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

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 [1–16]. These mainly include bifurcation [2], chaos [6, 7], chaotic synchronization [4, 8–10], 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, 4–15]. 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]:𝑥_𝑛+1(𝑖)=(1−𝜖)𝑓𝑥_𝑛(𝑖)+𝜖2𝑓𝑥^𝑛(𝑖−1)+𝑓𝑥_𝑛(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. 𝑥_𝑛(𝑖) represents the state variable for the 𝑖th site at time 𝑛. Throughout this paper, we adopt open boundary conditions [16, 17]: 𝑥𝑥_𝑛+1(1)=(1−𝜖)𝑓𝑥_𝑛(1)+𝜖𝑓𝑥_𝑛(2)_𝑛+1,(𝑁)=𝜖𝑓𝑥_𝑛(𝑁−1)+(1−𝜖)𝑓𝑥_𝑛(𝑁).(1.2) Here each of the lattice points in (1.1) and (1.2) is chosen to be the logistic map 𝑓(𝑥_𝑛(𝑖))=1−𝑎𝑥^2𝑛(𝑖), where 𝑎∈(0,2] and 𝑥_𝑛(𝑖)∈(−1,1). The logistic function 𝑓(𝑥)=1−𝑎𝑥² is equivalent to the well-known form 𝑔(𝑧)=𝑟𝑧(1−𝑧) [20] when the transformations 𝑎=𝑟(𝑟−2)/4 and 𝑥=2(2𝑧−1)/(𝑟−2)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: 𝑥_𝑘+1=𝐹𝑥_𝑘,𝑘=0,1,2,…,(2.1) where 𝑥_𝑘∈𝑅^𝑛 and 𝐹∶𝑅^𝑛→𝑅^𝑛 is continuous. Let 𝐵_𝑟(𝑥) denote the closed ball in 𝑅^𝑛 of radius 𝑟 centered at point 𝑥 and 𝐵^0𝑟(𝑥) its interior. Also, let ‖𝑥‖ be the usual Euclidean norm of 𝑥 in 𝑅^𝑛[22]. Then, if 𝐹 is differentiable in 𝐵_𝑟(𝑧), Marotto claimed that in the following, 𝐴⇒𝐵.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,𝑦∈𝐵_𝑟(𝑧),‖𝐹(𝑥)−𝐹(𝑦)‖>𝑠‖𝑥−𝑦‖.

Marotto thought that, if (A) is satisfied, then (B) can be derived, that is, F is expanding in 𝐵_𝑟(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 𝑅^𝑛 (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 𝑥₀≠𝑧 in a repelling neighborhood of 𝑧, such that 𝑥_𝑀=𝑧 and det(𝐷𝐹(𝑥_𝑘))≠0 for 1≤𝑘≤𝑀, where 𝑥_𝑘=𝐹^𝑘(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 𝑥_𝑘+1=𝐹𝑥_𝑘,𝑥_𝑘∈𝑅^𝑛,𝑘=0,1,2,…,(2.2) where 𝐹 is a map from 𝑅^𝑛 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 𝑅^𝑛 such that 𝐹 is expanding in 𝐵_𝑟(𝑥^∗), the closed ball of radius 𝑟 centered at 𝑥^∗ in (𝑅^𝑛,‖⋅‖),(2)𝑥^∗ is a snap-back repeller of F with 𝐹^𝑚(𝑥₀)=𝑥^∗,𝑥₀≠𝑥^∗, for some 𝑥₀∈𝐵_𝑟(𝑥^∗) and some positive integer 𝑚, where 𝐵_𝑟(𝑥^∗) is the open ball of radius 𝑟 centered at 𝑥^∗ in (𝑅^𝑛,‖⋅‖). Furthermore, 𝐹 is continuously differentiable in some neighborhoods of 𝑥₀,𝑥₁,…,𝑥_𝑚−1, respectively, and det[𝐷𝐹(𝑥_𝑗)]≠0, where 𝑥_𝑗=𝐹(𝑥_𝑗−1) 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 𝐴_𝑁×𝑁 with eigenvalues 𝜆₁,𝜆₂,…,𝜆_𝑁, the determinant of A is equal to ∏^𝑁𝑖=1𝜆_𝑖. Denote det(𝐴)=∏^𝑁𝑖=1𝜆_𝑖.

