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Coordinate transformation and matrix measure approach for synchronization of

complex networks

Jonq Juang and Yu-Hao Liang

Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 19, 033131 (2009); doi: 10.1063/1.3212941 View online: http://dx.doi.org/10.1063/1.3212941

View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/19/3?ver=pdfcov Published by the AIP Publishing

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Coordinate transformation and matrix measure approach

for synchronization of complex networks

Jonq Juanga兲 and Yu-Hao Liangb兲

Department of Applied Mathematics, National Chiao Tung University, Hsinchu, Taiwan 300, Republic of China

共Received 29 April 2009; accepted 6 August 2009; published online 8 September 2009兲

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability 共CGS兲 method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach共MMA兲 proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes. © 2009 American Institute of Physics.关DOI:10.1063/1.3212941兴

Synchronization of networks of dynamical systems is fre-quently observed in nature and technology.1,2 Recently, the study of synchronization phenomena in complex net-works with different topologies has received much attention.3–15There are mainly two analytical approaches for studying such time-varying networks. The first ap-proach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is lim-ited to the networks with cooperative couplings. The ma-trix measure approach (MMA) proposed by Chen, de-spite a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state cou-pling. In the current work, generalizing our previous work,26which considered time-independent networks, we are able to fix all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to deter-mine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

I. INTRODUCTION

During the past few decades the study of networks of dynamical systems has generated a rapidly growing interest in theoretical physics and other fields of science. Particularly, an increasing interest has been focused on complex networks with different topologies.3–30 Complex networks, including

the Internet, the World Wide Web, and electrical power grids, are prominent candidates to describe sophisticated collabora-tive dynamics in many sciences.3–17

As one of the basic characteristics of a dynamical net-work, synchronizing a crowd of dynamical nodes within the complex networks has become an important and interesting research topic in many fields.3–30General approaches to local synchronization of coupled chaotic systems have been pro-posed, including the master stability function-based criteria6,7,31,32 and the MMA.17,33 Typically, in networks of coupled chaotic systems, the synchronous solution becomes stable when the coupling strength between the oscillators exceeds a critical value. However, a few examples34,35 were reported to be inconsistent with this pattern. Among them is a lattice of the x-component coupled Rössler systems in which the stability of synchronization regime is lost with an increase in coupling strengths. Furthermore, even if the coupled system always stays in a compact set and local syn-chronization occurs, global synsyn-chronization can be absent due to the possible presence of different invariant sets lying outside the synchronous manifold共in certain cases, this is a multistability effect兲. As a result, global synchronization of coupled chaotic systems was also intensively studied.

The methods to deal with global synchronization include

but not limited to Lyapunov function-based

criteria12,14,23,35–38 as well as the MMA.26,33,39 Among the Lyapunov function-based criteria, the CGS22–25has the wid-est range of the applicability. Indeed, the method can be ap-plied to the asymmetrically coupled networks that are time varying. However, the couplings in the network are assumed to be non-negative. In fact, there exist some networks with both negative/competitive and positive/cooperative cou-plings. Recently, the MMA proposed by Chen17,33 has been

a兲Electronic mail: jjuang@math.nctu.edu.tw. b兲Electronic mail: moonsea.am96g@g2.nctu.edu.tw.

1054-1500/2009/19共3兲/033131/13/$25.00 19, 033131-1 © 2009 American Institute of Physics

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very successful in treating local synchronization with com-plex network topologies. In Refs. 33 and 39 some global synchronization theorems were also obtained via a similar MMA. Even though the theorems can be applied to a wider range of complex networks than those obtained by CGS, there are two drawbacks. First, the number of nodes consid-ered may be limited. The matrix measure of the diffusive synchronization stability matrix 共see, e.g., Refs.33and39兲, which is equivalent to our GC1共t兲 关see Eq.共5b兲兴, is size

de-pendent. Its corresponding matrix measure can go from negative to positive as the size of the nodes increases. In-deed, given a near-neighbor coupling with periodic boundary conditions if the number of nodes is greater than 7, then GC1

has positive matrix measure 共see TableI兲. Second, their ap-proach works better for systems of the full-state coupling between connected nodes as opposed to those of the partial-state coupling. It should be noted that the partial-partial-state cou-pling also finds applications in various fields. For instance, in self-pulsating laser diode equations共see, e.g., Ref.40兲, only the photon density can be coupled with the electron density of the active region. Moreover, in the case of coupled chaotic systems, the systems that are partial state coupled may ex-hibit different dynamic behavior. For instance, it is well known共see, e.g., Ref. 26兲 that for the coupled Lorentz sys-tems, if the x-component or y-component is coupled, the resulting system then achieves synchronization. In contrast, the network would fail to be synchronized provided that only the z-component is coupled.

The purpose of this paper is to give a different MMA, which was originated in Ref. 26to study global synchroni-zation in time-varying complex networks. In particular, a new and, in some cases, optimal coordinate transformation is introduced to remedy the first drawback of MMA. Moreover, by taking account of the structure of the uncoupled parts of the vector field of the individual oscillators, we are able to avoid the second drawback of MMA. In short, our approach fixes both drawbacks of Chen’s approach and preserves their salient feature of wider applicability of complex networks. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. Moreover, a rigorous lower bound on the coupling strength for global synchroni-zation of all oscillators is also obtained. The paper is orga-nized as follows. Section II is to lay down the foundation of our paper. The properties of the new coordinate transforma-tion and its resulting coupling matrix G共t兲 are studied in Sec. III. The main results are contained in Sec. IV. Some ex-amples to illustrate the effectiveness of our approach and to

compare with the existing methods are recorded in Sec. V. The examples include some complex networks such as the star type, the wavelet transformed type, the pristine world joining with some randomness, the generalized wheel type, and the prism type. In Sec. VI, we summarize our main results and give some concluding remarks. The needed defi-nitions and properties of matrix measures of matrices and some technical proofs leading to the main results of our pa-per are recorded in Appendices A and B, respectively.

