耦合混沌系統的網絡中之同步化與微波變換

國

立

交

通

大

學

應 用 數 學 系

博

士

論

文

耦合混沌系統的網絡中之同步化與微波變換

Synchronization and Wavelet Transform in

Networks of Coupled Chaotic Systems

研究生：李金龍

指導教授： 莊重 教授

耦合混沌系統的網絡中之同步化與微波變換

Synchronization and Wavelet Transform in

Networks of Coupled Chaotic Systems

研究生：李金龍 Student: Chin-Lung Li

指導教授：莊重 Advisor: Jonq Juang

國 立 交 通 大 學

應 用 數 學 系

博 士 論 文

A Dissertation

Submitted to Department of Applied Mathematics College of Science

National Chiao Tung University in Partial Fulfillment of Requirements

for the Degree of Doctor of Philosophy

in

Applied Mathematics

July 2007

耦合混沌系統的網絡中之同步化與微波變換

研究生：李金龍 指導教授： 莊 重 教授

國立交通大學

應用數學系

摘 要

本論文的目的分成兩個部分。第一部份是研究耦合混沌系統在網格中

的全域同步化。第二部份是理論地描述微波變換是如何影響所對應系

統的同步化。基於矩陣測度的概念，我們獲得在網絡上全域同步化的

穩定性。我們的結果可利用在十分廣義的拓樸連結上。更進一步地，

藉由檢驗單一系統的向量場結構，我們就可以決定此系統是否有全域

的同步化。不僅如此，我們也獲得對於所有系統全域同步化的耦合強

度的精確下界。同步化耦合強度的下界是與耦合矩陣的第二大固有值

λ

的絕對值倒數成正比的關係。然而，對於特有的拓樸連結就像是

擴散地耦合矩陣，當節點的個數增加時，λ

對零點越靠近。總結的

來說，為了實現同步化，較大的耦合強度是被要求的。在[48]，魏…

等人提出由微波轉換修改拓樸連接。做了這樣的處理後，λ

=λ

(α)

變成隨著微波常數α而變。他們還發現一個臨界的微波常數α

可以

被選擇使得λ

(α

)遠離零點，而不需要關心節點的個數。這重要地

減少了臨界耦合強度的大小。當耦合矩陣是擴散耦合且具有週期與諾

曼的邊界條件時，這種現象將被分析地證實。

Synchronization and Wavelet Transform in

Networks of Coupled Chaotic Systems

Student: Chin-Lung Li Advisor: Jonq Juang

Department of Applied Mathematics

National Chiao Tung University

Abstract

The purpose of this thesis is two-fold. First, global synchronization in

lattices of coupled chaotic systems is studied. Second, how wavelet

transforms affect the synchronization of the corresponding systems is

theoretically addressed. Based on the concept of matrix measures, global

stability of synchronization in networks is obtained. Our results apply to

quite general connectivity topology. 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. In addition, a rigorous

lower bound on the coupling strength for global synchronization of all

oscillators is also obtained. The lower bound on the coupling strength for

synchronization is proportional to the inverse of the magnitude of the second

largest eigenvalue λ

of the coupling matrix. However, for a typical

connectivity topology such as the diffusively coupled matrix, λ

moves

closer to the origin, as the number of nodes increases. Consequently, a larger

coupling strength is required to realize synchronization. In [48], Wei et al,

proposed a wavelet transform to alter the connectivity topology. In doing so,

λ

=λ

(α) becomes a quantity depending on wavelet parameter α. It is found

there that a critical wavelet parameter α

can be chosen to move λ

(α

) away

from the origin regardless the number of nodes. This in turn greatly reduces

the size of the critical coupling strength. Such phenomena are analytically

verified when the coupling matrix is diffusively coupled with periodic and

Neumann boundary conditions.

誌 謝

本論文得以完成，首先要感謝我的指導教授 莊 重教授在這四

年來給我的鼓勵、栽培、教導與包容。老師除了擁有卓越的專業知識

外，其嚴謹的研究態度與積極追求新知的熱誠，更是學生在研究旅程

上永遠的標竿。也感謝口試委員 林文偉教授、賴明治教授、郭忠勝

教授、李明佳教授以及許正雄教授，在論文口試時給我的寶貴建議，

使論文更甄完善。而這些建議更提供我作為未來研究生涯中的一個研

究方向。

在交大應數系求學的過程中，我要感謝 王夏聲教授、許義容教

授、石至文教授以及李榮耀教授給予我在課業上的教導。也感謝鄭昌

源學長、謝世峰學長、蔡宗龍學長、陳賢修學長、陳人豪學長、林英

杰學長、林吟衡學姐以及曾育豪適時的給予我幫助與鼓勵。還有許多

的朋友與學弟、妹：郁泉、明湟、雅文、士嘉、奐勛、靖尉、明誠、

志鴻、恭儉、梁育豪、俊銘，因為有你們讓我這四年的生活更多采多

姿。

感謝父親李秀彥先生、母親徐月娥女士的養育之恩、用心栽培、

無微不至的照顧與無怨無悔的付出。對於我的理想給予最大的支持與

肯定，在我疲倦、無助的時候，提供我最安全的依靠。也感謝姊姊佩

芳在辛苦工作之餘，替我分擔照顧父母親的責任，讓我能無後顧之憂

的完成學業。你們總是相信我可以克服所有的困難，同時你們關心話

語，讓我能夠繼續勇敢的面對挑戰。

最後要感謝的是陪我走了八年的女朋友心眉以及她的家人：張聰

騰先生、王金子女士以及景嵐，感謝你們的信任與支持，讓心眉在我

遇到挫折與壓力的時候默默的聆聽我抒發情緒的話語；一路上不斷的

鼓勵與陪伴著我，因為有她才讓我能有動力繼續的往前邁進完成學

業。我將永遠的銘記在心，也將在未來實現我給心眉幸福安定的承

諾。最後，我將這份榮耀獻給我的家人與心眉一家人。

Contents

1 Coupled Systems in Lattices 1

1.1 Introduction and Formulation . . . 1

1.2 Description of the Results . . . 6

1.3 Related Work . . . 7

2 Review of Local Synchronization and Global Synchronization 9 2.1 Master Stability Function . . . 9

2.2 Matrix Measure Criteria . . . 11

2.3 Definitions of Global Synchronization . . . 13

2.4 Lyapunov Function Approach . . . 15

2.4.1 Belykh . . . 15

2.4.2 Wu and Chua . . . 18

2.5 Partial Contraction Approach . . . 21

3 Global Synchronization via Matrix Measures Approach 24 3.1 Preliminaries . . . 25

3.2 Global synchronization results . . . 27

3.3 Applications . . . 38

3.3.1 Coupled Lorenz System . . . 38

3.3.2 Coupled Chaotic Walks System . . . 42

3.3.3 Coupled Duffing Oscillators . . . 46

4 Wavelet Method for Chaotic Control 56 4.1 Wavelet Method for the Diffusively Coupled with Mix Boundary

Con-ditions . . . 56 4.2 Perturbed Block Circulant Matrix and Their Eigenvalue Problems . . . 60 4.3 The Chaotic Control for Periodic and Neumann Boundary Conditions . 67 4.4 Numerical Illustrations for Periodic and Neumann Boundary Conditions 74 4.4.1 Periodic Boundary Conditions . . . 74 4.4.2 Neumann Boundary Conditions . . . 87

Chapter 1

Coupled Systems in Lattices

1.1

Introduction and Formulation

Coupled chaotic systems are typically synthesized from simpler, low-dimensional sys-tems to form new and more complex syssys-tems. This is often done with the intent of realistically modeling spatially extended systems, with the brief that dominant features of the underlying constituents will be retained. From an applications point of view this building up approach can also be used to create a novel system whose behavior is more flexible or richer than that of the constituents, but whose analysis and/or control re-mains tractable. These and other motivations have led to numerous studies of coupled systems in a wide range of disciplines. Synchronization has long been of interest in systems of identical or nearly identical coupled subsystems. The phenomenon of syn-chronization of coupled chaotic systems has recently become a topic of great interest, and is the focus of the present work. Systems that display this behavior are temporally chaotic, but spatially ordered or coherent. Here the coherence is of particular type-the dynamics is the same or nearly so for long periods of time for all coupled subsystems or large regions of them. The basic synchronization problem can be framed with the ques-tions, ”Will my system ever synchronize and, if so, under what conditions?” During the last few decades the study of networks of dynamical systems has attracted increasing attention [16-18,21-22,24,28-29,30-33,35-36,38-39,41-42,45-46,50-54]. 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 [17].

