網格型耦合混沌系統的全域同步化

國

立

交

通

大

學

應用數學系

碩

士

論

文

網格型耦合混沌系統的全域同步化

Global Synchronization in Lattices of Coupled

Chaotic Systems

研 究 生：梁育豪

指導教授：莊 重 教授

網格型耦合混沌系統的全域同步化

Global Synchronization in Lattices of Coupled Chaotic

Systems

研 究 生：梁育豪 Student：Yu-Hao Liang

指導教授：莊 重 Advisor：Jonq Juang

國 立 交 通 大 學

應 用 數 學 系

碩 士 論 文

A Thesis

Submitted to Department of Applied Mathematics

College of Science

National Chiao Tung University

in partial Fulfillment of the Requirements

for the Degree of

Master

in

Applied Mathematics

July 2007

Hsinchu, Taiwan, Republic of China

i

網格型耦合混沌系統的全域同步化

學生：梁育豪 指導教授

：

莊 重

國 立 交 通 大 學

應 用 數 學 系

碩 士 班

摘 要

於此，我們透過矩陣測度的概念來探討同步化現象的全域穩定性。這套方法將可被

應用於相當廣泛的系統連結模式上。此外，達成全域同步化所需的耦合力量下界可以被

嚴謹的求得。不但如此，我們僅需驗證被耦合的子系統型式便能辨別全域同步化的現象

能否發生。

ii

Global Synchronization in Lattices of Coupled Chaotic

Systems

Student：Yu-Hao Liang Advisor：Jonq Juang

Department of Applied Mathematics

National Chiao Tung University

Degree of Master

ABSTRACT

Based on the concept of matrix measures, we study global stability of synchronization in

networks. Our results apply to quite general connectivity topology. In addition, a rigorous

lower bound on the coupling strength for global synchronization of all oscillators is also

obtained. Moreover, by merely checking the structure of the vector field of the single

oscillator, we shall be able to determine if the system is globally synchronized.

iii

誌 謝

這篇論文得以完成，首先，感謝我的指導教授莊重老師在各方面給予的

幫助以及提供許多解決論文問題的關鍵切入點。在這兩年的碩士生涯中，

從老師的身上我看到了老師對於數學研究應有的三多態度：多問、多想、

多嘗試，樹立了我的模範，使我對於數學研究更加的充滿熱誠。

其次，感謝金龍、靖尉學長、郁泉學姊時常給予的意見與指教，讓我

的論文得以更順利的完成。也感謝我的大學同學：士傑、室友：介友、基

恩、建賢以及許多碩班同學，平時給予的照顧、歡笑，讓我在論文遇到挫

折後，還能有信心地站起來繼續前進。

最後，感謝我的媽媽：瓊珠、爸爸：有忠以及姊姊：翠洺給予的支持

與鼓勵讓我能無憂無慮的完成我的學業、追尋我的夢想。

iv

目 錄

中文摘要 ………

i

英文摘要 ………

ii

誌謝 ………

iii

目錄 ………

iv

1. Introduction………

1

2. Basic Framework and Preliminaries ………

2

3. Main Results………

5

4. Applications ………

13

5. Conclusion ………

19

Appendix ………

20

1. Introduction

Lattices of coupled chaotic oscillators model many systems of interest in physics, biology and engineering. In particular, complete chaotic synchronization, all oscilla-tors acquiring identical chaotic behavior, has received much attention analytically. There are, in general, two classes of results which give criteria for such synchro-nization. The first class of results linearizes around the synchronous manifold, and then computes the Lyapunov exponents or matrix measures of the variational equa-tions to get local synchronization [29,10] or use partial contraction principle to get global synchronization [33]. The second class of results uses Lyapunov method by constructing a Lyapunov function to give an analytical criteria for local or global synchronization [3-8,30,35-38]. This paper gives yet another approach by utilizing the concept of matrix measures to get global synchronization criteria. The coupling configuration of the networks is quite general, which includes asymmetric connec-tions between nodes and/or some competitive (gij < 0, i 6= j) couplings between

cells xi and xj, and partial-state coupling with nonzero off-diagonal connections.

Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized.

During the last few decades the study of networks of dynamical systems has at-tracted increasing attention [1-12, 14-31, 33-39]. The purpose to connect dynamical systems in networks is to get them to solve problems cooperatively. For instance, such networks are needed for information processing in the brain [15]. The sim-plest mode of the coordinated motion between dynamical systems is their complete synchronization when all cells of the network acquire identical dynamical behav-ior. Consequently, one asks questions such as: What are the conditions for the stability of the synchronous state, especially with respect to coupling strengths and coupling configurations of the network? Typically, in networks of continuous time oscillators, the synchronous solution becomes stable when the coupling strength between oscillators exceeds a critical value. In this context, a central problem is to find the bounds on the coupling strength so that the stability of synchronization is guaranteed.

General approaches to local synchronization of coupled chaotic systems have been proposed, including the master stability function (MSF)- based criteria [2,26-29,31], originated by Pecora and Carroll [29], and matrix measures approach [10]. The former computes the Lyapunov exponent of the variational equations, while the latter uses the concept of matrix measures to give criteria on the variation equations. Recently, local synchronization in a complex network of asymmetrically coupled units was also obtained [11, 19] via MSF-based criteria.

Global synchronization of coupled chaotic systems was also intensively studied. The methods include Lyapunov function- based criteria with symmetrical connec-tions [3-7,30,35-38] or asymmetrical connecconnec-tions [8, 37], and the partial contraction approach [33]. For Lyapunov-based criteria, the partial-state coupling matrix, de-termining which state variables are coupled, is assumed to have the form satisfying (2.4c). While the partial contraction approach needs to verify the contraction of the system, depending on the state variables and time t, which is not a small task. In developing the theory of global synchronization of coupled chaotic systems, one needs to assume bounded dissipation of the coupled system, that is, all solutions of the coupled system are, in some sense, eventually bounded. Such assumption plays the role of an a priori estimate. However, in obtaining the theory of local synchro-nization, one dose not need to know bounded dissipation of the coupled system. Thus, not surprisingly, the criteria in getting local synchronization are composed of a term that describes how chaotic the single system is and a term that depends on how the configuration of the networks is formed.