II. BASIC FRAMEWORK

In this paper, we will denote scalar variables in lower case, matrices in bold type upper case, and vectors 共or vector-valued functions兲 in bold type lower case. We con-sider an array of m nodes/oscillators, coupled linearly to-gether, with each node/oscillator being an n-dimensional sys-tem. The entire array is a system of nm ordinary differential equations. In particular, the state equations are

dxi

dt = f共xi,t兲 + d ·

j=1

m

gij共t兲Dxj, i = 1,2, . . . ,m, 共1a兲

where D =共dij兲n⫻n is the inner coupling matrix, xi =共xi1, xi2, . . . , xin兲T苸Rn, and f is a vector-valued function form Rn⫻R→Rndenoted by f共xi,t兲 =

f1共xi,t兲 ] fn共xi,t兲

. 共1b兲

Let x =共x1, x2, . . . , xm兲Tand G共t兲=共gij共t兲兲m⫻m. Then G共t兲 rep-resents the 共outer兲 coupling configuration of the network at time t. Equivalently, Eq. 共1a兲becomes

x˙ =

f共x1,t兲 ] f共xm,t兲

+ d共G共t兲D兲x ¬ F共x,t兲 + d共G共t兲D兲x, 共2兲 where 丢 denotes the Kronecker product. To study the syn-chronization of Eq. 共2兲, we assume, throughout the paper, that

G共t兲e = 0 ∀ t, 共3a兲

where e = 1/

m共1,1, ... ,1兲T. Such assumption above is to ensure the invariant property of the synchronization manifold M=兵x:xi= xj, 1ⱕi, jⱕm其.

We further assume that the inner coupling matrix D is, without loss of generality, of the form

D =

Ik 0

0 0

n⫻n

. 共3b兲

The index k, 1ⱕkⱕn, means that the first k components of the individual system are coupled. If k⫽n, then the system is said to be partial state coupled. Otherwise, it is said to be full state coupled.

Definition 1: System共1a兲is said to have global synchro-nization if for each initial condition x共0兲苸Rnm

, the trajectory x共t兲 satisfies

TABLE I. The table gives the matrix measures of GCi, i = 1 , 2 with various size of G, which is given in Eq.共9兲. Since G is a circular matrix, the matrix measures of G with respect to C1and C2are equal. Note that the matrix

measure of GCis␭2共G兲, ∀C苸O, which is negative regardless of the size

of G.

m 4 5 6 7 8 9

C1 ⫺1.78 ⫺1 ⫺0.51 ⫺0.19 0.05 0.23

C2 ⫺1.78 ⫺1 ⫺0.51 ⫺0.19 0.05 0.23

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lim

t→⬁1ⱕi⬍jⱕm

储xi共t兲 − xj共t兲储 = 0.

Permute the state variables in the following way:

x ˜i=

x1i ] xmi

, and ˜ =x

x ˜1 ] x ˜n

. 共4a兲

Then Eq.共2兲 can be written equivalently as

x ˜˙ =

f ˜ 1共x˜,t兲 ] f ˜n共x˜,t兲

+ d共DG共t兲兲x˜ ¬ F˜共x˜,t兲 + d共DG共t兲兲x˜, 共4b兲 where f ˜i共x˜,t兲 =

fi共x1,t兲 ] fi共xm,t

. 共4c兲

The purpose of such a reformulation is twofold. First, a transformation of coordinates of x˜ is to be applied to Eq.共4b兲

so as to isolate the synchronous manifold. Second, once the synchronous manifold is isolated, proving synchronization of Eq. 共2兲 is then equivalent to showing that the origin is as-ymptotically stable with respect to reduced system 共7a兲. To this end, we first make a coordinate change to isolate the synchronous subspace. Let C be an共m−1兲⫻m full-rank ma-trix with all its row sums being zero. Such a mama-trix is to be termed as coordinate transformation. Define

A =

C

eT

. 共5a兲

Then A−1=共CT共CCT兲−1, e兲 and AG共t兲A−1=

CG共t兲C T共CCT−1 0 eTG共t兲CT共CCT兲−1 0

¬

GC共t兲 0 h共t兲T 0

. 共5b兲 Let E = InA and y˜ = Ex˜. Multiplying E to both sides of Eq. 共4b兲, we get

y

˜˙ = EF˜ 共E−1˜,ty 兲 + d

D

GC共t兲 0 h共t兲T

0

冊冊

˜ .y Let y˜ =共y˜1, . . . , y˜n兲T. Then

y ˜i=

Cx˜i

j=1 m xji/

m

¬

yi ei

. 共6兲

Setting y =共y1, . . . , yn兲T, we have that the dynamics of y is now satisfied by the following equation:

y˙ = d共DGC共t兲兲y + F共y,t兲, 共7a兲

where

F共y,t兲 = 共InC兲 · F˜共E−1˜,ty 兲. 共7b兲

Since the rank and the row sums of C are m − 1 and 0, respectively, we conclude that the task of obtaining global synchronization of system 共1a兲 is now reduced to showing that the origin is globally and asymptotically stable with re-spect to system 共7a兲. The choice of a coordination transfor-mation will greatly influence how negative the matrix mea-sure of GC共t兲 could be, which plays the important role,

among others, to determine the global stability of Eq. 共7a兲 with respect to the origin.

III. MATRICES OF THE COORDINATE TRANSFORMATION

In what follows we shall address the question of how to choose a matrix C of the coordinate transformation and its corresponding properties. To make the origin an asymptoti-cally stable equilibrium of system 共7a兲, one would like to have the matrix measure of GC共t兲 as smaller a negative

num-ber as possible. In fact, such an optimal choice C can be achieved provided that the outer coupling matrix G共t兲 is symmetric, nonpositive definite.

Definition 2: Denote by C the set of共m−1兲⫻m

coordi-nate transformations, i.e.,

C=兵C 苸 R共m−1兲⫻m:C is full rank, and all its row sums are zero其. Let O債C be such that

O=兵C 苸 C:C such that matrix A = 共CT,e兲T is orthogonal其.

Theorem 1: Assume that all eigenvalues of outer cou-pling matrix G共t兲 have nonpositive real parts. Then

infC苸C␮2共GC共t兲兲ⱖRe ␭2共G共t兲兲. Here Re ␭2共G共t兲兲 is the

sec-ond largest real part of eigenvalues of G共t兲. If, in addition,

G共t兲 is symmetric for all t, then the above equality can be

achieved by choosing any C in O.

Proof: It follows from Eq. 共5b兲 that the spectrum

共GC共t兲兲 of GC共t兲 is equal to共G共t兲兲−兵0其. Using the fact

that Re␭共K兲ⱕ␭max共K+KT/2兲 for any real matrix K, we have, via Eq. 共A1兲, that␮2共GC共t兲兲ⱖRe ␭2共G共t兲兲. In particu-lar, if C苸O and G共t兲 is symmetric, then GC共t兲共=CG共t兲CT兲 is

symmetric and h共t兲=0. Here h共t兲 is given as in Eq. 共5b兲. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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Therefore,␮2共GC共t兲兲=␭2共G共t兲兲. We have just completed the proof of the theorem.