A particularly interesting form of dynamical behavior occurs in networks of coupled systems or oscillators when all of the subsystems behave in the same fashion; that is, they all do the same thing at the same time. Such behavior of a network simulates a continuous system that has a uniform movement, models neurons that synchronize, and coupled synchronized lasers and electronic circuit systems. The motion of the systems is described as follows. Let there be m nodes (oscillators). Assume xi is the n-dimensional vector of dynamical variables of the ith node. Let the isolated (un-coupling) dynamics be ˙xi = f(xi, t) for each node. We assume that xi has a chaotic dynamics in the sense that its largest Lyapunov exponent is positive. Let h : Rn_{→ R}n be an arbitrary function describing the coupling within the components of each node. The connectivity topology, indicating the coupling rules between nodes, is denoted by the coupling matrix G = (gij). Then the equation of the motion reads

dxi dt = f(xi, t) + d m X j=1 gijh(xj), i = 1, 2, ..., m, (1.1)

where d is a coupling strength. The m− 1 constraints x1 = x2 =· · · = xm define the synchronization manifold M. The sum

m X j=1

gij = 0 is required for the invariance of this synchronization manifold M. We further assume that 0 is a simple eigenvalue of G. Let F(x, t) = (f(x1, t), f(x2, t), . . . , f(xm, t))T_{, H(x) = (h(x1), h(x2), . . . , h(xm))}T_{, and} x=    x1 ... xm    , x_i =    xi,1 ... xi,n    . (1.2)

Then (1.1) can be written as the vector form dx

dt = F(x, t) + d(G⊗ In)H(x), (1.3)

where _{⊗ is the Kronecker product(see e.g.,[15]), and I}n is the n× n identity matrix. The simplest mode of the coordinated motion between dynamical systems is their

complete synchronization when all cells of the network acquire identical dynamical behavior. 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 state becomes stable when the coupling strength between the oscillators exceeds a critical value. In this context, a central problem is to find the bounds for the coupling strengths so that the stability of synchronization is guaranteed. It is well-known (see e.g., [4,48,49]) that the lower bound for the coupling strength for synchronization is proportional to the inverse of the magnitude of the second largest eigenvalue λ2 of the coupling matrix. However, for a typical connectivity topology such as the diffusively coupled matrix, λ2 moves closer to the origin, as the number of nodes increases. Consequently, a larger coupling strength is required to realize synchronization. As a result, controlling chaos is apparently of great interest and importance [20,34,37,39,48-49]. It is found in [48] that the modification of a tiny fraction of wavelet subspaces of a coupling matrix could lead to a dramatic change in the properties of chaotic synchronization. Specifically, in doing so, λ2 = λ2(α) becomes a quantity depending on wavelet parameter α. It is found there that a critical wavelet parameter αc can be chosen to move λ2(αc) away from the origin regardless the number of nodes. This in turn greatly reduces the size of the critical coupling strength.

To be self-contained, we briefly describe such wavelet transform. Let

A =    A11 · · · A1n ... ... ... An1 · · · Ann    n×n, (1.4a)

be a matrix with the dimension of each block matrix Akl being 2i _{× 2}i_{. By an i-scale} wavelet operator W [14,48], the matrix A is transformed into W (A) of the form

W (A) =    e A11 _{· · ·} A1ne .. . . .. ... e An1 _{· · ·} Anne    n×n, (1.4b)

where each entry of eAkl is the average of entries of Akl, 1≤ k, l ≤ n.

For a given matrix, the above wavelet transform allows a perfect reconstruction (in-verse wavelet transform), by which there is nothing to gain: A = W−1(W (A)). In [48], a simple operator Ok is introduced to attain a desirable coupling matrix. That is,

C = W−1(Ok(W (A))) = A + (k− 1)W (A) =: A + αW (A), (1.4c)

where Ok be the multiplication of a scalar factor α on each block matrix eAkl. After such reconstruction, the critical strength dc is again, determined in term of the second largest eigenvalue of C. A numerical simulation of a coupled system of 512 Lorenz oscillators in [48] shows that with h = I3 and G being diffusively coupled with periodic boundary conditions, the critical coupling strength dc decreases linearly with respect to the increase of α up to a critical value αc. The smallest dc is about 6, which is about 103 _{times smaller than the original critical coupling strength, indicating the efficiency} of the proposed approach.

To understand how such wavelet transform affects the critical coupling strength, we consider G to be diffusively coupled with mix boundary conditions. Let such mix boundary conditions be parameterized by a parameter β. Such reconstructed coupling matrix Aβ + αW (Aβ) is to be denoted by G = G(α, β). Let l = m₂i ∈ N, where i is a

fixed positive integer. Here G(α, β) is an l× l block matrix of the following form.

G(α, β) =          G1(α, β) G2(α, 1) 0 _{· · ·} 0 GT₂(α, β) GT₂(α, 1) G1(α, 1) G2(α, 1) · · · 0 0 .. . . .. . .. . .. ... .. . . .. . .. . .. ... 0 0 · · · GT₂(α, 1) G1(α, 1) G2(α, 1) G2(α, β) 0 _{· · ·} 0 GT₂(α, 1) ˆIG1(α, β) ˆI          l×l. (1.5a) Here

G1(α, β) =          −1 − β 1 0 · · · 0 1 ₋₂ 1 0 _{· · ·} 0 0 1 ₋₂ 1 _{· · ·} 0 .. . . .. ... ... ... ... 0 · · · 0 1 −2 1 0 · · · 0 1 −2          2j_×2j − α(1 + β) 22j ee T =: A1(β, 2j)− α(1 + β) 22j ee T_{, (1.5b)}

where e = (1, 1, ..., 1)T_{, j is a positive integer, α > 0 is a (wavelet) scalar factor and} β ∈ R represents a mixed boundary constant. Moreover,

G2(α, β) =      0 0 · · · 0 ... ... 0 0 β 0 _{· · · 0}      2j_×2j + αβ 22jee T =: A2(β, 2j) + αβ 22jee T_{, (1.5c)} ˆI =          0 0 _{· · · 0 1} 0 0 · · · 0 1 0 .. . _{· · ·} ... .. . _{· · ·} ... 0 1 0 · · · 0 0 1 0 _{· · · 0 0}          . (1.5d)

The dimension of G(α, β) is l2j_{× l2}j_{. From here on, we shall call l and j the block and} the wavelet dimensions of G(α, β), respectively. G(α, β) is a block circulant matrix (see e.g., [15]) only if β = 1. It is well-known, see e.g., Theorem 5.6.4 of [15], that for each α the eigenvalues of G(α, 1) consists of eigenvalues of a certain linear combinations of its block matrices. Such results are called the reduced eigenvalue problem for G(α, 1).