The purpose of this paper is yet to give another approach to study global syn-chronization of coupled chaotic systems. Our coupling rules are allowed to be asym-metric and/or some competitive (gij < 0, i 6= j) couplings between cells xi and xj

as long as the coupled system is bounded dissipative. In addition, the partial-state coupling in our approach is allowed to have the form satisfying (3.9a). Moreover, by merely checking the structure of the vector field of the single oscillator, we shall be able to determine if the system is globally synchronized. We also obtain a rigorous lower bound on the coupling strength for global synchronization of all oscillators with coupling configuration satisfying (2.4a), and (2.4b). Finally, the concept of matrix measures is introduced to obtain such global results.

We organize the paper as follows. Section 2 is to lay down the foundation of our paper. The main results are contained in Section 3. Coupled Lorenz systems and coupled Duffing systems are used as illustrations. We also compare our results with those in [7, 8].

2. Basic Framework and Preliminaries

In this paper, we will denote scalar variables in lower case, matrices in bold type upper case, and vectors (or vector-valued functions) in bold type lower case. We consider an array of m cells, coupled linearly together, with each cell being an n- dimensional system. The entire array is a system of nm ordinary differential equations. In particular, the state equations are

dxi dt = f (xi, t) + d · m X j=1 gijDxj, i = 1, 2, . . . , m, (2.1) 2

where xi∈ Rn, f : Rn× R → Rn and D is an n × n real matrix. Let x=     x1 .. . xm     , xi=     xi,1 .. . xi,n     , and G = (gij)_m×m. (2.2)

Then (2.1) can be written as

˙x =     f(x1, t) .. . f(xm, t)    + d(G ⊗ D)x =: F(x, t) + d(G ⊗ D)x , (2.3a)

where ⊗ denotes the Kronecker product, and

f(xi, t) =     f1(xi, t) .. . fn(xi, t)     . (2.3b) We next impose conditions on coupling matrices G and D. We assume that cou-pling matrix G satisfies the following:

(i) λ = 0 is a simple eigenvalue of G and e = [1, 1, . . . , 1]T 1×m is

its corresponding eigenvector. (2.4a)

(ii) All nonzero eigenvalues of G have negative real part. (2.4b) We further assume that coupling matrix D is, without loss of generality, of the form

D= Ik 0 0 0

!

n×n.

(2.4c) The index k, 1 ≤ k ≤ n, means that the first k components of the individual system are coupled. If k 6= n, then the system is said to be partial-state coupled. Otherwise, it is said to be full-state coupled.

From time to time, we will refer system (2.3) as the coupled system (D, G, F(x, t)).

To study synchronization of such system, we permute the state variables in the fol-lowing way: ˜ xi=     x1,i .. . xm,i     , and ˜x=     ˜ x1 .. . ˜ xn     . (2.5) 3

Then (2.3) can be written as ˙˜x =     ˜f1(˜x, t) .. . ˜fn(˜x, t)    + d(D ⊗ G)˜x =: ˜ F(˜x_{, t) + d(D ⊗ G)˜x,} (2.6a) where ˜fi(˜x, t) =     fi(x1, t) .. . fi(xm, t)     . (2.6b)

The purpose of such reformulation is two fold. First, a transformation of coordinates of ˜xis to be applied to (2.6) so as to decompose the synchronous manifold. Second, once the synchronous manifold is decomposed, proving synchronization of (2.3), is then equivalent to showing that the origin is asymptotically stable with respect to reduced system (3.3). From here on, we will treat ˜ as a function that takes x into ˜

xor xi into ˜xi.

We next give the definition of the bounded dissipation of a system.

Definition 2.1. (i) A system of n ordinary differential equations is called bounded dissipative provided that for any r > 0 and for any initial conditions x0 in Bn(r),

there exists a time t∗_{≥ t}

0 such that kx(t)k ≤ αr for all t ≥ t∗. (ii) If, in addition,

αris independent of r, then the system is said to be uniformly bounded dissipative

with respect to αr.

To prove global synchronization of coupled chaotic systems, one needs to assume bounded dissipation, which plays the role of an a priori estimate. Without such an a priori estimate, as in the case of the R¨ossler system, global synchronization is much more difficult to obtain. Only local synchronization was reported numerically in literature (see e.g., [4]). We remark that in certain cases of the R¨ossler system, the trajectory of each oscillator grows unbounded yet approaches each other (see e.g., [4]). An interesting question in this direction is how bounded dissipation of the coupled system is related to the uncoupled dynamics and its connectivity topology. Not much general theorems have been provided so far. In the case that G is diffusively coupled with periodic boundary conditions or zero-flux and D satisfies (2.4c), it was shown in [5] that bounded dissipation of the single oscillator implies that of the coupled chaotic oscillators. Moreover, the absorbing domain of the coupled system is a topological product of the absorbing domain of each individual system.

In our derivation of synchronization of system(2.3), we need the concept of matrix measures. For completeness and ease of references, we also recall the following definition of matrix measures and their properties (see e.g., [32]).

Definition 2.2. _{Let k · k}i be an induced matrix norm on Cn×n. The matrix

measure of matrix A on Cn×n_{is defined to be µ}

i(A) = lim ǫ→0+

kI + ǫAki− 1

ǫ .

Lemma 2.1. _{Let k · k}k be an induced k-norm on Rn×n, where k = 1, 2, ∞. Then

each of matrix measure µk(A) , k = 1, 2, ∞, of matrix A = (aij) on Rn×n is,

respectively, µ∞(A) = max i {aii+ X j6=i |aij|}, (2.7a) µ1(A) = max j {ajj+ X i6=j |aij|}, (2.7b) and

µ2(A) = λmax(AH+ A)/2. (2.7c)

Here λmax(A) is the maximum eigenvalue of A.