Lemma 3.2 (see [29, 30], the Gershgorin circle theorem). Let 𝐴 be an 𝑛×𝑛 matrix, and let 𝑅_𝑖 denote the circle in the complex plane with center 𝑎_𝑖𝑖 and radius ∑^{𝑛𝑗=1,𝑗≠}_i|𝑎_𝑖𝑗|; that is, 𝑅_𝑖=⎧⎪⎨⎪⎩𝑧∈𝒞∣||𝑧−𝑎_𝑖𝑖||≤^𝑛_{𝑗=1,𝑗≠𝑖}||𝑎_𝑖𝑗||⎫⎪⎬⎪⎭,(3.1) where 𝒞 denotes the complex plane. The eigenvalues of 𝐴 are contained within 𝑅=⋃^𝑛𝑖=1𝑅_𝑖. Moreover, the union of any 𝑘 of these circles that do not intersect the remaining (𝑛−𝑘)contains precisely 𝑘 (counting multiplicities) of the eigenvalues.

Theorem 3.3. If 0<𝜖<1/2 and 𝜖 is small enough, 𝑎∈{𝑎∣𝑎>(1+√2)/2≃1.2071}⋂{𝑎∣𝑎>(1−𝜖)²/(1−2𝜖)²−(1/4)}, and 𝑐=1/√(3𝑁+2)𝜖²−4𝑁𝜖+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: 𝐱_𝑘+1=𝐹𝐱_𝑘,(3.2) where 𝐱_𝑘=[𝑥_𝑘(1),𝑥_𝑘(2),…,𝑥_𝑘(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 𝑅^𝑛 such that 𝐹 is expanding in 𝐵_𝑟(𝐱^∗).(b)There exist a 𝐱₀∈𝐵(𝐱^∗,𝑟) and 𝐱₀≠𝐱^∗ such that 𝐹^𝑚(𝐱₀)=𝐱^∗ for some 𝑚∈ℕ and 𝑚≥2.(c)det[𝐷𝐹^𝑚(𝐱₀)]≠0.
The proof consists of four steps. The ideas are motivated chiefly by [13, 18, 19].
Step 1. Let 𝐱^∗=[(√4𝑎+1−1)/2𝑎,…,(√4𝑎+1−1)/2𝑎]^𝑇=𝑧^∗𝟏∈𝑅^𝑁, where 𝑧^∗=(√4𝑎+1−1)/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 𝐵_𝑟(𝐱^∗) for some 𝑟>0. Its Jacobian matrix at 𝐱^∗ is 𝐷𝐹𝐱^∗=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝(1−𝜖)𝑓^𝜖𝑧^∗𝜖𝑓^𝑧^∗00⋯02𝑓^0𝑧^∗(1−𝜖𝜖)𝑓^𝑧^∗2𝜖𝑓^𝑧^∗0⋯0⋮02𝑓^⋮0𝑧^∗(1−𝜖…)⋮𝑓^𝑧^∗2𝜖𝜖𝑓^⋮𝑧^∗⋯⋮⋮000⋯2𝑓^0𝑧^∗(1−𝜖𝑓𝜖_)𝑓^𝑧^∗2𝜖𝑓^𝑧^∗𝑧^∗(1−𝜖)𝑓^𝑧^∗⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.3) where 𝑓′(𝑧^∗)=1−√4𝑎+1<0. We denote 𝐷𝐹(𝐱^∗) by (1−√4𝑎+1)𝑀, where 