1.2

Description of the Results

The first results in the thesis are to give another approach to study global synchro-nization of coupled chaotic systems (1.3). Part of the results in this direction is based on the paper in [27]. Our coupling rules are allowed to be asymmetric and/or some competitive (gij < 0, i _{6= j) couplings between cells x}i 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 (3.31). 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 the global synchronization of all oscillators with coupling configuration satisfying (3.20a), and (3.20b). Finally, the concept of matrix measures is introduced to obtain such global results. The second part of the thesis is to prove analytically that the improvement by wavelet transform as described in section 1.1 is indeed true. Some new phenomena are also discovered via our analysis. The results in this part are re-organized from papers in [25,26]. In the following, we give a detailed description of the results. To understand the effectiveness of wavelet transform, it amounts to study-ing the eigencurve problem for a class of ”perturbed” block circulant matrices. That is,

G(α, β)b = λ(α, β)b. (1.6)

We prove that for m being a multiple of 4, then λ2(α, 1) = ( λ+1(α, 1), 0≤ α ≤ _sin12 π l, λ+l 2 (α, 1) =_−2, α_≥ 1 sin2 π l .

Let m = 2l be an even number which is not multiple of 4. We show that λ2(α, 1) = λ+_[l

2]

(α, 1) for α sufficiently large, where [₂l] = the largest positive integer that is less than or equal to n

2. Moreover, we prove that for such m that λ2(α, 1) < −2, whenever α > 1

sin2 π l

. With those results above, we get considerable more information than those obtained in [43]. Among other, such result suggests that if the number m of oscillators be even but not a multiple of 4, then the wavelet method works even better. Specifically, it is better in the sense that the corresponding second largest eigenvalue λ2(α, 1) is

further away from 0, and, hence, gives even smaller critical length. Our second main results of this part are the following. First, the reduced eigenvalue problem for G(α, 0) is completely solved. Some partial results for the reduced eigenvalue problem of G(α, β) are also obtained. Second, we are then able to understand behavior of λ2(α, 0) and λ2(α, 1) for any j and l∈ N.

1.3

Related Work

General approaches to local synchronization of chaotic systems have been proposed, including the master stability function (MSF)- based criteria [3,35-36,38-39,42], orig-inated by Pecora and Carroll [39], and recently the matrix measure approach in [12]. 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. Moreover, local synchronization in a complex network of asymmetrically coupled units was also obtained [11,24] via MSF-based criteria.

Global synchronization of chaotic systems was also intensively studied. The methods include Lyapunov function- based criteria with symmetrical connections [4,6-9,41,50-53] or asymmetrical connections [5,50], and the partial contraction approach [45]. For Lyapunov-based criteria, the partial-state coupling matrix, determining which vari-ables couple the oscillators, is assumed to have the form satisfying (3.20c). 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 chaotic systems, one needs to assume the bounded dissi-pation 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 synchronization, one dose not need to know the 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 uncoupled system is and a term that depends on how the configuration of the networks is formed. Some of their work are to be discussed in more details in Chapters 2 and

3. The first analytical work to understand the wavelet transform was done by Shieh, Wei, Wang and Lai et al. [43]. We summary their main results in the following. Theorem 1.3.1. Let N× N, N = 8k, k ∈ N, be the dimension of the matrix G(α, 1). Let the dimension of each block matrix in G(α, 1) is 2i_{× 2}i_{. Then the following} asser-tions hold.

(i) ρi := 2 cos π

2i − 2 is an eigenvalue of G(α, 1).

(ii) The second eigenvalue λ2(α, 1) of G(α, 1) is decreasing in α. Moreover, λ2(α, 1) = ρi whenever α≥ −2 i_ρ i 4 sin2 2iπ N .

Note that G(α, 1) is a block circulant matrix (see e.g., [15]). A classical result of a block circulant matrix states that its eigenvalues exactly consist of those of a certain linear combinations of its block matrices. (see e.g., Thm 5.6.4 of [15]). The proof of Theorem 1.3.1 was then reduced to working on the eigenvalues of those linear combinations of block matrices of G(α, 1).

Chapter 2

Review of Local Synchronization

and Global Synchronization

In this chapter, we shall review some of known results for local synchronization and global synchronization in networks of coupled chaotic systems. The local theory in-cludes the master stability function (MSF), originated by Pecora and Carroll [39], and the matrix measures approach by chen [12]. The theory of global synchronization under review includes Lyapunov function approaches by Belykh [4] and Wu and Chua [53], respectively, and partial contraction approach by W. Wang, and J-J. E. Slotine[45].

2.1

Master Stability Function

In this section, we introduce the master stability function to show the stability con-dition of local synchronization of coupled system (2.15), which is developed by L. M. Pecora and T. L. Corroll [39]. In determining the stability of the synchronous state, various criteria are possible. The weakest is that the maximum Lyapunov exponent or Floquet exponent be negative. In this respect, we get the variational equation of coupled system (1.3) by letting ξi be the variations on the ith node and the collection of variations is ξ = (ξ1, ξ2, . . . , ξm)T_{. Then,}

dξ

where DF, DH are the Jacobian matrices of F and H, respectively. Equation (2.1) is used to calculate Floquet or Lyapunov exponents. We really want to consider only vari-ations ξ which are transverse to the synchronization manifold M and G is a diagonal matrix. Moreover, the Jacobian matrix DF and DH are the same for each block, since they are evaluated on the synchronized state. If we rearrange the block diagonalized variational equation in equation (2.1), this leaves us with each block having the form

dξk

dt = [Df(t) + dλkDh(t)]ξk, (2.2)

where λk is an eigenvalue of G, k = 1, 2, . . . , m. Note that we order the eigenvalues of G with decreasing order λ1 = 0 > λ2 _{≥ · · · ≥ λ}m. For k = 1, we have the vari-ational equation for the synchronization manifold M (λ1 = 0), so we have succeeded in separating that from the other, transverse directions. All other k’s correspond to transverse eigenvectors.

Thus, for each k, the form of each block in equation (2.2) is the same with only the scalar multiplier dλk differing for each. Thus, one can reformulate the above equation as follows,

dζ

dt = [Df(t) + (α + iβ)Dh(t)]ζ, (2.3) that is the master stability equation (MSE). This equation depends on the two pa-rameters α and β, and the corresponding largest Floquet or Lyapunov exponent, which is also a function of α and β, represents the master stability function (MSF). We now give a property of the MSF as follows.

Theorem 2.1.1. If the function h in (1.1) is equal to the identity function, that is DH = I, then the Lyapunov exponents Li(α, β) of the master stability equation (2.3) are

The behavior of the largest Lyapunov exponent with respect to (α + iβ) fully accounts for linear stability of the synchronization manifold. Namely, the synchronized state (associated with λ1 = 0), is stable if all the remaining blocks (associated with λi, i = 2,· · · , m) have negative Lyapunov exponents. Moreover, if we suppose that the Lyapunov exponents of (2.3) are in the decreasing order

L1(α, β)≥ L2(α, β)≥ · · · ≥ Ln(α, β) for α, β ∈ R. (2.5) Then, the stability condition can be given by

L1(α, β) = L1(0, 0) + dλ2 =: Lmax+ dλ2 < 0, (2.6) As a consequence, the second largest eigenvalue λ2is dominant in controlling the stabil-ity of chaotic synchronization and the critical coupling strength dc can be determined in terms of λ2,

dc = Lmax

−λ2 . (2.7)

2.2

Matrix Measure Criteria

In this section, another criteria for (local) synchronization is provided by M. Y. Chen [12], which is based on the matrix measure and the Lyapunov converse theorem, an-alytically. Numerically, one of the local synchronization criteria by computing the Lyapunov exponent of the MSF have been introduced. Moreover, the matrix theory can also be used to analyze the stability conditions for the synchronized chaos.

