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

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 highdimensional 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.


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 : 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 .∈ 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 :

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 2 n i , where a ∈ 0, 2 and x n i ∈ −1, 1 .The logistic function f x 1−ax 2 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.

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: 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 x 0 / 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 x 0 .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
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 x * , x 0 / x * , for some x 0 ∈ 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 x 0 , x 1 , . . ., 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 .

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
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 n j 1, where C denotes the complex plane.The eigenvalues of A are contained within R n i 1 R 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.
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: 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 b There exist a x 0 ∈ B x * , r and x 0 / x * such that F m x 0 x * for some m ∈ N and m ≥ 2. c det DF m x 0 / 0. The proof consists of four steps.The ideas are motivated chiefly by 13, 18, 19 . Step 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 Under the conditions of Theorem 3.3, that is, 0 < < 0.5 and a > 1 − 2 / 1 − 2 2 − 1/4 , the following results are obtained: 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 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 With the conditions of Theorem 3.3, we get For any point x ∈ B r x * and small enough, there exists some s > 1 such that

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 : Moreover, the equation 3.9 has two solutions 2071 and 0 < ρ 2 < z * < 1, we take ρ ρ 2 c/ √ 2a.Then, z * −ρ < 1−z * , and we denote Thus, condition a of Definition 2.1 and Lemma 2.2 is satisfied.
Step 2. For all z z1 ∈ B r x * , we have

3.13
Assume that 3.12 has a solution, and denote Nz * , which has two solutions: Step 3. Now, let F x y 1 , that is,

3.15
Assume that the system of 3.14 has a solution, and denote We take z 2 2a √ 4a 1 − 1 /2a 2 .Thus, 3.17 So, y t is monotone increasing continuous function, and min y t x * .Steps 2 and 3 complete the proof of condition b .
2, all eigenvalues of DF y 2 lie in the interval Θ ≤ λ j ≤ Θ 1 − 2 < 0. Thus, with Lemma 3.1, det DF y 2 N j 1 λ j / 0.Then, we have F m x 0 x * and det DF m x 0 / 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.

Numerical Simulation of Chaos
When N 300, a 1.8, and 0.01, the conditions of Theorem 3.3 are satisfied.The DCML 1.1 can be denoted as follows:

3.18
The corresponding eigenvalues of DF x * lie in the interval 1 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 < 0.0582;

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.

3.19
Step 2. Define the mean square displacement M c n as follows:

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

3.22
Step 5. Steps 1-4 are 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, qplane 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 x 1 1 , x 1 2 , . . ., x 1 300 ; we carry out the 0-1 test with 0.03 and 0.12, respectively.Using the data set of x 111 in the system 3.18 , we get K 0.9981 at 0.03 and K 0.0030 at 0.12.The translation variables p, q are shown in Figures 3  and 4, respectively.We take N 300, a 1.8 and let 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 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.
Theorem 4.1.With the local controllers U n i α 1 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 < α 1 < 1.
Proof.Since the local controllers are given by we get the controlled DCML: where α 1 ∈ 0.6407, 1 .Expanding 4.3 around the fixed point X * , we obtain Then, by using 4.4 and 4.5 , we get

4.6
For the sake of simplicity, we denote Solving inequality 4.10 , we obtain −1.8263 1.2897α 1 < λ i < −1.8636 1.3270α 1 .Since 0.6407 < α 1 < 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 .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 higherdimension DCMLs using the Gershgorin circle theorem.

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.

Figure 3 :Figure 4 :
Figure 3: Plot of p versus q for the DCML 3.18 with 0.03.We used 30000 data points of x 111 .

Figure 5 :
Figure 5: Plot of K versus for the DCML 3.18 with ∈ 0.01, 0.058 increased in increments of 0.01.We used 30000 data points of x 111 .

Figure 6 :
Figure 6: The control result of the DCML 3.18 , with the parameter values a 1.8, 0.01, and α 1 0.85.The feedback control starts at the 71st iteration.
∈ 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 0 r 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. 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 .