chaotic or nonchaotic. Namely, bounded trajectories in the (𝑝,𝑞)-plane imply regular dynamics, whereas Brownian-like (unbounded) trajectories imply chaotic dynamics [31, 32]. With the sufficient length of the time series, 𝐾0.1 indicates that the dynamics is regular and 𝐾>0.1 indicates that the dynamics is chaotic [43].

Now, we apply the 0-1 test to the DCML (3.18). Fix 𝑁=300, 𝑎=1.8 and choose a random initial point (𝑥1(1),𝑥1(2),,𝑥1(300)); we carry out the 0-1 test with 𝜖=0.03 and 𝜖=0.12, respectively. Using the data set of 𝑥(111) in the system (3.18), we get 𝐾=0.9981 at 𝜖=0.03 and 𝐾=0.0030 at 𝜖=0.12. The translation variables (𝑝,𝑞) are shown in Figures 3 and 4, respectively.

174376.fig.003
Figure 3: Plot of 𝑝 versus 𝑞 for the DCML (3.18) with 𝜖=0.03. We used 30000 data points of 𝑥(111).
174376.fig.004
Figure 4: Plot of 𝑝 versus 𝑞 for the DCML (3.18) with 𝜖=0.12. We used 30000 data points of 𝑥(111).

We take 𝑁=300,  𝑎=1.8 and let 𝜖 vary from 0.01 to 0.058 in increments of 0.01. It is clear that the computed value of 𝐾 is effective for most values of 𝜖 in Figure 5. These 0-1 test results are consistent with numerical simulation in Section 3.2 and Theorem 3.3 in Section 3.1. Here we stress that the test results (chaos or nonchaos) are independent of the choices of initial point and changing the observable does not greatly alter the computed value of 𝐾.

174376.fig.005
Figure 5: Plot of 𝐾 versus 𝜖 for the DCML (3.18) with 𝜖(0.01,0.058) increased in increments of 0.01. We used 30000 data points of 𝑥(111).

4. Control Spatiotemporal Chaos

When 𝑁=300, 𝜖=0.01, and 𝑎=1.8, the system (3.18) displays chaotic dynamics. The DCML (3.18) has an unstable equilibrium point 𝑋=[(4𝑎+11)/2𝑎,,(4𝑎+11)/2𝑎]𝑇[0.5177,,0.5177]𝑇. The goal of this section is to control spatiotemporal chaotic motions in the DCML (3.18) to the equilibrium point 𝑋 using delay feedback [46, 47]. We rewrite the DCML (3.18) as𝑋𝑛+1𝑋(𝑖)=𝐹𝑛(𝑖),𝑋𝑛(𝑖1),𝑋𝑛(𝑖+1),(4.1) where 𝑋𝑛(𝑖)=(𝑥𝑛(1),𝑥𝑛(2),,𝑥𝑛(300))𝑇.

Theorem 4.1. With the local controllers 𝑈𝑛(i)=𝛼1[𝑋𝑛1(𝑖)𝐹(𝑋𝑛(𝑖),𝑋𝑛(𝑖1),𝑋𝑛(𝑖+1))], the chaotic motion in the DCML (4.1) (i.e., (3.18)) can be controlled to the fixed point 𝑋, where 0.6407<𝛼1<1.