First, we introduce the concept of matrix measure [44]. The matrix measure of matrix A = (aij)∈ Rn×n _is

µ.(A) = lim ǫ→0+

kIn+ ǫA_{k − 1}

ǫ (2.8)

where _{k · k is the matrix norm, and I}n is the identity matrix.

maxiPn_j=1|aij|, we can, respectively, obtain the matrix measures µ1(A) = max j {ajj+ n X i=1,i6=j |aij|} (2.9) µ2(A) = 1 2λmax(A T _{+ A) (2.10)} µ_∞(A) = max i {aii+ n X j=1,j6=i |aij|} (2.11)

where λmax(·) denotes the maximum eigenvalue. Let Ω = {1, 2, ∞} denote the set of the above matrix measures. If θ ∈ Ω, µθ(·) is one of the matrix measures µ1(·), µ2(·), and µ_∞(_·).

Now, we present a lemma on the manifold M.

Lemma 2.2.1. If the n-dimensional linear time-varying systems in (2.2) dξk

dt = [Df(t) + dλkDh(t)]ξk, 2≤ k ≤ m

are exponentially stable about its zero solution, then the manifold M is exponentially stable for coupled system (1.3).

To assure the zero solution 0 of system (2.2) is the exponentially stable, the matrix measure of the matrix is imposed. In the following two Theorems, we assume that Dh(t) is of the following two cases.

1) Dh(t) is a symmetric positive semidefinite matrix when θ = 2. 2) Dh(t) satisfies (Dh(t))ii≥

Pn

Theorem 2.2.2. If there exists a matrix measure µθ(·) such that Z ∞

t0

µθ(Df(t) + dλ2Dh(t)) dt =−∞, (2.12) then the manifold M can be exponentially stable.

Theorem 2.2.3. If there exists a diagonal matrix P > 0, a matrix measure µθ(·), and a constant ¯d < 0 such that

Z _∞ t0

µθ (Df(t) + ¯dDh(t))TP+ P(Df(t) + ¯dDh(t))
dt = _−∞, (2.13) then the manifold M can be exponentially stable provided that dλ2 _{≤ ¯}d.

From the above analysis, the criteria given in (2.12)-(2.13) require that Dh(t) must be either a symmetric positive semidefinite matrix or a matrix satisfying (Dh(t))ii ≥ Pn

j=1,j6=i|(Dh(t))ij| for 1 ≤ i ≤ n. In the following Theorem, it can be omitted these two conditions for Dh(t).

Theorem 2.2.4. The stability of the manifold M can be transformed into the master stability equation (2.3), and the stability condition is defined as

Z ∞ t0

µθ(Df(t) + σDh(t)) dt =−∞. (2.14) The ”synchronization region” S _{6= ∅ is the set of the parameter σ satisfying (2.14).} The manifold M is exponentially stable only if dλi _{∈ S for all 2 ≤ i ≤ n.}

Based on the concept of matrix measure, this brief provides some simple synchroniza-tion criteria of complex dynamical networks. If the coupling matrix and the largest nonzero eigenvalue of the coupling matrix satisfy certain conditions, the stability of the synchronization manifold can be ensured.

2.3

Definitions of Global Synchronization

We assume the system of ordinary differential equations under consideration has a unique solution for all time and for each initial condition. We write x(t, x0, t0) for

the unique solution at time t where x0 is the initial condition at time t0. This will sometimes be simplified as x(t). Let Bk(α) be the ball in Rk _{with center at 0 and} radius α. We define the system to be synchronized if the trajectories of all the cells approach each other. We define the system to be self-synchronized if the components xi,k of each subsystem xi approach each other. Various notions of synchronization and self-synchronization are given in the following.

Definition 2.3.1. (see e.g., Definition 1 of [53]) Let a ball Bn(α) be given. Sys-tem (1.3) is uniformly (resp., self-) synchronized if for each ǫ > 0, there exists a δ(ǫ) > 0 such that if kxi(t0)− xj(t0)k ≤ δ(ǫ) (resp., |xi,k(t0)− xj,k(t0)| ≤ δ(ǫ)), and xi(t0) and xj(t0) ∈ Bn(α) for all i, j (resp., i, j, k), then kxi(t)− xj(t)k ≤ ǫ (resp., |xi,k(t)− xj,k(t)| ≤ ǫ) for all t ≥ t0 and for all i, j (resp., i, j, k).

Definition 2.3.2. (see e.g., Definition 2 of [53]) Let a ball Bn(α) be given. System (1.3) is uniformly asymptotically (resp., self-) synchronized if the followings hold:

i. It is uniformly synchronized.

ii. There exists a δ > 0 such that for all ǫ > 0 there exists a tǫ ≥ 0 such that if kxi(t0)− xj(t0)k ≤ δ ( resp., |xi,k(t0)− xj,k(t0)| ≤ δ ), and xi(t0) and xj(t0) ∈ Bn(α) for all i, j ( resp., i, j, k ) and t≥ t0+ tǫ, then kxi(t)− xj(t)k ≤ ǫ. (resp., |xi,k(t)_{− x}j,k(t)| ≤ ǫ) for all i, j (resp., i, j, k).

Definition 2.3.3. Let a ball Bn(α) be given. System (1.3) is globally (resp., self-) synchronized if for all ǫ > 0, there exists a tǫ ≥ 0 such that kxi(t)− xj(t)k ≤ ǫ (resp., |xi,k(t)− xj,k(t)| ≤ ǫ) for all i, j (resp., i, j, k), all xi(t0) and xj(t0) ∈ Bn(α), and all t ≥ t0+ tǫ.

Proposition 2.3.4. If a system is globally (resp., self-) synchronized, then it is uni-formly asymptotically (resp., self-) synchronized.

Proof. If a system is as assumed, then given ǫ > 0, there exists a t′ _{such that for all} i, j and all xi(t0) and xj(t0) _{∈ B}n(α), we have _kxi(t)_{− x}j(t)k ≤ ǫ for t ≥ t′. Letting t0 = t′ _{and δ = ǫ, we see immediately that the corresponding system is uniformly}

synchronized. Obviously, the assumption in Definition 2.3.2.-(ii) can be fulfilled by choosing any δ > 0. The other assertion in the proposition can be similarly proved.

2.4

Lyapunov Function Approach

2.4.1

Belykh

In the last few years, many researchers try to give criteria for the global (or local) synchronization of coupled chaotic systems. Most of their methods based either on the eigenvalues of the coupling configuration matrix G or on the Lyapunov exponent of the coupled systems. In order to avoid calculating eigenvalues or Lyapunov exponent to determine global synchronization, the connection graph based stability method is developed by Belykh et al (2004) [4]. This method combines the Lyapunov function approach with graph theoretical reasoning, and elucidates the relation between syn-chronization and the form of the connected graph (the coupling configuration matrix G). The method can be applied to give a rigorous bound for the coupling strength including in the global, star, diffusive, 2K-nearest neighbor coupling cases, etc. More-over, the time-varying coupling configuration matrix G(t) is also discussed.