The theorem above amounts to saying that if G共t兲 is symmetric, nonpositive definite, then any choice of C in O yields the smallest possible matrix measure of GC共t兲. This, in

turn, gives one the best possible position to study the stabil-ity of Eq.共7a兲with respect to the origin.

Remark 1: In those earlier papers共see, e.g., Refs.12,26, and 33兲, the choice of the coordinate transformations is ei-ther C1=

1 − 1 0 ¯ 0 1 0 − 1  ] ] ]   0 1 0 ¯ 0 − 1

or 共8兲 C2=

1 − 1 0 ¯ 0 0 1 − 1  ] ]    0 0 ¯ 0 1 − 1

.

The drawback for such a choice of C is that even if G共t兲共⬅G兲 is the diffusive matrix with periodic boundary conditions, i.e., G共t兲 ⬅

− 2 1 0 ¯ 0 1 1 − 2 1 0 ¯ 0 0     ] ]     0 0 ¯ 0 1 − 2 1 1 0 ¯ 0 1 − 2

m⫻m , 共9兲

the corresponding matrix measure of GCi, i = 1 , 2 is positive

whenever m⬎7 共see TableI兲, while␮2共GC兲=␭2共G兲⬍0 for all C苸O regardless of the size of G.

Theorem 2: For any outer coupling matrix G共t兲 and any coordinate transformations Cp, Cq in O, ␮2共GCp共t兲兲

=␮2共GCq共t兲兲.

Proof: Since for any x苸Rm−1, there is z = CqCp T

x such that

xTCp共G共t兲 + G共t兲T兲CpTx = zTCq共G共t兲 + G共t兲T兲CqTz.

By the definition of matrix measure, we have that

␮2共GCp共t兲兲=␮2共GCq共t兲兲.

From here on, the matrix C in Eq.共7a兲is assumed to lie in D unless otherwise stated. For ease of the notations, we shall drop the subscript C of GC共t兲 if C苸O. The remainder

of the section is devoted to finding the matrix measure of G共t兲 where its corresponding coupling matrix G共t兲 appears often in many applications.

Proposition 1: Assume that for each t, G共t兲 is a node-balancing matrix, i.e., its row sums and column sums are equal. Then

␮2共G共t兲兲 = ␭2

G共t兲 + G共t兲T

2

, 共10兲

whenever all eigenvalues of G共t兲+G共t兲Tare nonpositive.

Proof: If G共t兲 is as assumed, then it follows from Eq.

共5b兲that

AG共t兲A−1=

G共t兲 0

0 0

.

Consequently, Eq. 共10兲holds as asserted.

In what follows, some outer coupling matrices are to be provided. Their corresponding matrix measures of G共t兲 and GCi共t兲, i=1 or 2, are to be compared.

Example 1:关Belykh et al. 共Ref.22兲兴 Consider the regu-lar coupled network by adding to the pristine world G 共the ring of 2K-nearest coupled oscillators兲 an additional global coupling such that the coupling p共t兲, 0ⱕp共t兲ⱕ1 is placed on all free spots of the matrix G共see, e.g., Ref.22兲. Specifically, the resulting coupling matrix G共t兲 can be represented by a circular matrix of the form

G共t兲 = circ共− g共t兲,1, ... ,1 K , p共t兲, ... ,p共t兲 m−2K−1 ,1, . . . ,1 K 兲, 共11兲 where g共t兲=2K+共m−2K−1兲p共t兲. Since G共t兲 is symmetric, we have that ␮2共G共t兲兲 = ␭2共G共t兲兲 = max 1ⱕjⱕm−1

− g共t兲 +

l=1 K 共␻lj +␻共m−l兲j+ p共t兲

l=K+1 m−K−1lj

.

Here ␻= exp共2␲i/m兲. The matrix measures ␮2共G共t兲兲 and

␮2共GCi共t兲兲, i=1,2, with p共t兲=t, t苸关0,1兴 are recorded in

Fig.1.

Example 2:关Wei et al. 共Ref.41兲; Juang et al. 共Ref.42兲兴

Let G = G共m兲, 0ⱕ␤ⱕ1 be the diffusive matrix of size m ⫻m with mixed boundary conditions. That is, if m⬎2,

0 0.2 0.4 0.6 0.8 1 -16 -14 -12 -10 -8 -6 -4 -2 t ma t. mea. µ2( ¯GCi(t)) µ2( ¯G(t))

FIG. 1. 共Color online兲 The matrix measures of G共t兲 and GCi共t兲, i=1,2 with G being given in Eq.共11兲and p共t兲=t are, respectively, represented by the solid line and the dotted lines above. Lines for GCi共t兲, i=1,2 are coincided since G共t兲 is circular for all t.

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G共m兲=

− 1 −␤ 1 0 ¯ 0 ␤ 1 − 2 1 0 ¯ 0 0     ] ]     0 0 ¯ 0 1 − 2 1 ␤ 0 ¯ 0 1 − 1 −␤

m⫻m , 共12兲 and if m = 2, G共2兲=

− 1 −␤ 1 +␤ 1 +␤ − 1 −␤

.

For such G,␮2共G兲=␭2共G兲⬍0. However, ␭2共G兲 would move closer to the origin as the number of nodes increases. As a result, synchronization of the network is more difficult to be realized as the number m of nodes increases. In Refs.41and 42a wavelet transformation method is proposed to alter the connectivity topology of the network. In doing so, ␭2共G共t兲兲 =␭2共p共t兲兲 becomes a quantity depending on wavelet param-eter p共t兲. By choosing suitable p共t兲, which is a wavelet trans-formation method41,42 applied to the coupling matrix G共m兲, one would expect that ␭2共p共t兲兲 will move away from the origin regardless of the number of the nodes. Under such a reconstruction, the resulting coupling matrix G共t兲 is of the following form:

G共t兲 = G共m兲+ p共t兲共G

共m/k兲eeT兲, 共13兲

where e =共1, ... ,1兲T. Here we assume p共t兲ⱖ0 and k=2lfor some l苸N, and m=Nk for some N苸N−兵1其. Since the re-constructed matrix G共t兲 is symmetric, ␮2共G共t兲兲=␭2共G共t兲兲 ⬍0. The matrix measures ␮2共G共t兲兲 and ␮2共GCi共t兲兲, i=1,2,

with p共t兲=t, t苸关0,1兴 are recorded in Fig.2.