𝑀=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝1−𝜖𝜖𝜖00⋯0201−𝜖𝜖2𝜖0⋯0⋮0⋮021…⋮−𝜖⋮2𝜖𝜖⋯⋮⋮000⋯021−𝜖𝜖1−2𝜖𝜖⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(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^∗), 𝜆_𝑗(𝑗=1,2,…,𝑁), are given by 1−√4𝑎+1≤𝜆_𝑗≤(1−√4𝑎+1)(1−2𝜖). Under the conditions of Theorem 3.3, that is, 0<𝜖<0.5 and 𝑎>(1−𝜖)²/(1−2𝜖)²−(1/4), the following results are obtained: 1<√4𝑎+1−1(1−2𝜖)≤||𝜆_𝑗||≤√4𝑎+1−1,∀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 𝐵_𝑟(𝐱^∗). That is, for any two distinct points 𝐱, 𝐲∈𝐵_𝑟(𝐱^∗), we have ‖𝐹(𝐲)−𝐹(𝐱)‖>𝑠‖𝐲−𝐱‖,(3.6) where 𝑠>1 and 𝐱,𝐲 are sufficiently close to 𝐱^∗. Since 𝐹(𝐲)−𝐹(𝐱)=𝐷𝐹(𝐱)(𝐲−𝐱)+𝛼, where ‖𝛼‖/‖𝐲−𝐱‖→0 as ‖𝐲−𝐱‖→0 [19], specially, ‖𝐹(𝐱)−𝐹(𝐱^∗)‖=‖𝐷𝐹(𝐱^∗)(𝐱−𝐱^∗)+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=1∑^𝑁𝑖=1𝑎^2𝑖𝑗)^1/2. With the conditions of Theorem 3.3, we get |(1−√4𝑎+1)(1−𝜖)|>1. For any point 𝐱∈𝐵_𝑟(𝐱^∗) and 𝜖 small enough, there exists some 𝑠>1 such that ‖𝐷𝐹(𝐱^∗)𝐱‖_𝐹=‖‖1−√4𝑎+1𝑀𝑥₁,𝑥₂,𝑥₃,…,𝑥_𝑁−2,𝑥_𝑁−1,𝑥_𝑁^𝑇‖‖_𝐹=‖‖‖‖‖‖‖‖‖‖‖‖1−√4𝑎+1⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝(1−𝜖)𝑥¹+𝜖𝑥₂𝜖2𝑥¹+(1−𝜖)𝑥₂+𝜖2𝑥³𝜖⋮2𝑥^𝑁−2+(1−𝜖)𝑥_𝑁−1+𝜖2𝑥^𝑁𝜖𝑥_𝑁−1+(1−𝜖)𝑥_𝑁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠‖‖‖‖‖‖‖‖‖‖‖‖_𝐹≥𝑠‖‖‖‖‖‖‖‖‖‖‖⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑥¹𝑥₂𝑥⋮_𝑁−1𝑥_𝑁⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠‖‖‖‖‖‖‖‖‖‖‖_𝐹.(3.7) Since 𝐹(𝐱) is continuously differentiable, 𝐷𝐹(𝐱) is also expanding for 𝐱∈𝐵_𝑟(𝐱^∗). Let the bound of the maximal open expanding ball 𝐵_𝑟(𝐱^∗) be denoted by 𝜌𝟏, where 𝜌 satisfies the following inequality [13]: ‖𝐷𝐹(𝜌𝟏)‖=4𝑎²(1−𝜖)²𝑁𝜌²+2𝜖²𝑎²(𝑁−2)𝜌²+8𝜖²𝑎²𝜌²>1.