Theorem 2.1. (see e.g., 3.5.32 of [32]) Consider the differential equation ˙x(t) = A_{(t)x(t)+v(t), t ≥ 0, where x(t) ∈ R}n

,A(t) ∈ Rn×n,and A(t), v(t) are

piecewise-continuous. Let k · ki be a norm on Rn, and k · ki, µi denote, respectively, the

corresponding induced norm and matrix measure on Rn×n_{. Then whenever t ≥}

t0≥ 0, we have kx(t0)k exp Z t t0 −µi(−A(s))ds  − Z t t0 exp Z t s −µi(−A(τ))dτ  kv(s)kds ≤ kx(t)k ≤ kx(t0)k exp Z t t0 µi(A(s))ds  + Z t t0 exp Z t s µi(A(τ ))dτ  kv(s)kds. (2.8)

To conclude this section, we define global synchronization as in the following. Definition 2.3. (i) System (2.3) is said to be globally synchronized if for any given initial values x0 there exists a d = dx₀ such that system (2.3) is synchronized for

the initial conditions x0. Here dx₀ is a constant depending on x₀. (ii) System (2.3)

is said to be uniformly, globally synchronized if there exists a d = d1 such that

system (2.3) is synchronized for all initial values x0.

3. Main Results

To study synchronization of (2.3), we first make a coordinate change to decom-pose the synchronous subspace. Let A be an m × m matrix of the form

A=          1 −1 0 · · · 0 0 . .. ... ... ... .. . . .. ... ... 0 0 · · · 0 1 −1 1 · · · 1 1          m×m =: C eT ! , (3.1a)

where e is given as in (2.4a). It is then easy to see that CCT _{is invertible and that}

A−1=CT(CCT₎−1_| e m  . (3.1b) Setting E= In⊗ A, (3.1c) we see that

E_{(D ⊗ G)E}−1= (In⊗ A)(D ⊗ G)(In⊗ A−1)

= D ⊗ AGA−1= D ⊗ CGC T_(CCT₎−1 ₀ ∗ 0 ! =: D ⊗ G¯ 0 ∗ 0 ! . (3.1d) We remark, via (3.1d), that σ(G) − {0} = σ( ¯G), where σ(A) is the spectrum of matrix A. Multiplying E to the both side of equation (2.6a), we get

˙˜y =: E ˙˜x = E˜F(˜x, t) + dE(D ⊗ G)E−1_y˜

= E ˜F(E−1_{y˜, t) + d( D ⊗} G¯ 0 ∗ 0 ! )˜y. (3.2) Let ˜y =     ˜ y1 .. . ˜ yn     . Then ˜yi =       x1,i− x2,i .. . xm−1,i− xm,i Pm j=1xj,i       . Setting ˜yi = ¯ yi Pm j=1xj,i ! , and ¯y=     ¯ y1 .. . ¯ yn     ,

we have that the dynamics of ¯yis satisfied by following equation ˙¯y = d(D ⊗ ¯G)¯y+ ¯F(¯y, t). (3.3)

Here ¯Fis obtained from E ˜F(E−1y˜, t) accordingly.

The task of obtaining global synchronization of system (2.3) is now reduced to showing that the origin is globally and asymptotically stable with respect to system (3.3). To this end, the space ¯y is broken into two parts ¯yc, the coupled space, and

¯

yu, the uncoupled space.

¯ y= y¯c ¯ yu ! , and ¯F(¯y, t) = F_¯¯c(¯y, t) Fu(¯y, t) ! , respectively. (3.4) Here ¯yc =     ¯ y1 .. . ¯ yk     , and ¯yu =     ¯ yk+1 .. . ¯ yn    

. The dynamics on the coupled space with respect to the linear part is under the influence of ¯G, which is asymptotically stable. The dynamics of the nonlinear part on coupled space can then be controlled by choosing large coupling strength. As a matter of fact, it is easier to obtain synchronization of coupled chaotic systems with a larger coupled space. On the other hand, the uncoupled space has no stable matrix ¯G to play with. Thus, its corresponding vector field ¯Fu(¯y, t) must have a certain structure to make the

trajectory stay closer to the origin as time progresses. As we shall explain latter. Now, assume that ¯Fc(¯y, t) satisfies a dual-Lipschitz condition with a dual-Lipschitz

constant b1. That is,

k¯Fc(¯y, t)k ≤ b1k¯yk (3.5a)

whenever ¯y in the ball B(m−1)n(α), and for all time t. Since the estimate in the

right-hand side of (3.5a) depends on the whole space ¯y, condition (3.5a) is a mild assumption provided that the coupled system is bounded dissipative. Write ¯Fu(¯y, t)

as

¯

Fu(¯y, t) = U(t)¯yu+ ( ¯Fu(¯y, t) − U(t)¯yu)

=: U(t)¯yu+ ¯Ru(¯y, t). (3.5b)

Assume that U(t) is a block diagonal matrix of the form U(t) = diag(U1(t), · · · , Ul(t))

where Uj(t), j = 1, . . . , l, are matrices of size (m − 1)kj × (m − 1)kj . Here l

X

j=1

kj= n − k, and kj∈ N. We assume further that the followings hold. 7

(i) The matrix measures µi(Uj(t)) are less than −γ for all t and all j, where γ > 0. (3.5c) (ii) Let ¯Ru(¯y, t) =     Ru1(¯y, t) .. . Rul(¯y, t)     .

Then Ruj(¯y, t), j = 1, . . . , l, satisfy a strong

dual-Lipschitz condition with a strong dual-Lipschitz constant b2. Specifically, let

¯ yu=     ¯ yu1 .. . ¯ yul    

, written in accordance with the block structure of U(t). Then we assume that kRuj(¯y, t)k ≤ b2k       ¯ yc ¯ yu1 .. . ¯ yu j−1       k (3.5d)

whenever ¯yin the ball B(m−1)n(α), and for all j = 1, . . . , l and all time t.