Example 3: Let

Gt = circ− 2,2,0, ... ,0

m

.

Since G共t兲 is a node-balancing matrix, ␮2共G共t兲兲=␭2共G共t兲兲

⬍0. Note that the values of ␮2共GCi兲, i=1,2, are positive

provided that m⬎5 共see TableII兲.

Proposition 2: Let C =共c1, . . . , cm−1兲T苸O. If, in addition, 兵ci其i=1m−1 are pairwise G共t兲-conjugate, i.e., ci

TG共t兲cj = 0, ∀1 ⱕi⫽ jⱕm−1, then G共t兲 is a diagonal matrix. Moreover,

␮2共G共t兲兲 = ␭2共G共t兲兲, 共14兲

whenever all eigenvalues of G共t兲 are nonpositive.

Proof: Note that G共t兲=CG共t兲CT=共ciTG共t兲cj兲. Hence, G共t兲 is a diagonal matrix. Therefore, the assertion in Eq. 共14兲 holds as asserted.

Example 4: 关Chen 共Ref. 33兲兴 Let G共t兲 describe a

star-typed coupled network of the form

G共t兲 =

− d1共t兲 d1共t兲  ] − d1共t兲 d1共t兲 1 ¯ 1 −共m − 1兲

m⫻m . 共15兲

Here d1共t兲 is a real number. We next show that a set 兵ci其i=1 m−1 of column vectors can be chosen so that C =共ci, . . . , cm−1兲 苸O and that 兵ci其i=1

m−1 are pairwise G共t兲-conjugate. Define i =共i共i+1兲兲−1/2, i = 1 , . . . , m − 1. Let ci T =i, . . . ,i i ,− ii,0, . . . ,0 m−i−1 

for all i = 1 , . . . , m − 1. Then ci, i = 1 , . . . , m − 1 are orthonor-mal vectors. Moreover, they are also G共t兲-conjugate. To see this, we first note that d1共t兲 is an eigenvalue of G共t兲 and its associated eigenvectors are ci, i = 1 , . . . , m − 2. Therefore, ci

TG共t兲cj= 0 for all 1ⱕi⫽ jⱕm−2. Some direct computation would yield that ci

TG共t兲cm−1

= 0 for i = 1 , . . . , m − 2 and that cm−1T G共t兲cm−1= −d1共t兲−共m−1兲. By Proposition 2, we have that

␮2共GC共t兲兲 = max兵− d1共t兲,− d1共t兲 − 共m − 1兲其 = − d1共t兲. 共16兲 The matrix measures ␮2共G共t兲兲 and ␮2共GCi共t兲兲, i=1,2, with p共t兲=t, t苸关0,1兴 are demonstrated in Fig.3.

The remainder of the section is to address the system with even more complex topology.

Proposition 3: Let G共t兲=O共t兲+P共t兲 with O共t兲 and P共t兲 having all its row sums zero. Suppose further that P共t兲 is node balancing. Then

␮2共G共t兲兲 ⱕ␮2共O共t兲兲 + ␭2

P共t兲 + P共t兲T

2

,

whenever all eigenvalues of P共t兲+P共t兲Tare nonpositive.

Proof: Noting that G共t兲=CG共t兲CT= O共t兲+CP共t兲CT, we easily conclude that the above inequality holds as asserted.

0 0.2 0.4 0.6 0.8 1 - .4 5 -4 - .3 5 -3 - .2 5 -2 - .1 5 -1 - .0 5 0 t ma t. mea. µ2( ¯GC1(t)) µ2( ¯GC2(t)) µ2( ¯G(t))

FIG. 2.共Color online兲 The matrix measures of G共t兲 and GCi共t兲, i=1,2 with G given in Eq.共13兲and p共t兲=t are, respectively, represented by the solid line and the dotted lines above.

TABLE II. The table gives the matrix measures of GCi, i = 1 , 2 with various size of G, which is given in Example 3.

m 4 5 6 7 8 9

C1 ⫺0.83 ⫺0.17 0.24 0.54 0.78 0.98

C2 ⫺0.83 ⫺0.17 0.24 0.54 0.78 0.98

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Example 5:关Belykh et al. 共Ref.22兲兴 Consider the outer

coupling matrix G共t兲 to be of random type. Specifically, G共t兲 is of the form G共t兲 = circ共− 2K,1, ... ,1 K ,0, . . . ,0 m−2K−1 ,1, . . . ,1 K 兲 + P共t兲 ¬ O + P共t兲, 共17兲

where P共t兲¬共pij共t兲兲 is a symmetric matrix with all its row sums being zero and satisfies pij共t兲⬅0 for 共i, j兲 with i − j mod mⱕK or j−i mod mⱕK, and pij共t兲=Sij共q兲 for 共q − 1兲␶ⱕt⬍q␶ for all remaining pairs 共i, j兲 with i⫽ j. Here each of Sij共q兲 is a random variable that takes the value 1 with probability p and 0 with probability 1 − p.

The random variables Sij共q兲 are assumed to be all inde-pendent. To each realization ␻ of this stochastic process

S共1兲,S共2兲,..., where S共q兲=兵Sij共q兲,i=1, ... ,n, j=i

+ l mod n , l = K + 1 , . . . ,关n/2兴其, i.e., to each switching se-quence␻, there corresponds a time-varying system described by Eq. 共2兲.

Since P共t兲 is symmetric, by Proposition 3,

␮2共G共t兲兲 ⱕ␮2共O兲 + ␭2共P共t兲兲 ⱕ␮2共O兲 = ␭2共O兲 ⬍ 0. Let G共t兲⬅G. Generally speaking, infC苸C␮2共GC

⫽␮2共GC兲 for any C苸O. Nevertheless,␮2共GC兲 produces a

good upper bound of infC苸C␮2共GC兲.

To support the observation, we conclude this section by providing some additional network topologies where the ma-trix measure of its corresponding GC共t兲, C苸O is smaller

than that of GCi, i = 1 , 2. As a matter of fact, ␮2共GCi兲, i

= 1 , 2, switch signs as the number of nodes increases. In con-trast,␮2共GC兲 mostly remains negative as the size of the

sys-tem grows.