In equation (1.3), let H : Rmn → Rmn _{be defined by H(x) = (Im ⊗ D)x, and} G = (gij(t))m×m is a time-varying, symmetric matrix with vanishing row-sums, non-negative off-diagonal elements. Then, we have the following equation,

dx dt =    f(x1, t) ... f(xm, t)    + d(G ⊗ In)(Im⊗ D)x =: F(x, t) + d(G ⊗ D)x, (2.15a) where D = diag(1,_{· · · , 1, 0, · · · 0) is a diagonal matrix with k elements equal to 1, ⊗} denotes the Kronecker product, and

f(xi, t) =    f1(xi, t) .. . fn(xi, t)    . (2.15b) .

Remark 2.4.1. (i)For all time t, we denoted that the number of the nonzero elements of the off-diagonal elements of G is 2p. (ii) The matrix G is meaningful in the graph theory. It refers to the connected graph with m vertices and p edges, and if the edge from vertex i to vertex j exists, then gj,i(t) = gi,j(t) > 0, 1≤ i, j ≤ m, for all time t. Before starting the study of the transversal stability of the synchronization manifold M_{, we need also one additional assumption on the eventually dissipativeness of coupled} system (2.15). Assume that the individual system dxi

dt = f(xi, t) is eventually dissipa-tive, i.e. there exists a compact set B which attracts all trajectories of the system from the outside. Therefore, there are no trajectories which go to infinity.

Now, we introduce the notation for the differences Xij = xj_{− x}i, we obtain the differ-ence equation system as follows,

˙ Xij = Z 1 0 Df(βxj+ (1_{− β)x}i) dβ  Xij + d m X l=1 {gjlDXjl− gilDXil}, (2.16) for all i, j = 1, . . . , m. To study the stability of difference equation system (2.16), we introduce the auxiliary system by adding an uncharted matrix A,

˙ Xij = Z 1 0 Df(βxj+ (1_{− β)x}i) dβ_{− A}  Xij, i, j = 1, . . . , m, (2.17) where A = diag(a1,· · · , ak, 0,· · · , 0) is a matrix with ai ≥ 0 for 1 ≤ i ≤ k. Moreover, we assume that there exist Lyapunov functions of the form,

Vij = XT_ijHXij, i, j = 1,_{· · · , n.} (2.18) where the vector variables Xij ={X(1)ij ,· · · , X

(k)

ij }, H = diag(h1, . . . , hk, H1) with h1 > 0,_{· · · , hk}> 0 and the matrix H1 is positive definite. Furthermore, their derivatives of Lyapunov functions in (2.18) with respect to auxiliary system (2.17) are required to be negative, ˙ Vij = XT_ijH Z 1 0 Df(βxj + (1_{− β)x}i) dβ_{− A}  Xij < 0, Xij 6= 0, i, j = 1, · · · , n. (2.19) Hence, we can study global stability of the synchronization manifold M by the following main Theorem.

Theorem 2.4.2. Under the assumption on the eventual dissipativeness of the individ-ual oscillator system dxi

dt = f(xi, t) and assumption (2.19), the synchronization mani-fold M of coupled system (2.15) is global asymptotically stable if the following inequality holds d p X l=1 gil,jlX (d)2 il,jl > ad m m−1 X i=1 m X j>i X(d)_i,j2, d = 1,_{· · · , k} (2.20) where the index (il, jl) is the pair of satisfying gil,jl> 0.

Theorem 2.4.2. indeed gives sufficient conditions for the global synchronization. How-ever, these inequalities in (2.20) are not easily to be applied. To get rid of it and find a rigorous bound of gil,jl, the following theorem is given.

Theorem 2.4.3. Under the assumption of Theorem 2.4.2, the synchronization mani-fold M of coupled system (2.15) is global asymptotically stable if

dgil,jl>

al

mbil,jl(m, p) f or l = 1,· · · , p and for all time t. (2.21)

where bil,jl(m, p) =

P

m1>m2; il,jl∈Pm1m2z(Pm1m2) is the sum of the lengths of all chosen

paths Pm1m2 which pass through the edge from vertex il to vertex jl.

Several coupling configuration could be given a general rigorous bound of the coupling strength via above two theorems. In the following, some examples are listed.

Example 1. Suppose the coupled system satisfies the sufficient conditions in the The-orem 2.4.2. Let

1. (Global coupling) G has all off-diagonal element nonzero. Then the global syn-chronization reaches provided dgi,j > _ma for all i 6= j.

2. (Star coupling) G =        −g g12 g13 · · · g1m g12 _−g12 0 · · · 0 g13 0 −g13 · · · 0 .. . ... ... . .. ... g1m 0 0 · · · −g1m       

, where g = Pm_i=2g1i.

Then the global synchronization reaches provided dg1i > a

2m_{− 3}

2, . . . , m. 3. (Diffusive coupling) G=        −(g12+ g1m) g12 0 0 g1m g12 _−(g12+ g23) g23 0 0 0 g23 . .. . .. 0 0 0 . .. . .. gm_−1,m g1m 0 0 gm_−1,m −(g1m+ gm_−1,m)        . Then the

global synchronization reaches provided dgi,j >    am₂₄2 − 1 24  for odd m am2 24 + 1 12  for even m , for all i, j.

4. (2K-nearest neighbor coupling) G has its off-diagonal elements of the form dgi,j >



g for 1≤ |j − i| mod m ≤ K

0 otherwise _.

Then the global synchronization reaches provided g > a m  m 2K 3 1 + 65 4 K m  .

2.4.2

Wu and Chua

In this section, we introduce the Lyapunov’s direct method to prove uniformly asymp-totical synchronization of coupled system (2.15), which is developed by C. W. Wu, and L. O. Chua [53]. A typical results states that coupled system (2.15) will synchronize if the nonzero eigenvalues of the coupling matrix have real parts that are negative enough. Moreover, sufficient conditions for synchronization for several coupling configurations will be considered.

It will mainly use Lyapunov’s direct method to prove uniform asymptotical synchro-nization of the coupled system in (2.15). We use d(x) to denote a nonnegative real-valued function that measures the distance between the various nodes. We also define the following class of matrices:

• M1(k) are matrices M (not necessarily square) with entries in Fk such that each row of M contains zeros and exactly one αIk and one −αIk for some nonzero α. • M2(k) are matrices M in M1(k) such that for any pair of indices i and j there exist indices i1, i2,_{· · · , i}l with i1 = i and il = j such that for all 1 ≦< l, M(p, iq) 6= 0 and M(p, iq+1)_{6= 0 for some p.}

In particular, we define d(x) to have the following form: d(x) =kMxk2 _{= x}T

MTMx, M∈ M2(n) (2.22)

where M is an m_{× m matrix in M}2(n) (but considered as an nm_{× nm real-valued} matrix).

Because of the assumptions on M, the crucial property of d(x) is that d(x)_{→ 0 if and} only if kxi− xjk → 0 for all i and j. One possible choice for d(x) is

d(x) = m_X−1 i=1 kxi− xi+1k2 (2.23) which corresponds to M=      I _−I 0 _{· · ·} 0 0 I _−I . .. ... ... ... ... ... 0 0 _{· · ·} 0 I _−I      (2.24)

Definition 2.4.4. A function α : R→ R is said to belong to class K if 1) α(_{·) is continuous and nondecreasing,}

2) α(0) = 0,

3) α(p) > 0 whenever p > 0.