Specifically, we break the vector field ¯Fu into (time dependent) linear part

U(t)¯yu and nonlinear part ¯Ru(¯y, t). We will further break U(t) into certain block

diagonal form if necessary. Note that form (3.5b) can always be achieved since the remainder term ¯Rustill depends on the whole space ¯y. To take control of the

dy-namics on the linear part, we assume that the matrix measure of each diagonal block Uj(t) is negative. As to contain corresponding dynamics on the nonlinear part, we

assume that (3.5d) holds. Note that though the nonlinear terms Ruj(¯y, t) could

possibly depend on the whole space, their norm estimates are required to depend only on the coupled space and uncoupled subspaces with their indexes proceeding j. In this set up, the nonlinear dynamics on uncoupled space can be iteratively controlled by choosing large coupling strength. We also remark that if (3.5c) and (3.5d) are satisfied for l, the number of diagonal blocks, being one, then we do not need to further break U(t). Such further breaking is needed only if (3.5c) and (3.5d) are not satisfied. The proof in the following theorem gives exactly how the above strategy can be realized.

Theorem 3.1. Let G and D be given as in (2.4). Assume that ¯F satisfies (3.5a-d), and system (3.3) is uniformly bounded dissipative with respect to α. Let λ1=

max{λj|λj ∈ Re(σ( ¯G))}. If d > cb1 −λ1+ ǫ  1 + (b2 γ) 2 2l =: dc, (3.6)

where ǫ ≥ 0 and c is some constant depending on G and ǫ, then lim

t→∞y(t) = 0.¯ 8

Proof. Since system (3.3) is uniformly bounded dissipative with respect to α, with-out loss of generality, we may assume that k¯y(t)k ≤ α for all time t ≥ t0. Using

(3.5b), we write (3.3) as ˙¯yc ˙¯yu ! = d(Ik⊗ ¯G) 0 0 U(t) ! ¯ yc ¯ yu ! + F_¯¯c(¯y, t) Ru(¯y, t) ! . (3.7a) Applying the variation of constant formula to (3.7a) on ¯yc, we get

¯ yc(t) = e(t−t0)d(Ik⊗ ¯ G₎ ¯ yc(t0) + Z t t0 e(t−s)d(Ik⊗ ¯G)_F¯ c(¯y(s), s)ds.

Let λ1= max{ λj|λj∈ Re( σ(G) − {0} ) }. Then

ketd(Ik⊗ ¯G)_{k ≤ ce}tdν _(3.7b)

for ν = λ1+ ǫ and some constant c. Here 0 < ǫ < −λ1. Thus,

k¯yc(t)k ≤ ce(t−t0)dνk¯yc(t0)k + cb1 Z t t0 ed(t−s)νk¯y(s)kds ≤ ce(t−t0)dν_{α +}α d cb1 |ν| =: ce (t−t0)dν_{α +}α dc0. Let δ > 1, we see that

k¯yc(t)k ≤

α

dc0δ, (3.8a) whenever t ≥ t0,1 for some t0,1 > 0. We then apply Theorem 2.1 on ¯yu1 and the

resulting inequality is

k¯yu1(t)k ≤ k¯yu1(t0,1)k exp

( Z t t0,1 µi(U1(s))ds ) + Z t t0,1 exp Z t s µi(U1(τ ))dτ  kRu1(¯y(s), s)kds.

It then follows from (3.5c-d) and (3.8a) that k¯yu1(t)k ≤ αe−γ(t−t0,1)+ α d b2 γc0δ ≤ α d b2 γc0δ 2_=:α dc1δ 2 , (3.8b) 9

whenever t ≥ t1,1 for some t1,1 ≥ t0,1. Inductively, we get k¯yuj(t)k ≤ α d   b2 γ v u u t j−1 X i=0 c2 i  δj+1=: α dcjδ j+1_{, j = 2, . . . , l, (3.8c)}

whenever t ≥ tj,1(≥ tj−1,1). Letting tl,1 = t1 and summing up (3.8a), (3.8b) and

(3.8c), we get k¯y(t)k = v u u t l X j=1 k¯yuj(t)k2+ k¯yc(t)k2 ≤ α d  1 + (b2 γ) 2 2l _cb 1 |ν|δ l+1_{=: hα,} whenever t ≥ t1. Choosing d >  1 + (b2 γ) 2 l 2 _cb 1 |ν|δ

l+1_{, we see that the contraction}

factor h is strictly less than 1, and k¯y(t)k contracts as time progresses. To complete the proof of the theorem, we note that δ > 1 can be made arbitrary close to 1. Consequently, if d > 1 + (b2 γ) 2 l 2 _cb 1

|ν|, then h can still be made to be less than

1. 

Remark 3.1. (i) In case that ¯Gis symmetric, then c and ǫ can be chosen to be one and zero, respectively. (ii) b1and b2could possibly depend on α. (iii) If system

(3.3) is only bounded dissipative, then the estimate in (3.6) is still valid. However, in this case, b1 and b2 depend not only on α but also on x0.

Corollary 3.1. Suppose ¯Fand G are given as in Theorem 3.1. Let D= D¯k×k 0

0 0 !

n×n,

where Re( σ( ¯D) ) > 0. (3.9a) Assume, in addition, that either σ(G) or σ( ¯D) has no complex eigenvalue. Then assertions in Theorem 3.1 still hold true, except dc needs to be replaced by

dc= c b1 |ν| min{Re( σ( ¯D_{) )}}  1 + (b2 γ) 2 2l . (3.9b)

Proof. Assumption on D is to ensure that (3.7b) is still valid. Other parts of the proof are similar to those in Theorem 3.1 and are thus omitted.  We next turn our attention to finding conditions on the nonlinearities fi(u, t),

i = 1, . . . , n, u ∈ Rn_{, so that assumptions (3.5a-d) are satisfied. To this end, we}

need the following notations. Let ˜xi and ˜xbe given as in (2.5). Define

[˜xi]−=     x1,i .. . xm−1,i     , and [˜x]−=     [˜x1]− .. . [˜xn]−     . (3.10) 10

We then break ˜Fas given in (2.6a) into two parts so that the breaking is in consis-tent with ¯yin (3.4). Specifically, we shall write

˜ F(˜x, t) = F˜_˜c(˜x, t) Fu(˜x, t) ! . (3.11) We are now in the position to state the following propositions.

Proposition 3.1. Suppose that fi(x, t), i = 1, 2, . . . , k satisfy a Lipschitz condition

in Bn(α₂) with a Lipschitz constant b1. That is

|fi(u, t) − fi(v, t)| ≤

b1

k ku − vk,i = 1, 2, . . . , k, (3.12) for all u, v in Bn(α₂) and all time t. Then (3.5a) holds true.