Example 6: Consider a generalized wheel-typed coupled network of the form as illustrated in Fig. 4共a兲. The inner

nodes have the strong all-to-all connections. The outer nodes are only directly connected with their nearest neighbors. The communications between the inner and outer nodes are

through one way going from each inside node to its nearest outside node. Specifically, such a network can be written as the following: G共t兲 ⬅

G1 G2 G3 G4

m⫻m , 共18兲 where G1=

m 2 − 1

1 ¯ 1 1   ] ]   1 1 ¯ 1 −

m 2 − 1

m/2⫻m/2 ,

corresponding to the all-to-all coupling, G2= 0, G3= 0.1I, and G4= G1

共m/2兲− 0.1I. Here G

1

共m/2兲is the diffusive matrix with

pe-riodic boundary conditions and of size m/2⫻m/2. The nu-merical computation suggests that the matrix measures of GCi, i = 1 , 2, are positive provided that mⱖ4 while that of

GC, C苸O, remains negative 共see Table III兲.

Example 7: Consider the prism-typed coupled network

of the form as illustrated in Fig.4共b兲. The difference between the generalized wheel-typed network and the one considered here lies only on how the inner nodes communicate with each other 共see Fig.4兲. Specifically, such a network can be written as the following:

0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2 3 4 5 t mat. mea.

µ

2

( ¯

G

C1

(t))

µ

2

( ¯

G

C2

(t))

µ

2

( ¯

G(t))

FIG. 3.共Color online兲 The matrix measures of G共t兲 and GCi共t兲, i=1,2 with G being given in Eq.共15兲and p共t兲=t are, respectively, represented by the solid line and the dotted lines above.

1 2 N 3 N-1 N+1 N+2 N+i 2N 2N-1 N+3 i (a) 1 2 N 3 N-1 N+1 N+2 N+i 2N 2N-1 N+3 i (b)

FIG. 4. 共Color online兲 Coupling topologies: 共a兲 generalized wheel-typed coupled network with m = 2N and 共b兲 prism-typed coupled network with

m = 2N. Networks共a兲 and 共b兲 appear in Examples 6 and 7, respectively.

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G共t兲 ⬅

G1 G2 G3 G4

m⫻m

,

where G1= G1共m/2兲, G2= 0, G3= 0.1I, and G4= G1共m/2兲− 0.1I. The numerical computation suggests 共see TableIV兲 that the matrix measures of GCi, i = 1 , 2, are positive provided that mⱖ4, while that of GC, C苸O, stays negative until m=86.

The example demonstrates that a coordinate transformation C, C苸O, is indeed a good candidate among all coordinate transformations.

IV. MAIN RESULTS

In the section, we turn our attention back to the dynam-ics of Eq. 共7a兲and analyze the stability of the origin of the system. As in Ref. 26, we break the space y into two parts: yc, the coupled space, and yu, the uncoupled space. Specifi-cally, let

y =

yc yu

, and F共y,t兲 =

Fc共y,t兲

Fu共y,t兲

. 共19兲 Here yc=

y1 ] yk

, and yu=

yk+1 ] yn

.

Then Eq.共7a兲 can be rewritten in the form

y˙c y˙u

=

d共IkG共t兲兲 0 0 U共t兲

冊冉

yc yu

+

Fc共y,t兲 Ru共y,t兲

, 共20兲 where Ru共y,t兲¬Fu共y,t兲−U共t兲yufor some matrix U共t兲. Note that form 共20兲can always be achieved since the remainder term Rustill depends on the whole space y. In what follows, we shall give some intuitive explanations as to why assump-tions on system共20兲 would make the origin into a globally attracting equilibrium.

The dynamics on the coupled space with respect to the linear part is under the influence of G共t兲, which is assumed to have the negative matrix measure. The dynamics of the non-linear part on coupled space can then be controlled by

choos-ing a large couplchoos-ing strength. On the other hand, the un-coupled space has no stable matrix G共t兲 to play with. Thus, its corresponding vector field Fu共y,t兲 must have a certain structure to make the trajectory stay closer to the origin as time progresses, which we shall explain more later. Specifi-cally, the following list of assumptions is needed for our first main theorem:

共H1†兲 System共20兲or Eq.共7a兲is bounded dissipative with respect to ␣. By that, we mean that there is a bounded region B¬兵y:储y储ⱕ␣其 such that for each parameter d⬎0, and each initial value y共0兲, there is a time t0, such that y共t兲 lies in B whenever tⱖt0. 共H2†兲 There is some ␭⬎0 such that

2共G共t兲兲ⱕ−␭, ∀t ⱖ0.

共H3†兲 For any 0⬍, 储Fc共y,t兲储ⱕb

1␤ whenever 储y储 ⱕ␤. Here b1 is independent of␤ and t.

共H4†兲 Matrix U共t兲 is of block diagonal form, i.e., U共t兲 = diag共U1共t兲, ... ,Ul共t兲兲. Here the sizes of Uj共t兲, j = 1 , . . . , l, are 共m−1兲kj⫻共m−1兲kj. Moreover, there is some ␥⬎0 such that the matrix measures

␮2共Uj共t兲兲ⱕ−for all t sufficiently large and all j. 共H5†兲 Let

Ru共y,t兲 =

Ru1共y,t兲

] Rul共y,t兲

with each Ruj共y,t兲苸R共m−1兲⫻kj,∀j=1, ... ,l, where l,

kjare given as in共H4†兲. There is some b2⬎0 such that for each j = 1 , . . . , l, 储Ruj共y,t兲储ⱕb2␤ whenever 储共yc, yu1, . . . , yuj−1兲储ⱕ␤and储y储ⱕ␣. Here

yui=

y共m−1兲·

k+ j=1 i−1k j

+1 ] y共m−1兲·

k+ j=1 i k j

for all i = 1 , . . . , l.

Remark 2: 共i兲 Although the nonlinear terms Ruj共y,t兲

could possibly depend on the whole space, their norm esti-mates are required to depend only on the coupled space and the uncoupled subspaces with their indices proceeding j.共ii兲 The size of the partition matrices Uj共t兲, j=1, ... ,l from U共t兲 depends on how the uncoupled part of the vector field of the single oscillator is structured. To determine how to partition U共t兲, we begin with checking the case for l=1. That is, if for

l = 1, hypotheses 共H4†兲 and 共H5兲 are satisfied, then no fur-ther partition is necessary. Ofur-therwise, we furfur-ther partition U共t兲 into a set of smaller pieces to see if the resulting in-equalities in共H4†兲 and 共H5兲 are fulfilled.