(3.8) Moreover, the equation 2𝑎²𝜌²(3𝑁+2)𝜖²−4𝑁𝜖+2𝑁=1(3.9) has two solutions 𝜌_1,2=∓1√2𝑎×√(3𝑁+2)𝜖²−4𝑁𝜖+2𝑁=∓√𝑐2𝑎,(3.10) where 𝑐=1/√(3𝑁+2)𝜖²−4𝑁𝜖+2𝑁. One has 𝑐∈(0,1) (because 𝑓(𝜖)=(3𝑁+2)𝜖²−4𝑁𝜖+2𝑁 is a quadratic function, the discriminant Δ=−8𝑁²−16𝑁<0, when 𝜖=4𝑁/2(3𝑁+2), min𝑓(𝜖)=(4(3𝑁+2)2𝑁−16𝑁²)/4(3𝑁+2)>1). In fact, 𝑐∈(1/√2𝑁,1/√(3/4)𝑁+(1/2)). Since 𝑎>(1+√2)/2≈1.2071 and 0<𝜌₂<𝑧^∗<1, we take 𝜌=𝜌₂=𝑐/√2𝑎. Then, 𝑧^∗−𝜌<1−𝑧^∗, and we denote 𝑟=𝑧^∗−𝜌=√4𝑎+1−1−√2𝑐2𝑎>0.(3.11) Thus, condition (a) of Definition 2.1 and Lemma 2.2 is satisfied.Step 2. For all 𝐳=𝑧𝟏∈𝐵_𝑟(𝐱^∗), we have |𝑧−𝑧^∗|<𝑟, that is, 𝜎₁<𝑧<𝜎₂, where 𝜎₁=𝑧^∗−𝑟=𝑐/√2𝑎,𝜎₂=𝑧^∗+𝑟=(√4𝑎+1−1−(√2/2)𝑐)/𝑎. Now let 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑁)^𝑇 and 𝐹(𝐱)=𝐱^∗, that is, (1−𝜖)1(−1−𝑎𝑥𝜖)_2𝑖+11−𝑎𝑥²¹+𝜖1−𝑎𝑥²²=𝑧^∗,𝜖1−𝑎𝑥+_2𝑁−12𝜖1−𝑎𝑥^2𝑖+2𝜖1−𝑎𝑥^2𝑖+2=𝑧^∗,+(1−𝜖)1−𝑎𝑥^2𝑁=𝑧^∗,(3.12) where 𝑖=1,2,…,𝑁−2. Summing all the above equations, we obtain 1−+2𝜖1+1−𝜖𝑎𝑥²¹+1+2𝜖1−𝑎𝑥²²+^𝑁−2_𝑘=31−𝑎𝑥^2𝑘21−𝑎𝑥^2𝑁−1+1−2𝜖1−𝑎𝑥^2𝑁=𝑁𝑧^∗.(3.13) Assume that (3.12) has a solution, and denote 𝐲₁=𝑧₁𝟏, that is 𝑁(1−𝑎𝑧²¹)=𝑁𝑧^∗, which has two solutions: 𝑧₁=±√(1−𝑧^∗)/𝑎=±(√4𝑎+1−1)/2𝑎. We choose 𝑧₁=(1−√4𝑎+1)/2𝑎 since 𝑧₁−𝜎₁=(1−√4𝑎+1−√2𝑐)/2𝑎<0,thatis,𝑧₁<𝜎₁ and 𝑧₁∉(𝜎₁,𝜎₂).Step 3. Now, let 𝐹(𝐱)=𝐲₁, that is, (1−𝜖)1(−1−𝑎𝑥𝜖)_2𝑖+11−𝑎𝑥²¹+𝜖1−𝑎𝑥²²=𝑧₁,𝜖1−𝑎𝑥+_2𝑁−12𝜖1−𝑎𝑥^2𝑖+2𝜖1−𝑎𝑥^2𝑖+2=𝑧₁,+(1−𝜖)1−𝑎𝑥^2𝑁=𝑧₁,(3.14) where 𝑖=1,2,…,𝑁−2. Summing the above 𝑁 equations, we get 1−+2𝜖1+1𝜖−𝑎𝑥²¹+1+2𝜖1−𝑎𝑥²²+∑^𝑁−2_𝑘=31−𝑎𝑥^2𝑘21−𝑎𝑥^2𝑁−1+1−2𝜖1−𝑎𝑥^2𝑁=𝑁𝑧₁.