Proof. Note that E ˜F(˜x, t) =     A˜f1(˜x, t) .. . A˜fn(˜x, t)     ,

where A is given as in (3.1a), and so

[A˜fi(˜x, t)]−=     fi(x1, t) − fi(x2, t) .. . fi(xm−1, t) − fi(xm, t)     , i = 1, 2, . . . , n. (3.13) Since ¯ Fc(¯y, t) =     [A˜f1(˜x, t)]− .. . [A˜fk(˜x, t)]−     ,

we conclude that (3.5a) holds.  From the above proposition, we see that the nonlinearities on the corresponding coupled space are only assumed to be Lipchitz. The following proposition is very useful in the sense that by checking how each component fi of the nonlinearity f is

formed, one would then be able to conclude whether (3.5c-d) are satisfied.

Proposition 3.2. Let u = (u1, . . . , un)T and v = (v1, . . . , vn)T be vectors in

Bn(α₂). Let wp= p

X

i=0

ki, p = 1, . . . , l, where k0= k, the dimension of coupled space,

and k1, . . . , kl and l are given as in (3.5c). Write fwp−1+i(u, t) − fwp−1+i(v, t),

i = 1, . . . , kp,as fwp−1+i(u, t) − fwp−1+i(v, t) = kp X j=1

qwp−1+i,wp−1+j(u, v, t)(uwp−1+j− vwp−1+j) + rwp−1+i(u, v, t).

(3.14a) We further assume that the followings are true.

(i) For p = 1, . . . , l, let Qu,v,p= (qw_p−1+i,wp−1+j(u, v, t)), where 1 ≤ i, j ≤ kp.

Then µ∗(Vp) < −γ for all p, u, v in Bn(α₂) and all time t, where ∗ =

1, 2, ∞. (3.14b) (ii) Let rp= rwp−1+1(u, v, t), . . . , rwp(u, v, t)

T . We have that krpk ≤ b2k     u1− v1 .. . uw_p−1− vw_p−1    k (3.14c) for all p, u, v in Bn(α₂) and all time t.

Then (3.5c) and (3.5d) hold true for ∗ = 1, 2, ∞.

Proof. Since ri(u, v, t) depend on the whole space, fi(u, t) − fi(v, t) can always be

written as the form in (3.14a). Using (3.14a) and (3.13), we have that the matrices Up(t) in the linear part of ¯Fu(¯y, t) take the form

Up(t) = m−1

X

w=1

Qx_w,xw+1,p(t) ⊗ Dw, (3.15)

where xw are given as in (2.2), and

(Dw)_ij=

(

1 i = j = w,

0 otherwise, 1 ≤ i, j ≤ m − 1.

It then follows from (2.7a,b), and (3.15) that µ∗(Up(t)) < −γ for ∗ = 1 or ∞. For

∗ = 2, we have that m−1 [ w=1 σ{Qx_w,xw+1,p(t) + Qxw,xw+1,p(t)
T } = σ (_m−1 X w=1  Qx_w,xw+1,p(t) ⊗ Dw+ Qxw,xw+1,p(t)
T ⊗ Dw  ) = σ Up(t) + UTp(t)
, 12

where σ(A) is the spectrum of A. We remark that the first equality above can be verified by the definition of eigenvalues due to the structure of Up(t). It then

follows from (2.7c) that µ2(Up(t)) < −γ. The remainder of the proof is similar to

that of Proposition 3.1, and is thus omitted.  Remark 3.2. The upshot of Proposition 3.2 is that by only checking the “struc-ture” of the vector field f of the single oscillator, one should be able to determine if our main result can be applied. To be precise, we begin with saving notations by setting f as f = f (x, t) = (f1(x, t), . . . , fn(x, t))T. We then check the form of the

difference of “uncoupled” part of dynamics. That is, we write fi(u, t) − fi(v, t) in

the form of (3.14a) with i = k + 1, . . . , n. If (3.14b, c) can be satisfied, then l = 1 gets the job done. Otherwise, we further break the uncoupled states into a set of smaller pieces to see if the resulting (3.14b, c) are satisfied.

We are now ready to state the main theorems of the paper.

Theorem 3.2. Assume that system (2.3) is (resp., uniformly) bounded dissipative. Let coupling matrices G and D satisfy (2.4) and the nonlinearities fi(x, t), i =

1, 2, . . . , n, satisfy (3.12) and (3.14). Suppose d is greater than dc, as given in

(3.6). Then system (2.3) is (resp., uniformly,) globally synchronized.

Proof. The proof is direct consequences of Propositions 3.1 and 3.2, and Theorem

3.1. 

Remark 3.3. From here on, we will refer the assumptions in Theorem 3.2 as synchronization hypotheses.

Theorem 3.3. Coupled system (D, G, F(x, t)), given as in Corollary 3.1, is also (resp., uniformly,) globally synchronized provided that its coupled system is (resp., uniformly) bounded dissipative and that d is greater than dc. Here dc is given in

(3.9b).

4. Applications

To see the effectiveness of our main results, we consider two examples in this section. These are coupled Lorenz equations [7, 20], and coupled Duffing oscillators [39].

(I)We shall begin with Lorenz equations. Let x = (x1, x2, x3)T,

f(x, t) = f (x) = (σ(x2− x1), rx1− x2− x1x3, −bx3+ x1x2)T

=: (f1(x), f2(x), f3(x))T. 13

Here σ = 10, r = 28 and b = 8₃. In the following cases (a), (b), (c) and (d), G denotes the diffusive coupling with zero flux and D is, respectively,

   1 0 0 0 0 0 0 0 0    ,    0 0 0 0 1 0 0 0 0    ,    0 0 0 0 0 0 0 0 1    , and    0 0 0 0 1 1 0 0 1    .

For the first three cases, it was shown in [5] that such the coupled system (D, G, F(x)) have the topological product of an absorbing domain

B = {x21+ x22+ (x3− r − σ)2<

b2_{(r + σ)}2

4(b − 1) =: β}. (4.1) Hence, in each case, we will concentrate on the illustration of how our main results may or may not be applied.