We are now in a position to state our first main theorem.

Theorem 3: Let the outer coupling matrix G共t兲 satisfy-ing Eq. (3a)and the inner coupling matrix D be given as in Eq. (3b). Suppose hypotheses 共H1†兲, 共H2兲, 共H3兲, 共H4兲,

and共H5†兲 hold true, then limt→⬁y共t兲=0 for any initial value provided that the coupling strength d satisfies the following inequality:

TABLE III. The table gives the matrix measures of GCi, i = 1 , 2 and GC, C 苸D with various size of G, which is given in Eq.共18兲.

m 4 6 8 10 5000

C1 0.11 0.32 0.53 0.74 517.47

C2 0.23 0.56 0.96 1.44 34 843.01

C ⫺0.1 ⫺0.1 ⫺0.1 ⫺0.1 ⫺0.1

TABLE IV. The table gives the matrix measures of GCi, i = 1 , 2 and GC, C 苸D with various size of G, which is given in Example 7.

m 4 6 8 86 88

C1 0.34 0.32 0.35 4.65 4.72

C2 0.34 0.56 0.73 4.79 4.86

C ⫺0.1 ⫺0.1 ⫺0.1 ⫺0.0006 0.0004

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db1 ␭

1 + b22 ␥2

l/2 . 共21兲

Proof: For any initial condition y共0兲, there is t0⬎0 such that 储y共t兲储ⱕfor all tⱖt0. Without loss of generality, t0 is chosen sufficiently large so that the inequalities in 共H4† hold. Applying the matrix measure inequality 共A2兲 and hy-potheses共H2†兲, 共H3兲 on yc, for any tⱖt

0, we have that 储yc共t兲储 ⱕ 储yc共t0兲储e␭d共t−t0兲+

b1␭d

e−␭d共t−t0兲+ b1 ␭d

␣¬

e−␭d共t−t0兲+ c0 1 d

␣.

Let␦⬎1. We see that 储yc共t兲储 ⱕ

dc0␦ 共22a兲

whenever tⱖt0,1 for some t0,1⬎t0. Similarly, applying in-equality 共A2兲and hypotheses共H4†兲, 共H5兲 on yu1,

储yu1共t兲储 ⱕd

b2c0

␦2¬ ␣ dc1␦ 2, 共22b兲

whenever tⱖt1,1for some t1,1⬎t0,1. Inductively, we have 储yuj共t兲储 ⱕ

dcj

j+1 共22c兲

whenever tⱖtj,1 for all j = 2 , . . . , l. Here cj= b2/␥

i=0 j−1c

i

2.

Letting t1= tl,1 and summing up Eqs.共22a兲–共22c兲, we get 储y共t兲储 ⱕd

1 + b22 ␥2

l/2 b1 ␭␦l+1¬ hwhenever t⬎t1. Choosing d⬎共1+b2 2/2l/2共b 1/␭兲␦l+1, we

see that the contraction factor h is strictly less than 1, and 储y共t兲储 contracts to zero as time progresses. Since␦⬎1 can be made arbitrarily close to 1, consequently, if d is chosen as assumed, then h can still be made to be less than 1. The assertion of the theorem now follows.

Note that the verification of hypotheses 共H3†兲, 共H4†兲, and共H5†兲 is a nontrivial matter since those assumptions de-pend on the coordinate transformation C. Furthermore, these hypotheses are made for system共7a兲or Eq.共20兲. Hence, it is desirable to derive some easily verifiable hypotheses for sys-tem 共1a兲. Indeed, we are able to derive a set of hypotheses for system共1a兲that can be easily checked. In fact, by merely checking the structure of the vector field f of the individual oscillator, one would be able to verify if those hypotheses hold true. Since the derivation of such a new set of hypoth-eses is rather long and technical, we shall refer the interested readers to Appendix B, which contains Propositions 4 and 5. We summarize these derived hypotheses in the following: 共H1兲 System共1a兲is bounded dissipative with respect to␣. 共H2兲 There is some ␭⬎0 such that␮2共G共t兲兲ⱕ−␭, ∀tⱖ0. 共H3兲 Functions fi共·,t兲, i=1, ... ,k in Eq.共1a兲are uniformly Lipschitz in region B given in共H1兲. That is, there is a constant r⬎0 such that 兩fi共u,t兲− fi共v,t兲兩ⱕr储u−v储, whenever t is sufficiently large, and u,v in B.

共H4兲 The matrix Q共v,t兲, which is given as in Eq. 共B1兲, is

of block diagonal form, i.e., Q共v,t兲

= diag共Q1共v,t兲, ... ,Ql共v,t兲兲. Here the sizes of Qj共v,t兲, j=1, ... ,l, are kj⫻kj. Moreover, there is some ␥⬎0 such that matrix measures ␮2共Qj共v,t兲兲 ⱕ−␥ for all j, whenever t is sufficiently large, andv in B.

共H5兲 Denoted by s1= k and sj= k +兺i=1 j−1

ki, j = 2 , . . . , l, where

ki and l are defined in 共H4兲. Suppose, for any 1ⱕ j ⱕl, there is a␦⬎0 such that

储关r共u,v,t兲兴s

j+1

sj+kj储 ⱕ␦储关u − v兴

1 sj

for t sufficiently large, and u, v in B. Here 关u兴ij is defined to be共ui, . . . , uj兲T.

Remark 3:共i兲 Using the similar techniques as developed

in the proof of Propositions 4 and 5, we may also conclude that the global theorems obtained in Ref. 33 may still be valid by using the coordinate transformation developed here in this paper. Consequently, the first drawback of their ap-proach can be removed.共ii兲 Examples are given in Sec. V to illustrate how hypotheses 共H4兲 and 共H5兲 can be easily checked.

The main result of the paper is now stated in the follow-ing. The proof of the main theorem follows directly from Theorem 3 and Propositions 4 and 5.

Theorem 4: Let the outer coupling matrix G共t兲 satisfy-ing Eq. (3a)and the inner coupling matrix D be given as in Eq. (3b). Suppose hypotheses (H1), (H2), (H3), (H4), and (H5) hold true, then coupled system (1a) achieves global synchronization whenever dr

k cond共C1C T

1 + ␦2储C˜储2储C 1CT储2 ␥2

l/2 , 共23兲

where C, C1, and C˜ are given as in Theorem 1, Eq.(8), and

Eq. (B4c), respectively.