(3.15) Assume that the system of (3.14) has a solution, and denote 𝐲₂=𝑧₂𝟏, that is, 𝑁(1−𝑎𝑧²²)=𝑁𝑧₁, that is,1−𝑎𝑧²²=(1−√4𝑎+1)/2𝑎, which has two solutions: 𝑧₂=±(2𝑎+√4𝑎+1−1)/2𝑎². We take 𝑧₂=(2𝑎+√4𝑎+1−1)/2𝑎². Thus, 𝜎₂−𝑧₂=√4𝑎+1−1−√2/2𝑐𝑎−2𝑎+√4𝑎+1−12𝑎²=√8𝑎+2−√2−𝑐−2𝑎−1+√4𝑎+1√2𝑎.(3.16) Denote √4𝑎+1=𝑡; since 𝑎>(1+√2)/2,thatis,3>𝑡>3+2√2≈2.4142 and 𝑎=(𝑡²−1)/4, then √8𝑎+2−√2−𝑐−2𝑎−1+√4𝑎+1=√2𝑡−√2−𝑐−𝑡²−1=2𝑡−√2+𝑡−1𝑡²+2𝑡−3−2−√2𝑐√2.(3.17) Denoting 𝑦(𝑡)=2𝑡−√𝑡²+2𝑡−3, we get 𝑦′(𝑡)=2−(𝑡+1)/(√(𝑡+1)²−4)>0,𝑡∈(3+2√2,3). So, 𝑦(𝑡) is monotone increasing continuous function, and min𝑦(𝑡)=23+2√2−2(√2+3+2√2)≈2.0613. We get 2𝑡−√𝑡²+2𝑡−3−2−√2𝑐≥0.0613−√2𝑐>0 (since the condition 𝑐<(0.0613/√2)). Therefore, 𝑧₂<𝜎₂. On the other hand, 𝑧₂−𝜎₁=(2𝑎+√4𝑎+1−1)/2𝑎²−(𝑐/√2𝑎)=(2𝑎+√4𝑎+1−1−𝑐)/√2𝑎>0,thatis,𝑧₂>𝜎₁. Thus, 𝜎₁<𝑧₂<𝜎₂,𝐲₂∈𝐵_𝑟(𝐱^∗), and 𝑧₂≠𝑧^∗,thatis,𝐲₂≠𝐱^∗. 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+1−1)𝑀≜𝜔𝑀, where 𝜔=(√4𝑎+1−1)>0, with Lemma 3.2, all eigenvalues of 𝐷𝐹(𝐲₁) lie in the interval 0<𝜔(1−2𝜖)≤𝜆_𝑗≤𝜔. Thus, with Lemma 3.1, det[𝐷𝐹(𝐲₁)]=∏^𝑁𝑗=1𝜆_𝑗≠0. Moreover, according to  𝐷𝐹(𝐲₂)=−4𝑎+2√4𝑎+1−2𝑀≜Θ𝑀, where Θ=−4𝑎+2√4𝑎+1−2<0, with Lemma 3.2, all eigenvalues of 𝐷𝐹(𝐲₂) lie in the interval Θ≤𝜆_𝑗≤Θ(1−2𝜖)<0. Thus, with Lemma 3.1, det[𝐷𝐹(𝐲₂)]=∏^𝑁𝑗=1𝜆_𝑗≠0. Then, we have 𝐹^𝑚(𝐱₀)=𝐱^∗ and det[𝐷𝐹^𝑚(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:𝑥𝑥_𝑛+1(1)=(1−𝜖)1−𝑎𝑥^2𝑛(1)+𝜖1−𝑎𝑥^2𝑛(2)_𝑛+1,(2)=(1−𝜖)1−𝑎𝑥^2𝑛(2)+𝜖21−𝑎𝑥^2𝑛𝑥⋮(1)+1−𝑎𝑥^2𝑛(3)_𝑛+1,(299)=(1−𝜖)1−𝑎𝑥^2𝑛(299)+𝜖21−𝑎𝑥^2𝑛𝑥(298)+1−𝑎𝑥^2𝑛(300)_𝑛+1,(300)=𝜖1−𝑎𝑥^2𝑛(299)+(1−𝜖)1−𝑎𝑥^2𝑛(300).(3.18) The corresponding eigenvalues of 𝐷𝐹(𝐱^∗) lie in the interval (1−√4𝑎+1,(1−√4𝑎+1)(1−2𝜖)), that is, 𝜆_𝑖∈(−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.