(a) Let D = D1=    1 0 0 0 0 0 0 0 0    .

For “coupled” nonlinearity f1, we get that

|f1(u) − f1(v)| = σ|(u2− v2) − (u1− v1)| ≤

√

2σku − vk.

Hence, condition (3.5a) is satisfied. For “uncoupled” nonlinearities f2 and f3, we

see that

f2(u) − f2(v) = (−u2− u1u3+ ru1) − (−v2− v1v3+ rv1)

= [−(u2− v2) − u1(u3− v3)] + (r − v3)(u1− v1) (4.2a)

and

f3(u) − f3(v) = (u1u2− bu3) − (v1v2− bv3)

= [u1(u2− v2) − b(u3− v3)] + v2(u1− v1). (4.2b)

Writing (4.2a,b) in the vector form, we get f2(u) − f2(v) f3(u) − f3(v) ! = −1 −u1(t) u1(t) −b ! u2− v2 u3− v3 ! + (r − v3)(u1− v1) v2(u1− v1) ! =: Qu,v,1(t) u2− v2 u3− v3 ! + r1. (4.2c) 14

Clearly, µ2(Qu,v,1(t)) = max{−1, −b} = −1 < 0, and kr1k ≤ (σ +√β) · |u1− v1|,

where its estimate depends only on coupled space. Hence, conditions (3.14b,c) are satisfied. (b) Let D = D2 =    0 0 0 0 1 0 0 0 0    .

As in the case (a), the “coupled” nonlinearity f2 is clearly Lipschitz on the absorbing domain. The difference of “uncoupled”

nonlinearities f1and f3are given as follows.

f1(u) − f1(v) = [−σ(u1− v1)] + σ(u2− v2),

f3(u) − f3(v) = [−b(u3− v3)] + u1(u2− v2) + v2(u1− v1).

If l = 1 is chosen, then (3.14c) is violated. For in the case, the norm estimate in the right hand side of (3.14c) can only depend on u2− v2. Now, if we choose

l = 2 and pick the space of the first diagonal block being the one associated with the nonlinearity f1, then Qu,v,1 = (−σ) and r1 = σ(u2 − v2). Consequently,

(3.14b) and (3.14c) are satisfied. Moreover, we have Qu,v,2 = (−b) and r2 =

u1(u2− v2) + v2(u1− v1), which depends only on the coupled space and the first

uncoupled space. Thus, r2 satisfies (3.14c).

(c) For illustration, we also consider D = D3 =

   0 0 0 0 0 0 0 0 1    .

In this case, the uncoupled nonlinearities of f1 and f2 both contain the terms x2 and x1. The only

feasible choice to break the uncoupled space is not to do any breaking. Conse-quently, Qu,v,1 = −σ σ

r − u3(t) −1

!

. For such Qu,v,1, its matrix measure can

not stay negative for all time. An indicated, see e.g., [20], synchronization fails for this type of partial coupling.

(d) Let D = D4 =    0 0 0 0 1 1 0 0 1    .

To apply Theorem 3.3, we first note that for

D = D5 =    0 0 0 0 1 0 0 0 1    ,

the corresponding coupled system (D5, G, F(x)) is

indeed globally synchronized, and hence, so is the system (D4, G, F(x)). Note that

bounded dissipation of the system (D4, G, F(x)) can be verified similarly as in [20]. 15

(e) The work that are most related to ours are those in [7,8]. While their esti-mates for dc seems to be sharper than ours, which we shall illustrate in case (f),

their connectivity topology requires that off-diagonal entries be nonnegative. We only assume our connectivity topology satisfies (2.4a,b). Consider for instant the following matrix: G=      −1 2 0 −1 −1 −1 0 2 2 −1 −3 2 0 0 3 −3      .

Such G has some negative off-diagonal entries and satisfy (2.4a,b). In fact, the eigenvalues of G are 0, −1 ±√5i, and −6. Clearly, applying our results, we see immediately that the coupled system (Di, G, F(x)), i = 1, 2, 4 are globally

synchro-nized. Numerical results (see Figure 4.1.) indeed confirm synchronization of such connectivity topology. We remark that by constructing the Lyapunov function as given in [20], one would be able to show bounded dissipation of the coupled system with this particular connectivity topology.

0 10 20 30 40 50 60 70 80 −20 0 20 40 time x−different 0 20 40 60 80 100 −40 −20 0 20 time y−different 0 20 40 60 80 100 −50 0 50 time z−different

Figure 4.1. _{The difference of each component of two coupled} os-cillators in case (e).

(f) In this part, we shall compute the lower bound for global synchronization for case (a) by using our method, those obtained in [7] and MSF, respectively. To compute dc, given in (3.6), we note that ¯G = CGCT(CCT)−1 = C(CTC)CT(CCT)−1

= CCT_{. Since ¯Gis symmetric, c and ǫ, given as in (3.7b), can be chosen to be 1,}

and 0, respectively. Consequently, dc = √ 2σp1 + β + 2σ√β + σ2 4 sin2(π 2n) . (4.3) Here 4 sin2(π

2n) = |λ1|. Applying Theorem 3.3, we see that the coupled

sys-tem (D, G, F(x)) is uniformly, globally synchronized provided that the coupling strength d is greater than dc. For n = 4, dc ≈ 1189. In [7], the bound ¯dc for

threshold of uniformly global synchronization is ¯ dc= ( _a 8n2 if n is even a 8(n 2_{− 1) if n is odd} .

Here a = b(b+1)(r+σ)_16(b−1) 2 − σ. For n = 4, ¯dc ≈ 1039, which is slightly better than dc.

Using the MSF-criteria, we numerically (see Figure 4.2.) compute the maximum Lyapunov exponent of the variational equations with respect to the parameter α. We have in this example that if

α = dλ1< −7.778, (4.4)

then its maximum Lyapunov exponent is negative. Here λ1 = −4 sin2 π₈ is the

largest nonzero eigenvalues of G. Hence if d > −7.778

λ1 ≈ 13.3, then local

synchro-nization of the coupled system (D, G, F(x)) can be realized.