Remark 4: The small price to pay by introducing the

coordinate transformation C is that the lower bound given as in the right hand side of Eq.共23兲on the coupling strength d is size dependent.

V. APPLICATIONS AND COMPARISONS

To see the effectiveness of our main results and to com-pare our results with existing methods, we consider coupled Lorentz equations with various coupling configurations. The vector field of the individual Lorentz oscillator under consid-eration is recognized as f共x兲=共共x2− x1兲,rx1− x2− x1x3, −bx3+ x1x2兲T¬共f1共x兲, f2共x兲, f3共x兲兲T. Here ␴= 10, r = 28, and

b = 8/3. We shall illustrate, via the first three cases, how one

should examine the structure of f共x兲 to see if hypotheses 共H3兲–共H5兲 are fulfilled or not.

Case 1: Let the inner coupling matrix D correspond to y-component partial-state coupling, i.e.,

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D =

0 0 0 0 1 0

0 0 0

. 共24兲

Let the outer coupling matrix G共t兲 be either of the forms given as in Eqs. 共9兲, 共11兲–共13兲, and 共17兲. Then hypothesis 共H1兲 of bounded dissipation of system 共1a兲 is well known 共see, e.g., Ref.23兲, which is the ball B containing the topo-logical product of an absorbing domain

B =

x12+ x22+共x3− r −␴兲2⬍

b2共r +2 4共b − 1兲

.

Hypothesis 共H2兲 of matrix measure␮2共G兲 is clearly held as shown in Sec. III. Since the “coupled” nonlinearity f2共·兲 satisfies

储f2共u兲 − f2共v兲储 = 储共r − v3兲共u1−v1兲 − 共u2−v2兲 − u1共u3−v3兲储 ⱕ r储u − v储

for some constant r⬎0 in region B, hypothesis 共H3兲 holds true. Moreover, the difference of “uncoupled” nonlinearities

f1and f3is given as follows:

f1共u兲 − f1共v兲 = 关−共u1v1兲兴 +␴共u2v2兲,

f3共u兲 − f3共v兲 = 关− b共u3−v3兲兴 + u1共u2−v2兲 + v2共u1−v1兲. It is readily seen that one should break the uncoupled space into two parts. That is, if we choose l = 2 and pick the space of the first共respectively, second兲 diagonal block being the one associated with the nonlinearity f1共respectively, f3兲, then Q1共v,t兲=共−兲 and Q2共v,t兲=共−b兲, it then follows that hypothesis 共H4兲 is held. Furthermore, since 关r共u,v,t兲兴22 = r2共u,v,t兲=共u2−v2兲, which depends only on coupled space, and 关r共u,v,t兲兴33= r3共u,v,t兲=u1共u2−v2兲+v2共u1−v1兲, which depends on the coupled system and the uncoupled subspace with the proceeding index, hypothesis共H5兲 is ful-filled as well. Hence, by Theorem 4, coupled system共1a兲has global synchronization provided coupling strength d is large enough. A numerical result is also presented to support our analytic result, see Fig.5共a兲.

Case 2: Let the inner coupling matrix D correspond to

either x-component partial-state coupling or full-state cou-pling, i.e., D =

1 0 0 0 0 0 0 0 0

. 共25兲

Let the outer coupling matrix G共t兲 be either of the forms given as in Eqs. 共9兲, 共11兲–共13兲, and 共17兲. In this case, the coupled nonlinearity f1satisfies the following uniformly Lip-schitz condition:

储f1共u兲 − f1共v兲储 = 储−共u1−v1兲 +␴共u2−v2兲储 ⱕ

2␴储u − v储.

For uncoupled nonlinearities, we see that

f2共u兲 − f2共v兲 f3共u兲 − f3共v兲

=

− 1 −v1 v1 − b

冊冉

u2−v2 u3v3

+

共r − u3兲共u1−v1兲 u2共u1−v1兲

¬ Q共v,t兲

u2−v2 u3v3

+ r共u,v,t兲.

Clearly, ␮2共Q共v,t兲兲=max兵−1,−b其=−1⬍0 and 储r共u,v,t兲储 ⱕr兩u1−v1兩 for some constant r⬎0 in region B. Hence, hy-potheses 共H4兲 and 共H5兲 are satisfied, and we can conclude that coupled system共1a兲has global synchronization provided coupling strength d is large enough. A numerical result is also presented to support our analytic result, see Fig.5共b兲.

Case 3: Consider the z-component partial-state coupling.

Since the remainder term in the difference of uncoupled non-linearities f1 and f2 contains each other, the only feasible breaking is to be given in the following:

0 5 −50 0 50 time x−different 0 5 −100 0 100 time y−di ff erent 0 5 0 50 100 time z−different (a) 0 5 −50 0 50 time x−different 0 5 −100 0 100 time y−di ff erent 0 5 −100 0 100 time z−di ff erent (b)

FIG. 5.共Color online兲 The difference of components of the first two coupled oscillators:共a兲 the y-component partial-state coupling addressed in Case 1 and共b兲 the x-component partial-state coupling addressed in Case 2. In both cases m = 8 and the outer coupling matrix is given as in Eq.共9兲.

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f1共u兲 − f1共v兲 f2共u兲 − f2共v兲

=

−␴ ␴ r −v3 − 1

冊冉

u1−v1 u2−v2

+

0 − u1共u3−v3兲

.

With r = 28, the matrix measure of the associated Q共v,t兲 can-not stay negative. Consequently, the conclusion of our main theorem cannot be assumed, which is in consistence with the numerical results.

In the next two cases various 共outer兲 coupling matrices addressed in Sec. III are considered.

Case 4: Let the coupling matrix G共t兲 be given in Eq.

共15兲with d1共t兲= 3

2− sin共t兲. The numerical results are demon-strated in Fig.6. The synchronization of the coupled Lorentz systems with the coupling matrices G共t兲 studied in Examples 5–7 of Sec. III is also verified numerically. To save the space, we will not provide such figures. It should be noted that for G共t兲 considered in Example 7, our numerical results demon-strate that the synchronization of the corresponding system still occurs with m = 88.

Comparison 1:关Chen 共Ref. 33兲兴 Consider the case that

the inner coupling matrix D is given in Eq.共25兲and the outer coupling matrix G共t兲 is of the form

G =

− 2 1 1

1 − 2 1

1 1 − 2

.