We assume that all Lyapunov functions we consider are continuous. For a Lyapunov Function V (t, x), the generalized (Dini) derivative along the trajectories of the system

dx dt = f(x, t) is defined as D+_{V (t, x) = lim sup} h→0+ 1 h[V (t + h, x + hf(x, t))− V (t, x)] (2.25)

Theorem 2.4.5. Suppose that D is an open set such that if Xi(t0) ∈ Bα∗ for all i,

then x(t, x(t0), t0)∈ D for all t ≥ t0. Suppose that a Lyapunov function V (t, x), locally Lipschitzian in x, exists on R× D such that for all t ≥ t0 and x∈ D,

a(d(x))≤ V (t, x) ≤ b(d(x))

where a(_{·) and b(·) are functions in class K. Suppose that there exists µ > 0 such that} for all t _{≥ t}0 and d(x) ≥ µ,

D+V (t, x)_{≤ −c}

for some constant c > 0 where D+_{V (t, x) is the generalized derivative of V along the} trajectories of the coupled system in (2.15). If there exists δ > 0 such that a(δ) > b(µ), then for each X(t0)∈ Bα∗ there exists t₁ ≥ t0 such that for all t≥ t1,

d(x(t, x(t0), t0))_{≤ δ} . Furthermore, if d(x(t0))_{≤ µ, then}

d(x(t, x(t0), t0))≤ δ for all t ≥ t0.

Theorem 2.4.6. Suppose that D is an open set such that if xi(t0)∈ Bα∗ for all i, then

x(t, x(t0), t0) ∈ D for all t ≥ t0. Suppose that a Lyapunov function V (t, x), locally Lipschitzian in x, exists on R_{× D such that for all t ≥ t}0, x _{∈ D,}

a(d(x))_{≤ V (t, x) ≤ b(d(x))} where a(_{·) and b(·) are in class K, and for all t ≥ t}0,

D+V (t, x)≤ −c(d(x))

for some function c(·) in class K where D+_{V (t, x) is the generalized derivative of V} along the trajectories of the coupled system in (2.15). Then the coupled system in (2.15) is uniformly asymptotically synchronized with respect to α∗_.

2.5

Partial Contraction Approach

In this section, we shall describe the partial contraction approach for studying global synchronization of coupled chaotic systems. This approach was given by [45]. method to analyze networks of coupled identical nonlinear oscillators, and study applications to synchronization. Specifically, we use nonlinear partial contraction theory to derive exact and global results on synchronization. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive-definite diffusion coupling, it can be shown that synchronization always occur globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis.

Basically, a nonlinear time-varying dynamic system will be called contracting if initial conditions or temporary disturbances are forgotten exponentially fast, i.e., if trajec-tories of the perturbed system return to their nominal behavior with an exponential convergence rate. The concept of partial contraction allows one to extend the applica-tions of contraction analysis to include convergence to behaviors or to specific properties (such as equality of state components, or convergence to a manifold) rather than tra-jectories. We briefly summarize the basic definitions and main results of Contraction Theory here. Consider a nonlinear system

dx

dt = f(x, t)

where x_{∈ R}n _{is the state vector and f is an n× 1 vector function. Assuming f(x, t) is} continuously differentiable, we have

d dt(δx T_{δx) = 2δx}T_δdx dt = 2δx T ∂f ∂xδx≤ 2λmaxδx T_{δx (2.26)}

where δx is a virtual displacement between two neighboring solution trajectories, and λmax(x, t) is the largest eigenvalue of the symmetric part of the Jacobian J = _∂x∂f. Hence, if λmax(x, t) is uniformly strictly negative, any infinitesimal length kδxk converges exponentially to zero. By path integration at fixed time, this implies in turn that all solutions of system (2.26) converge exponentially to a single trajectory, independently

of the initial conditions. Note that differential analysis yields global results.

We now introduce the concept of partial contraction, which forms the basis of the contraction analysis. It derives from the very simple but very general result which follows.

Theorem 2.5.1. Consider a nonlinear system of the form ˙x = f(x, x, t)

and assume that the auxiliary system

˙y = f(y, x, t)

is contracting with respect to y. If a particular solution of the auxiliary y-system verifies a smooth specific property, then all trajectories of the original x-system verify this property exponentially. The original system is said to be partially contracting. Let us now move to networked systems under a very general coupling structure. Con-sider a coupled system containing m identical nodes

dxi

dt = f(xi, t) + X j∈Ni

Kji(xj− xi), i = 1,· · · , m, (2.27) where Ni denotes the set of indices of the active links of elements i. It is equivalent to

dxi dt = f(xi, t) + X j∈Ni Kji(xj_{− x}i)_{− K}0 m X j=1 xj + K0 m X j=1 xj, i = 1,_{· · · , m,} (2.28) where K0 is chosen to be a constant symmetric positive definite matrix (we will discuss its function later). Again, construct an auxiliary system

dyi dt = f(yi, t) + X j∈Ni Kji(yj− yi)− K0 m X j=1 yj + K0 m X j=1 xj, i = 1,· · · , m, (2.29) that has a particular solution y1 =· · · = ym = y_∞ with

dy_∞ dt = f(y∞, t)− mK0y∞+ K0 m X j=1 xj, i = 1,· · · , m, (2.30) According to Theorem 2.5.1, if the auxiliary system in (2.29) is contracting, then all system trajectories will verify the independent property x1 =· · · = xm exponentially.

Theorem 2.5.2. Regardless of initial conditions, all the elements within a generally coupled system in (2.27) will reach synchrony or group agreement exponentially if

1) the network is connected, 2) λmax(Jis) is upper bounded, 3) the couplings are strong enough. where Jis =∂f ∂y(y, t)  s, and Fs = 1 2(F + F T_).

Chapter 3

Global Synchronization via Matrix

Measures Approach

This chapter contains the main results of the first part of the thesis. In particular, we use matrix measures approach to study global synchronization of coupled chaotic systems. Our coupling rules are allowed to be asymmetric and/or some competitive (gij < 0, i _{6= j) couplings between cells x}i 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 (3.14). 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 the global synchronization of all oscillators with coupling configuration satisfying (3.3a), and (3.3b). Part of the results in this direction is bases on the paper [27]. To conclude this section, we define the global synchronization as in the following.

Definition 3.0.3. (i) System (2.15) is said to be globally synchronized if for any given initial values x0 there exists a d = dx0 such that system (2.15) is synchronized for the

initial conditions x0. Here dx0 is a constant depending on x0. (ii) System (2.15) is

said to be uniformly, globally synchronized if there exists a d = d1 such that system (2.15) is synchronized for all initial values x0.

3.1

Preliminaries

Chaotic synchronization is a fundamental phenomenon in physical systems with dissi-pation. In this section, we introduce the concept of the bounded dissipation to coupled system (2.15). Then, we use this concept of the matrix measure theory to achieve the behavior of global synchronization in coupled system (2.15). Hence, we give the definition of bounded dissipation as follows.

Definition 3.1.1. (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∗ ≥ t0 and αr such that kx(t)k ≤ αr for all t ≥ t∗_{. (ii) If, in addition,} αr is 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 of the system, which plays the role of an a priori estimate. Without such an a priori estimate, as in the case of R¨ossler system, the 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 (3.3c), 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. Moreover, it often requires to construct an approximate Lyapunov function to prove the bounded dissipation of the system. The following proposition gives the type of Lyapunov functions that would ensure the bounded dissipation of the system.

Proposition 3.1.2. Let a system of n ordinary differential equations be given. Let V be a continuous real-valued function V : Rn_{→ R}+ _{so that V is strictly decreasing along}

the solution of the system on Rn_{− Γ, where Γ is homeomorphic to an open ball in R}n_. Suppose

lim

kxk→∞V (x) =∞. (3.1)

Then the system is bounded dissipative.