−15 −10 −5 0 0 1 2 3 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 β α −0.4 −0.2 0 0.2 0.4 0.6 0.8

Figure 4.2. _{The vertical axis denotes the maximum Lyapunov} exponent of the variational equations. While the horizontal axis represents α = dλ.

(II)Another formulation not considered in [7,8] is the Duffing oscillators. Specif-ically, the individual system considered is defined by

˙x1= −αx1− x32+ a cos wt (4.5a)

˙x2= x1, (4.5b)

where α and a are positive constants. Letting x = (x1, x2)T, we have

f(x, t) = (f1(x, t), f2(x)) = (−αx1− x32+ a cos wt, x1). (4.6a)

Assume coupling matrices D and G are, respectively, D(c) = 1 c 0 0 ! (4.6b) and G(ǫ, r) =              −2ǫ ǫ − r 0 · · · 0 ǫ + r ǫ + r −2ǫ ǫ − r . .. 0 0 . .. . .. . .. . .. ... .. . . .. . .. . .. . .. 0 0 . .. . .. −2ǫ ǫ − r ǫ − r 0 · · · 0 ǫ + r −2ǫ              , (4.6c)

where ǫ > 0 and r are scalar diffusive and gradient coupling parameters, respec-tively. Note that

f2(u) − f2(v) = 0(u2− v2) + (u1− v1)

and so the matrix measure of the corresponding Qu,v,1 is zero. To apply our

theorem, we need to make the following coordinate change.

Letting y2= x2 and y1= qx1+ px2, we see that (4.5a,b) becomes

˙y1= (p

q − α)y1+ p(α − p

q)y2− qy

3

2+ qa cos wt =: ¯f1(y) (4.7a)

˙y2=−p

q y2+ 1

qy1=: ¯f2(y), (4.7b) and the corresponding coupled system (3.2) becomes

˙˜y1= (p q− α)˜y1+ p(α − p q)˜y2− q˜y 3 2+ g(t) + d(qc − p)G(ǫ, r)˜y2+ dG(ǫ, r)˜y1 (4.8a) ˙˜y2= − q py˜2+ 1 qy˜1, (4.8b) 18

where ˜y3₂ = (y3

1,2, . . . , ym,23 )T and g(t) = a cos(wt) (1, · · · , 1)T. In the following, we

choose (p, q) to be (1, c −1d) as c > 0, and to be (−1, − 1

d) as c = 0, respectively.

Then in the case of c > 0, (4.8) becomes ˙˜y1= dG(ǫ, r)˜y1+ (c − α − 1 d)˜y1+ (α − c + 1 d)˜y2− ˜y 3 2+ g(t) + G(ǫ, r)˜y2 =: dG(ǫ, r)˜y1+ ˜Fc(˜y, t) ˙˜y2= − 1 c −1 d ˜ y2+ ˜y1.

The purpose of the coordinate transformation is two-fold. First, to make the dy-namics of the linear part on the uncoupled space stable. In this case, the coefficient of ˜y2 becomes negative when d > 2_c. Second, to make sure the parameters in the

nonlinear part of coupled space contain no bad influence of d, coupling strength. Otherwise, we may not be able to control its corresponding dynamics by choosing d large.

It is then easy to check that assumptions for Theorem 3.1 are all satisfied, and similar arguments can be followed for the case of c = 0. Finally, in Appendix, we will show that if _4+αm4α 2 > c ≥ 0, ǫ > 0 and r ∈ R, then the coupled system

(D(c), G(ǫ, r), F(x, t)) is bounded dissipative. Thus, we can summarize the results as follows

Theorem 4.1. Let f , D(c) and G(ǫ, r) be given as in (4.6a), (4.6b) and (4.6c), respectively. Let 0 ≤ c < 4+α4α2_m. Then the coupled system (D(c), G(ǫ, r), F(x, t))

is globally synchronized provided that d is chosen sufficiently large.

Proof. It remains only to verify that G(ǫ, r) satisfies assumptions (2.4a,b). Indeed G(ǫ, r) is a circulant matrix (see e.g., [13]), the eigenvalues λk of G(ǫ, r) are

λk = −2ǫ(1 − cos 2kπ n ) − i 2r sin 2kπ n , k = 0, . . . , m − 1.  Remark 4.1. (i) It was shown in [17] that there are positive constants d0 and

c0 such that, for d ≥ d0, c ≥ c0, the system (D(c), G(ǫ, 0), F) given in (4.7) is

synchronized. Our results also work for the case that c0 is zero or small or G(ǫ, r),

r 6= 0. (ii) It was shown in [1] that there are positive constants d0 and c0such that

for d ≥ d0, c ≥ c0, the system (D(c), G, F) is synchronized. Here −G is a positive

definite matrix.

5. Conclusion

We have developed theory to prove global synchronization in lattices of coupled chaotic systems. The results can be applied to quite general connectivity topol-ogy. In fact, it needs only to satisfy (2.4). In addition, a rigorous lower bound on the coupling strength to acquire global synchronization of the coupled system is

obtained. Moreover, by merely checking the structure of the vector field of single oscillator and verifying bounded dissipation of the coupled system, we shall be able to determine if the coupled system is synchronized or not. We conclude this paper by mentioning some possible future work. First, it is of great interest to extend our method to study the real world topology. Second, it is certainly worthwhile to study how bounded dissipation of the coupled system is related to the uncoupled dynamics and its connectivity topology. Third, it is interesting to study (global) synchronization of coupled system which lacks bounded dissipation such as the R¨ossler system.

ACKNOWLEDGMENT

We thank referees for suggesting numerous improvements to the original draft. Some future work from one of the referees is also greatly appreciated.