Figure 1 

Spatiotemporal chaos in the DCML (3.18) without any control, with parameters 𝑁=300, 𝑎=1.8, 𝜖=0.01.

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 [31–36]. 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 [37–44] 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 𝑝_𝑐(𝑛)=^𝑛_𝑗=1𝜙(𝑗)cos(𝑗𝑐),𝑞_𝑐(𝑛)=^𝑛_𝑗=1𝜙(𝑗)sin(𝑗𝑐).(3.19)

Step 2. Define the mean square displacement 𝑀_𝑐(𝑛) as follows: 𝑀_𝑐(𝑛)=lim_𝑁→∞1𝑁^𝑁_𝑗=1𝑝_𝑐(𝑗+𝑛)−𝑝_𝑐(𝑗)²+𝑞_𝑐(𝑗+𝑛)−𝑞_𝑐(𝑗)².(3.20) Note that this definition requires 𝑛≪𝑁. In practice, 𝑛≤𝑁/10 yields good results. Denote 𝑛_cut=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 𝐷_𝑐(𝑛)=𝑀_𝑐(𝑛)−𝑉_osc(𝑐,𝑛),(3.21) where 𝑉_osc(𝑐,𝑛)=(𝐸𝜙)²(1−cos𝑛𝑐)/(1−cos𝑐) and 𝐸𝜙=lim_𝑁→∞(1/𝑁)∑^𝑁𝑗=1𝜙(𝑗).

Step 4. Form the vectors 𝜉=(1,2,…,𝑛_cut) and Δ=(𝐷_𝑐(1),𝐷_𝑐(2),…,𝐷_𝑐(𝑛_cut)). Then define the correlation coefficient 𝐾_𝑐=corr(𝜉,Δ)∈[−1,1].(3.22)

Step 5. Steps 1–4 are performed for 𝑁_𝑐 values of 𝑐 chosen randomly in the interval (𝜋/5,4𝜋/5). In practice, 𝑁_𝑐=100 is sufficient. We then compute the median of these 𝑁_𝑐 values of 𝐾_𝑐 to compute the final result 𝐾=median(𝐾_𝑐). 𝐾≈0 indicates regular dynamics, and 𝐾≈1 indicates chaotic dynamics.

Note that the (𝑝_𝑐(𝑛),𝑞_𝑐(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, 𝐾≤0.1 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.

Figure 3 

Plot of 𝑝 versus 𝑞 for the DCML (3.18) with 𝜖=0.03. We used 30000 data points of 𝑥(111).

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.

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+1−1)/2𝑎,…,(√4𝑎+1−1)/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) as𝑋_𝑛+1(𝑖)=𝐹𝑋_𝑛(𝑖),𝑋_𝑛(𝑖−1),𝑋_𝑛(𝑖+1),(4.1) where 𝑋_𝑛(𝑖)=(𝑥_𝑛(1),𝑥_𝑛(2),…,𝑥_𝑛(300))^𝑇.