Proof. Since the local controllers are given by 𝑈𝑛(𝑖)=𝛼1𝑋𝑛1𝑋(𝑖)𝐹𝑛(𝑖),𝑋𝑛(𝑖1),𝑋𝑛(𝑖+1),(4.2) we get the controlled DCML: 𝑋𝑛+1𝑋(𝑖)=𝐹𝑛(𝑖),𝑋𝑛(𝑖1),𝑋𝑛(𝑖+1)+𝑈𝑛=(𝑖)1𝛼1𝐹𝑋𝑛(𝑖),𝑋𝑛(𝑖1),𝑋𝑛(𝑖+1)+𝛼1𝑋𝑛1(𝑖),(4.3) where 𝛼1(0.6407,1). Expanding (4.3) around the fixed point 𝑋, we obtain 𝑋𝑛+1𝑋=𝜕𝐹𝜕𝑋𝑛||||𝑋𝑋𝑛𝑋+𝜕𝐹𝜕𝑋𝑛1||||𝑋𝑋𝑛1𝑋.(4.4) Since 𝑥𝑛(𝑖)𝑓[𝑥𝑛1(𝑖)],𝑓(𝑥)=1𝑎𝑥2, and 𝑥=(4𝑎+11)/2𝑎, we have 𝑥𝑛(𝑖)𝑥=𝜕𝑓/𝜕𝑥𝑛1|𝑥(𝑥𝑛1(𝑖)𝑥). Thus, we get 𝑥𝑛1(𝑖)𝑥=(1/(𝜕𝑓/𝜕𝑥𝑛1|𝑥))(𝑥𝑛(𝑖)𝑥) and 𝑋𝑛1𝑋=1𝜕𝑓/𝜕𝑥𝑛1||𝑥𝑋𝑛𝑋.(4.5) Then, by using (4.4) and (4.5), we get 𝑋𝑛+1𝑋=𝜕𝐹𝜕𝑋𝑛||||𝑋𝑋𝑛𝑋+𝜕𝐹𝜕𝑋𝑛1||||𝑋1𝜕𝑓/𝜕𝑥𝑛1||𝑥𝑋𝑛𝑋=𝜕𝐹𝜕𝑋𝑛||||𝑋+1𝜕𝑓/𝜕𝑥𝑛1||𝑥𝜕𝐹𝜕𝑋𝑛1||||𝑋𝑋𝑛𝑋.(4.6) For the sake of simplicity, we denote 𝐽 by 𝜕𝐹/𝜕𝑋𝑛|𝑋+(1/(𝜕𝑓/𝜕𝑥𝑛1|𝑥))(𝜕𝐹/𝜕𝑋𝑛1)|𝑋; then 𝐽=Λ𝑓(𝑥(1))2Θ𝑓(𝑥(2))00Θ𝑓(𝑥(1))Λ𝑓(𝑥(2))Θ𝑓(𝑥(3))00Θ𝑓(𝑥(298))Λ𝑓(𝑥(299))Θ𝑓(𝑥(300))002Θ𝑓(𝑥(299))Λ𝑓(𝑥(300))+𝐽2,(4.7) where 𝐽2=𝛼1𝑓0𝛼(𝑥(1))0001𝑓(𝛼𝑥(2))00001𝑓0𝛼(𝑥(299))0001𝑓(𝑥(300)),(4.8)Θ=(𝜖/2)(𝜖𝛼1/2), Λ=1𝜖𝛼1+𝛼1𝜖, and 𝑓(𝑥(𝑖))=2𝑎𝑥(𝑖). With the Gershgorin circle theorem (Lemma 3.2), we get ||||𝜆𝑖1𝛼1𝛼(1𝜖)𝑓(𝑥(𝑖))+1||||<𝑓(𝑥(𝑖))1𝛼1𝜖𝑓(𝑥(𝑖)),(4.9) that is, 1𝛼1𝑓(𝑥(𝑖))2𝜖1𝛼1𝛼𝑓(𝑥(𝑖))+1𝑓(𝑥(𝑖))<𝜆𝑖<1𝛼1𝛼𝑓(𝑥(𝑖))+1.𝑓(𝑥(𝑖))(4.10) Solving inequality (4.10), we obtain 1.8263+1.2897𝛼1<𝜆𝑖<1.8636+1.3270𝛼1. Since 0.6407<𝛼1<1, we get |𝜆𝑖|<1. The proof is completed.

The simulation result is shown in Figure 6. Chaotic motions are quickly controlled to the fixed point 𝑋[0.5177,,0.5177]𝑇.

174376.fig.006
Figure 6: The control result of the DCML (3.18), with the parameter values 𝑎=1.8, 𝜖=0.01, and 𝛼1=0.85. The feedback control starts at the 71st iteration.

Remark 4.2. In the process of proving Theorems 3.3 and 4.1, we only need to know that eigenvalues are greater (or less) than one in absolute and it is not necessary to compute explicitly the eigenvalues. These ideas avoid difficulties in calculating eigenvalues in higher-dimension DCMLs using the Gershgorin circle theorem.

5. Conclusion

With the Marotto theorem and the Gershgorin circle theorem, we have theoretically analyzed the chaos in the DCML with open boundary conditions, which presents a theoretical foundation for chaos analysis of the DCML. What is more is that the 0-1 test further confirms the existence of chaos and we control spatiotemporal chaotic motions in the DCML to period-1 orbits. Stability analysis is presented. The results of simulations are consistent with theoretical analysis. We wish to emphasize that the methods of this paper can be used in all those cases where the eigenvalues of Jacobi matrix are difficult to calculate in CMLs.

Acknowledgment

The paper was supported by NSFC (no. 10871074).

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