Appendix _A

In this appendix, we prove bounded dissipation of the systems considered in (4-II). Setting ˜x3₂ = (x3

1,2, . . . , x3m,2)T, and g(t) = a cos(wt) (1, · · · , 1)T. We see that

(2.6) becomes

˙˜x1= −α˜x1− ˜x32+ g(t) + dcG(ǫ, r)˜x2+ dG(ǫ, r)˜x1 (A.1a)

˙˜x2= ˜x1. (A.1b)

We consider the following scalar-valued function as the Lyapunov function of the coupled system (D(c), G(ǫ, r), F(x, t)) U (˜x1, ˜x2) =1 2 < ˜x1, ˜x1> + m X i=1 x4 i,2 4 + c < ˜x2, ˜x1>, (A.2)

Taking the time derivative of U along solutions of the coupled system (D(c), G(ǫ, r), F(x, t)), we have dU dt =< ˜x1, ˙˜x1> + m X i=1 x3i,2xi,1+ c < ˜x1, ˜x1> +c < ˜x2, ˙˜x1> = (c − α) < ˜x1, ˜x1> −cα < ˜x2, ˜x1> −c < ˜x2, ˜x32> + < ˜x1+ c˜x2, g(t) > + d < ˜x1, G(ǫ, r)˜x1> +2dc < ˜x1, G(ǫ, r)˜x2> +dc2< ˜x2, G(ǫ, r)˜x2> = (c − α) < ˜x1, ˜x1> −cα < ˜x2, ˜x1> −c < ˜x2, ˜x32> + < ˜x1+ c˜x2, g(t) > + d (˜x1, ˜x2) 1 c c c2 ! ⊗ G(ǫ, r) ! ˜ x1 ˜ x2 ! ≤ (c − α) < ˜x1, ˜x1> −cα < ˜x2, ˜x1> −c < ˜x2, ˜x32> + < ˜x1+ c˜x2, g(t) > 20

Note that the last inequality holds true since 1 c c c2 ! ⊗ G(ǫ, r) ! + 1 c c c2 ! ⊗ G(ǫ, r) !T = 1 c c c2 ! ⊗ (G(ǫ, r) + G(ǫ, r)T₎ ,

and G(ǫ, r) + G(ǫ, r)T _{is a nonpositive definite matrix. On the other hand, since}

< ˜x2, ˜x32>= m X i=1 x4 2,i≥ 1 m m X i=1 x2 i,2 !2 ≥ 1 mk˜x2k 4 2, we have dU dt ≤ (c − α)k˜x1k 2 2+ cαk˜x2k2k˜x1k2− c mk˜x2k 4 2+ √ ma(k˜x1k2+ ck˜x2k2) =: u(k˜x2k1, k˜x2k2).

We are now in a position to show bounded dissipation of the coupled system (D(c), G(ǫ, r), F(x, t)).

Proposition A.1.

(i) If c satisfies the inequality

0 < c < min{_{4 + α}4α2_m, α} =

4α 4 + α2_m

.

(A.3) Then there exists a constant c0 so that dUdt < 0 for k˜x2k

2

1+ k˜x2k22≥ c0.

(ii) If c = 0, then the first assertion of the proposition still holds true. Proof. Suppose k˜x2k2≥ 1. Then

u(k˜x1k2, k˜x2k2) ≤ (c − α)k˜x1k22+ cαk˜x2k2k˜x1k2− c mk˜x2k 2 2+ √ ma(k˜x1k2+ ck˜x2k2) =: ¯u(k˜x1k2, k˜x2k2).

It then follows from (A.3) that the the level curve of ¯u is a bounded closed curve. We shall call such curve ellipse-like is an elliptic in the plane. Thus, there exists a c1so that dU_dt < 0 whenever k˜x2k21+ k˜x2k22≥ c1and k˜x2k2≥ 1.Let k˜x2k2< 1 and

k˜x2k21+ k˜x2k22≥ c2. Here c2 is a constant to be determined. Then

u(k˜x1k2, k˜x2k2) ≤ (c − α)k˜x1k22+ (cα +

√

ma)k˜x1k2+√mac =: h(k˜x1k2). 21

Since h(k˜x1k2) is a parabola-like curve which is open downward, there exists a

c3 > 1 such that h(k˜x1k2) < 0 whenever k˜x1k2 ≥ c3. Thus, if c2 ≥ c23+ 1, then

u(k˜x1k2, k˜x2k2) < 0 whenever k˜x2k2 < 1 and k˜x1k22+ k˜x2k22 ≥ c2. Picking c0 =

max{c1, c2}, we have that the assertion of the proposition holds true. 

Proposition A.2. Assume (A.3) holds true. Then lim

r→∞U (˜x1, ˜x2) = ∞, where

r =_pk˜x1k2+ k˜x2k2.

Proof. From (A.2), we have that U (˜x1, ˜x2) = 1 2k˜x1k 2₊ m X i=1 x4 i,2 4 + c < ˜x2, ˜x1> ≥ 1₂k˜x1k2+ 1 4mk˜x2k 4 − ck˜x2k · k˜x1k, Let 1 4mb 2

1> c2. Then suppose k˜x2k > b1, we have

U (˜x1, ˜x2) ≥

1 2k˜x1k

2_{+ c}2_k˜x

2k2− ck˜x2kk˜x1k =: h1(k˜x1k, k˜x2k).

Since the level curve of h1(k˜x1k, k˜x2k) is elliptic-like in the plane. Thus, for

any given M > 0, there exists a d1 > 0 such that U (˜x1, ˜x2) > M whenever

k˜x1k2+ k˜x2k2≥ d21 and k˜x2k > b1. Let k˜x2k ≤ b1. Then U (˜x1, ˜x2) ≥1 2k˜x1k 2 − cb1k˜x1k =: h2(k˜x1k, k˜x2k),

since h2(k˜x1k, k˜x2k) is a parabola-like curve which is open upward in the plane.

Thus, for any given M > 0, there exists a d2 > 0 such that U (˜x1, ˜x2) > M

whenever k˜x1k2+k˜x2k2≥ d22and k˜x2k ≤ b1. Picking δ = max{d1, d2}, we have that

U (˜x1, ˜x2) > M for all k˜x1k2+ k˜x2k2≥ δ2. Thus, the assertion of the proposition

holds true. 

Theorem A.1. The coupled system (D(c), G(ǫ, r), F(x, t)) is bounded dissipative if condition (A.3) holds true.

Proof. The proof is direct consequences of Propositions A.1 and A.2.  References

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