Theorem 4.1. With the local controllers 𝑈_𝑛(i)=𝛼₁[𝑋_𝑛−1(𝑖)−𝐹(𝑋_𝑛(𝑖),𝑋_𝑛(𝑖−1),𝑋_𝑛(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 𝑈_𝑛(𝑖)=𝛼₁𝑋_𝑛−1(𝑖)−𝐹𝑋_𝑛(𝑖),𝑋_𝑛(𝑖−1),𝑋_𝑛(𝑖+1),(4.2) we get the controlled DCML: 𝑋_𝑛+1(𝑖)==𝐹1−𝑋𝛼_𝑛1(𝑖),𝑋_𝑛(𝑖−1),𝑋_𝑛(𝑖+1)+𝑈_𝑛(𝑖)𝐹𝑋_𝑛(𝑖),𝑋_𝑛(𝑖−1),𝑋_𝑛(𝑖+1)+𝛼₁𝑋_𝑛−1(𝑖),(4.3) where 𝛼₁∈(0.6407,1). Expanding (4.3) around the fixed point 𝑋^∗, we obtain 𝑋_𝑛+1−𝑋^∗=𝜕𝐹𝜕𝑋_𝑛||||_𝑋∗𝑋_𝑛−𝑋^∗+𝜕𝑋𝜕𝐹_𝑛−1||||_𝑋∗𝑋_𝑛−1−𝑋^∗.(4.4) Since 𝑥_𝑛(𝑖)≈𝑓[𝑥_𝑛−1(𝑖)],𝑓(𝑥)=1−𝑎𝑥², and 𝑥^∗=(√4𝑎+1−1)/2𝑎, we have 𝑥_𝑛(𝑖)−𝑥^∗=𝜕𝑓/𝜕𝑥_𝑛−1|_𝑥∗(𝑥_𝑛−1(𝑖)−𝑥^∗). Thus, we get 𝑥_𝑛−1(𝑖)−𝑥^∗=(1/(𝜕𝑓/𝜕𝑥_𝑛−1|_𝑥∗))(𝑥_𝑛(𝑖)−𝑥^∗) and 𝑋_𝑛−1−𝑋^∗=1𝜕𝑓/𝜕𝑥_𝑛−1||_𝑥∗𝑋_𝑛−𝑋^∗.(4.5) Then, by using (4.4) and (4.5), we get 𝑋_𝑛+1−𝑋^∗=𝜕𝐹𝜕𝑋_𝑛||||_𝑋∗𝑋_𝑛−𝑋^∗+𝜕𝑋𝜕𝐹_𝑛−1||||_𝑋∗1𝜕𝑓/𝜕𝑥_𝑛−1||_𝑥∗=𝜕𝐹𝑋_𝑛−𝑋^∗𝜕𝑋_𝑛||||_𝑋∗+1𝜕𝑓/𝜕𝑥_𝑛−1||_𝑥∗𝜕𝐹𝜕𝑋_𝑛−1||||_𝑋∗𝑋^𝑛−𝑋^∗.(4.6) For the sake of simplicity, we denote 𝐽 by 𝜕𝐹/𝜕𝑋_𝑛|_𝑋∗+(1/(𝜕𝑓/𝜕𝑥_𝑛−1|_𝑥∗))(𝜕𝐹/𝜕𝑋_𝑛−1)|_𝑋∗; then 𝐽=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝Λ𝑓^Θ𝑓_(𝑥(1))2Θ𝑓^(𝑥(2))0⋯0⋮(0𝑥(1))Λ𝑓^…⋮(𝑥(2))ΘΘ𝑓𝑓^_⋱(𝑥(3))⋱⋯⋮00⋯(𝑥0(298))2ΛΘ𝑓𝑓^_(𝑥(299))Θ𝑓^(𝑥(300))(𝑥(299))Λ𝑓^(𝑥(300))⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠+𝐽²,(4.7) where 𝐽₂=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝛼¹𝑓^(𝑥0(1))𝛼00⋯0₁⋮0𝑓^(…⋮𝑥(2))⋱00⋱⋯𝛼₁⋮00⋯0𝑓^(𝑥0(299))𝛼0₁𝑓^(𝑥(300))⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(4.8)Θ=(𝜖/2)−(𝜖𝛼₁/2), Λ=1−𝜖−𝛼₁+𝛼₁𝜖, and 𝑓′(𝑥(𝑖))=−2𝑎𝑥(𝑖). With the Gershgorin circle theorem (Lemma 3.2), we get ||||𝜆_𝑖−1−𝛼₁(1−𝜖)𝑓′(𝑥(𝑖))+𝛼¹𝑓′(𝑥(𝑖))||||<1−𝛼¹𝜖𝑓′(𝑥(𝑖)),(4.9) that is, 1−𝛼₁𝑓′(𝑥(𝑖))−2𝜖1−𝛼₁𝑓′(𝑥(𝑖))+𝑓′(𝛼𝑥¹(𝑖))<𝜆^𝑖<1−𝛼₁𝑓′(𝑥(𝑖))+𝑓′(𝛼𝑥¹(𝑖)).(4.10) Solving inequality (4.10), we obtain −1.8263+1.2897𝛼₁<𝜆_𝑖<−1.8636+1.3270𝛼₁. Since 0.6407<𝛼₁<1, we get |𝜆_𝑖|<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.

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).