Proof. For any x0 ∈ Rn_{, we first prove that x(t) must enter Γ at a certain time.} Otherwise, the values of V at the points of the ω-limit set of x(t) must be the same, a contradiction. The contradiction comes from the facts that the ω-limit set is closed and invariant and V is strictly decreasing along the solution trajectory, which stays in Rn_{− Γ. We then find a ball Bn(r) so that Bn(r)⊃ Γ. Let k1 = maxx}

∈ ¯Bn(r)V (x), and

Bn(αr) be a ball satisfying V (x) > k2 whenever x_{∈ R}n_{−Bn(αr), where k2 > k1. Then} we conclude that if x0 _{∈ B}n(r), x(t) stays in Bn(αr) for all time t. We just complete the proof of the proposition.

In our derivation of synchronization of system(3.1), we need the concept of matrix measure. For the completeness and ease of references, we also recall the following definition of the matrix measure and its properties (see e.g., [44]).

Theorem 3.1.3. (see e.g., 3.5.32 of [44]) Consider the differential equation ˙x(t) = A(t)x(t) + v(t), t ≥ 0, where x(t) ∈ Rn

, 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} corre-sponding 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. (3.2)

3.2

Global synchronization results

Our main result in the first part of the thesis is contained in this section. We begin with imposing conditions on coupling matrices G and D. We assume that the coupling 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. (3.3a)

(ii) All nonzero eigenvalues of G have negative real part. (3.3b) We further assume that the matrix D is, without loss of generality, of the form

D=  Ik 0 0 0  n×n. (3.3c)

The index k, 1 _{≤ k ≤ n, means that the first k components of the subsystem 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 (3.1) as coupled system (D, G, F(x, t)). To study the synchronization of such system, we permute the state variables in the following way: ˜ x_i =    x1,i .. . xm,i    , and ˜x=    ˜ x1 .. . ˜ xn    . (3.4)

Then (3.1) can be written as

˙˜x =    ˜f1(˜x, t) ... ˜fn(˜x, t)    + d(D ⊗ G)˜x =: ˜F(˜x, t) + d(D_{⊗ G)˜x,} (3.5a) where

˜fi(˜x, t) =    fi(x1, t) ... fi(xm, t)    . (3.5b)

Note that such reformulation is certainly not new (see e.g., [29, 53]). From here on, we will treat ˜ as a function that takes x into ˜xor xi into ˜xi. A transformation of coordi-nates of ˜x is then to be applied to (3.4) so as to decompose the synchronous manifold. The problem of synchronization of (2.15), and hence (3.5) is then equivalent to proving the asymptotical stability of reduced system (see (3.8)). To study the synchronization of (2.15), we first make a coordinate change to decompose 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  , (3.6a)

where e is given as in (3.3a). It is then easy to see that CCT _{is invertible and that}

A−1 =CT(CCT₎−1_| e m  . (3.6b) Setting E = In⊗ A, (3.6c) we see that

E(D⊗ G)E−1 = (In⊗ A)(D ⊗ G)(In⊗ A−1) = D_{⊗ AGA}−1 = D_⊗  CGCT(CCT₎−1 ₀ ∗ 0  =: D⊗ ¯G 0_{∗ 0}  . (3.6d)

We remark, via (3.6d), that σ(G)_{−{0} = σ( ¯}G), where σ(A) is the spectrum of matrix A. Multiplying E to the both side of equation (3.5a), we get

˙˜y =: E ˙˜x = E˜F(˜x, t) + dE(D ⊗ G)E−1_y˜ = E ˜F(E−1y˜, t) + d( D⊗ ¯G 0_{∗ 0}  )˜y. (3.7) Let ˜y =    ˜ y1 ... ˜ yn    . Then ˜yi =      x1,i− x2,i .. . xm_Pm−1,i− xm,i j=1xj,i      . Setting ˜yi =  ¯ yi Pm j=1xj,i  , and ¯ y=    ¯ y1 .. . ¯ y_n    ,

we have that the dynamics of ¯yis satisfied by the following equation

˙¯y = d(D ⊗ ¯G)¯y+ ¯F(¯y, t). (3.8) Here ¯Fis obtained from E ˜F(E−1_{y, t) accordingly.˜}

The task of obtaining the global synchronization of system (2.15) is now reduced to showing that the origin is globally and asymptotically stable with respect to system (3.8). To this end, the space ¯y is broken into two parts ¯yc, the coupled space, and ¯yu, the uncoupled space.

¯ y=  ¯ y_c ¯ y_u  , and ¯F(¯y, t) = ¯F_¯c(¯y, t) F_u(¯y, t)  , respectively. (3.9)

Here ¯yc =    ¯ y₁ ... ¯ y_k    , and ¯yu =    ¯ y_k+1 ... ¯ y_n  

. 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. In short, this part of the dynamics is easy to contain. In fact, the larger k, the number of state variables being coupled, gives the better chance of the synchronization of the coupled system. 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 ≤ b}1k¯yk (3.10a) whenever ¯y in the ball B(m−1)n(α), and for all time t. Since the estimate in the

right-hand side of (3.10a) depends on the whole space ¯y, condition (3.10a) is a mild assump-tion provided that coupled system is bounded dissipative. Write ¯Fu(¯y, t) as

¯

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

=: U(t)¯yu+ ¯Ru(¯y, t). (3.10b)

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.}

(i) The matrix measures µi(Uj(t)) are less than _{−γ for all t and all j,}

(ii) Let ¯Ru(¯y, t) =    R_u1(¯y, t) ... R_ul(¯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 =    ¯ y_u1 ... ¯ 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 (3.10d)

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

Specifically, we break the vector field ¯F_u into (time dependent) linear part U(t)¯y_u and nonlinear part ¯R_u(¯y, t). We will further break U(t) into certain block diagonal form if necessary. Note that the form (3.10b) can always be achieved. Since the remainder term ¯Rstill depends on the whole space ¯y. To take control of the dynamics 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 the (3.10d) holds. Note that though the nonlinear terms Ruj(¯y, t) could possibly depend on the whole space, their norm estimate 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 (3.10c) and (3.10d) 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 (3.10c) and (3.10d) are not satisfied. The proof in the following theorem gives exactly how the above strategy can be realized.

Theorem 3.2.1. Let G and D be given as in (3.3). Assume that ¯F satisfies (3.10a-d), and system (3.8) 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 l 2 =: dc, (3.11)

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

t→∞y(t) = 0.¯

Proof. Since system (3.8) is uniformly bounded dissipative with respect to α, without loss of generality, we may assume that _{k¯y(t)k ≤ α for all time t ≥ t0.} Using (3.10b), we write (3.8) as  ˙¯yc ˙¯yu  =  d(Ik_{⊗ ¯}G) 0 0 U(t)   ¯ yc ¯ yu  + ¯F_¯c(¯y, t) Ru(¯y, t)  . (3.12a)

Applying the variation of constant formula to (3.12a) on ¯y_c, we get

¯ yc(t) = e(t−t0)d(Ik⊗ ¯G)y¯c(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 _(3.12b)

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

k¯yc(t)k ≤ ce(t−t0)dν_{k¯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

whenever t ≥ t0,1 for some t0,1 > 0. We then apply Theorem 3.1.3 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 (3.10c-d) and (3.13a) that

k¯yu1(t)_{k ≤ αe}−γ(t−t0,1)₊α

d b2 γc0δ≤ α d b2 γc0δ 2 _=: α dc1δ 2 , (3.13b)

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, (3.13c)}

whenever t ≥ tj,1(≥ tj−1,1). Letting tl,1 = t1 and summing up (3.13a), (3.13b) and (3.13c), 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 l 2 cb1 |ν|δ l+1 _{=: hα,} whenever t_{≥ t}1. Choosing d >  1 + (b2 γ) 2 l 2 _cb 1

|ν|δl+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

Remark 3.2.2. (i) In case that ¯G is symmetric, then c and ǫ can be chosen to be one and zero, respectively. (ii) b1 and b2 could possibly depend on α. (iii) If system (3.8) is only bounded dissipative, then the estimate in (3.11) is still valid. However, in this case, b1 and b2 depend not only on α but also on x0.

Corollary 3.2.3. Suppose ¯Fand G are given as in Theorem 3.2.1. Let

D= ¯Dk×k 0

0 0

 n×n,

where Re( σ( ¯D) ) > 0. (3.14a) Assume, in addition, that either σ(G) or σ( ¯D) has no complex eigenvalue.

Then assertions in Theorem 3.2.1 still hold true, except dc needs to be replaced by

dc = c b1 |ν| min{Re( σ( ¯D) )_}  1 + (b2 γ ) 2 l 2 . (3.14b)

Proof. Assumption on D is to ensure that (3.29b) is still valid. Other parts of the proof are similar to those in Theorem 3.2.1 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 (3.10a-d) are satisfied. To this end, we need the} following notations. Let ˜xi and ˜xbe given as in (3.4). Define

[˜xi]−=    x1,i .. . x_m−1,i    , and [˜x]− =    [˜x1]− .. . [˜x_n]−    . (3.15)

We then break ˜Fas given in (3.5a) into two parts so that the breaking is in consistent with ¯y in (3.9). Specifically, we shall write

˜ F(˜x, t) = ˜F_˜c(˜x, t) Fu(˜x, t)  . (3.16) We are now in the position to state the following propositions.

Proposition 3.2.4. 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, (3.17) for all u, v in Bn(α

2) and all time t. Then (3.10a) holds true.

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

where A is given as in (3.23a), and so

[A˜fi(˜x, t)]− =    fi(x1, t)− fi(x2, t) .. . fi(xm−1, t)− fi(xm, t)    , i = 1, 2, . . . , n. (3.18) Since ¯ Fc(¯y, t) =    [A˜f1(˜x, t)]− ... [A˜fk(˜x, t)]−    , we conclude that (3.10a) 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 (3.10c-d) are satisfied.

Proposition 3.2.5. 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 (3.10c). Write fwp−1+i(u, t)−fwp−1+i(v, t), i = 1, . . . , kp,

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).

(3.19a) We further assume that the followings are true.

(i) For p = 1, . . . , l, let Qu,v,p = (qwp−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, ∞. (3.19b)

(ii) Let rp = rwp−1+1(u, v, t), . . . , rwp(u, v, t)

T . We have that krpk ≤ b2k    u1− v1 .. . uwp−1 − vwp−1    k (3.19c)

for all p, u, v in Bn(α₂) and all time t.

Then (3.10c) and (3.10d) hold true for ∗ = 1, 2, ∞.

Proof. Since ri(u, v, t) depend on whole space, fi(u, t)− fi(v, t) can always be written as the form in (3.19a). Using (3.19a) and (3.18), we have that the matrices Up(t) in the linear part of ¯F_u(¯y, t) take the form

Up(t) = m_X−1

w=1

Qxw,xw+1,p(t)⊗ Dw, (3.20)

where xw are given as in (1.2), and

(Dw)_ij = 

1 i = j = w,

It then follows from (2.9-2.11), and (3.20) that µ_∗(Up(t)) < −γ for ∗ = 1 or ∞. For ∗ = 2, we have that m_[−1 w=1 σ{Qxw,xw+1,p(t) + Qxw,xw+1,p(t)
T } = σ (_m−1 X w=1  Qxw,xw+1,p(t)⊗ Dw+ Qxw,xw+1,p(t)
T ⊗ Dw ) = σ Up(t) + UT_p(t)
,

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 (2.11) that µ2(Up(t)) < −γ. The remainder of the proof is similar to that of Proposition 3.2.4, and is thus omitted.

Remark 3.2.6. The upshot of Proposition 3.2.5 is that by only checking the ”structure” 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)_{− f}i(v, t) in the form of (3.19a) with i = k + 1, . . . , n. If (3.19b, 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 (3.19b, c) are satisfied.

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

Theorem 3.2.7. Assume that system (2.15) is (resp., uniformly) bounded dissipative. Let the coupling matrices G and D satisfy (3.3) and the nonlinearities fi(x, t), i = 1, 2, . . . , n, satisfy (3.17) and (3.19). Suppose d is greater than dc, as given in (3.11). Then system (2.15) is (resp., uniformly,) globally synchronized.

Proof. The proof is direct consequences of Propositions 3.2.4 and 3.2.5, and Theorem 3.2.1.

Remark 3.2.8. From here on, we will refer the assumptions in Theorem 3.2.7 as synchronization hypotheses.

Theorem 3.2.9. The coupled system (D, G, F(x, t)), given as in Corollary 3.2.3, 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 (3.14b).

3.3

Applications

To see the effectiveness of our main results, we consider four examples in this section. These are coupled Lorenz equations [4,29], coupled chaotic walk system [56], coupled Duffing oscillators [54] and coupled Lorenz-like system.

3.3.1

Coupled Lorenz System

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 .

Here σ = 10, r = 28 and b = 8

3. In the following cases (a), (b), (c) and (d), G denotes the diffusive coupling with zero flux and D is, respectively,

  1 0 00 0 0 0 0 0   ,   0 0 00 1 0 0 0 0   ,   0 0 00 0 0 0 0 1   , and   0 0 00 1 1 0 0 1   .

For the first three cases, it was shown in [8] that such coupled system (D, G, F(x)) have the topological product of an absorbing domain

B =_{x2₁+ x2₂+ (x3_{− r − σ)}2 < b

2_{(r + σ)}2

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 the corresponding ”coupled” nonlinearity f1, we get that

|f1(u)− f1(v)| = σ|(u2− v2)− (u1− v1)| ≤√2σku − vk.

Hence, condition (3.10a) is satisfied. For the corresponding ”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) (3.22a)

and

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

= [u1(u2_{− v}2)_{− b(u}3− v3)] + v2(u1− v1). (3.22b)

Writing (3.22a,b) in the vector form, we get  f2(u)− sf2(v) f3(u)_{− f}3(v)  =  −1 −u1(t) u1(t) _−b   u2− v2 u3_{− v}3  +  (r− v3)(u1− v1) v2(u1_{− v}1)  =: Qu,v,1(t)  u2− v2 u3− v3  + r1. (3.22c)

Clearly, µ2(Qu,v,1(t)) = max_{{−1, −b} = −1 < 0, and kr}1k ≤ (σ +√β)· |u1−v1|, where its estimate depends only on coupled space. Hence, conditions (3.19b,c) are satisfied.

(b) Let D = D2 =   0 0 0 0 1 0 0 0 0   .

As in the case (a), the corresponding ”coupled” nonlinearity f2 is clearly Lipschitz on the absorbing domain. For the corresponding ”uncoupled” nonlinearities f1 and f3, we get

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 (3.19c) is violated. For in the case, the norm estimate in the right hand side of (3.19c) 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 r}1 = σ(u2−v2). Consequently, (3.19b) and (3.19c) are satisfied with p = 1. For p = 2, we have Qu,v,2 = (_{−b) and r}2 = u1(u2− v2) + v2(u1− v1), which depends only on the coupled space and the preceding uncoupled space. Thus, r2 can also be satisfied with (3.19c).

(c) For illustration, we also consider D = D3 =   0 0 0 0 0 0 0 0 1   .

In this case, the uncou-pled 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. That is, pick l = 1. Otherwise, (3.19c) is isolated. For l = 1, we have that 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., [29], synchronization fails for this type of partial coupling.

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

To apply Theorem 3.2.9, 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 [29].