To apply the global theorem proposed by Chen,33one needs to verify the following:

共a兲 There exists a matrix K苸R3⫻3 such that f共x兲+Kx is V-uniformly decreasing for some symmetric positive definite matrix V苸R3⫻3. That is, there is a positive constant ␮ such that 共x−y兲TV共f共x兲+Kx−f共y兲−Ky兲 ⱕ−␮储x−y储2for all x, y苸R3.

共b兲 There exists a diagonal matrix U苸R2⫻2 and M 苸R6⫻6with MT

M = UV such that

␮2共M共dGC1D − I2丢K兲M−1兲 ⬍ 0. Here GC1= −3I2.

For the choices of U = I2and V = I3, we shall show that conditions 共a兲 and 共b兲 cannot be satisfied simultaneously. Indeed, suppose condition 共b兲 holds true for some K, then ␮2共M共dGC1D − I2丢K兲M−1兲=␮2共−3dD−K兲⬍0 and, hence, zTKzⱖ−3dzTDz for any z苸R3. Let x =共x

1, x2, x3兲T

with x2ⱖ10, x1, x3苸R, and y=x−k共1,0,1兲T with k苸R −兵0其. Then there is an␣, 0⬍␣⬍1 such that

共x − y兲T共f共x兲 + Kx − f共y兲 − Ky兲

=共x − y兲T共K + Df共x +共1 −␣兲y兲兲共x − y兲 ⱖ 共x − y兲T共Df共x +共1 −兲y兲 − 3dD兲共x − y兲 ⱖ2

3k2. Here Df共x兲 =

− 10 10 0 28 − x¯3 − 1 − x¯1 x ¯2 ¯x1 − 83

.

And so, condition 共a兲 fails.

Comparison 2:关Chen 共Ref. 39兲兴 Let the outer coupling

matrix G共t兲 be given in Eq.共9兲, and the inner coupling ma-trix D be I3. To verify the criterion for global synchroniza-tion in Ref. 39it suffices to show

␮2共Im−1A + GC1In兲 ⬍ 0, 共26兲

where A = diag共a1, . . . , an兲 and aiⱖ0, i=1,2, ... ,n. How-ever, if the number of oscillators is greater than 7, i.e., m ⬎7, then ␮2共GC1兲⬎0 共see TableI兲. And so

␮2共Im−1A + GC1In兲 ⱖ␮2共GC1兲 −␮2共− A兲 =␮2共GC1兲 + min兵a1, . . . ,an其 ⬎ 0.

VI. CONCLUSIONS

A general framework for determining the global stability of synchronous chaotic oscillations in coupled oscillator sys-tems with complex networks has been discussed. This frame-work allows one to address very large array of oscillators with complex topology including time-varying networks, networks with asymmetric positive and negative coupling, and networks with some randomness. Furthermore, the veri-fication of our framework can be easily checked. The vehicle for providing such general synchronization theory is the ma-trix measure as well as the newly introduced coordinate transformation.

Theoretical studies of globally synchronous chaos have been conducted with the coupled Lorentz oscillators. The x,

y, and z-component couplings of the system have been used

as illustrations on how to apply the main theorem. The net-works such as the star type, the wavelet transformed type, the pristine world joining with some randomness, the

general-0 5 −10 0 10 time x−different 0 5 −50 0 50 time y−different 0 5 −50 0 50 time z−different

FIG. 6.共Color online兲 The difference of components of the first two coupled oscillators considered in Case 4. Here the x-component partial-state cou-pling is considered with m = 8 and the outer coucou-pling matrix given as in Eq. 共15兲.

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ized wheel type, and the prism type have been discussed. The comparisons with the existing methods have also been provided.

We would like to conclude our paper with the following remarks. To prove global synchronization of the coupled cha-otic systems, one needs to assume the bounded dissipation of the systems, which plays the role of an a priori estimate. Such an assumption is also implicitly required in both CGS and MMA. Unfortunately, there have not many general theo-rems been provided for the bounded dissipation of coupled chaotic systems with complex topology. Note that the bounded dissipation of the individual oscillator does not nec-essarily imply that the coupled systems with complex net-works would share the same property. Therefore, it would certainly be of great interest to develop a theory of the bounded dissipation of the coupled chaotic systems with complex network topologies.

ACKNOWLEDGMENTS

We thank referees for suggesting numerous improve-ments to the original draft.

APPENDIX A: CONCEPTS OF MATRIX MEASURE

In our derivation of synchronization of system 共2兲, 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共see, e.g., Ref.43兲.

Definition 3: 关Vidyasagar 共Ref. 43兲兴 Let 储·储i be an in-duced matrix norm onRn⫻n. The matrix measure of matrix K onRn⫻n

is defined to be␮i共K兲=lim⑀→0+储I+K储i− 1/⑀.

Lemma 1: [Vidyasagar (Ref.43)] Let储·储kbe an induced k-norm onRn⫻n, where k = 1 , 2 ,⬁. Then each of matrix mea-surek共K兲, k=1,2,⬁, of matrix K=共kij兲 on Rn⫻nis, respec-tively, ␮⬁共K兲 = max i

kii+

j⫽i 兩kij兩

, ␮1共K兲 = max j

kjj+

i⫽j 兩kij兩

, and ␮2共K兲 = ␭max共KT+ K兲/2. 共A1兲

Heremax共K兲 is the maximum eigenvalue of K.

Theorem 5: [Vidyasagar (Ref.43)] Consider the differ-ential equation x˙共t兲=K共t兲x共t兲+v共t兲, tⱖ0, where x共t兲苸Rn, K共t兲苸Rn⫻n

, and K共t兲, v共t兲 are piecewise continuous. Let 储·储i be a norm onRn

and储·储i,idenote, respectively, the

corre-sponding induced norm and matrix measure on Rn⫻n

. Then whenever tⱖt0ⱖ0, we have 储x共t0兲储iexp

t0 t −␮i共− K共s兲兲ds

t0 t exp

s t −␮i共− K共兲兲d

储v共s兲储ids

ⱕ 储x共t兲储iⱕ 储x共t0兲储iexp

t0 ti共K共s兲兲ds

+

t0 t exp

s ti共K共兲兲d

储v共s兲储ids. 共A2兲

APPENDIX B: PROOFS OF MAIN RESULTS

The following notation is needed. Let u

=共u1, . . . , ui, ui+1, . . . , uj, . . . , un兲T. Denote by 关u兴ij =共ui, ui+1, . . . , uj兲T. Write the difference of f共·,t兲 at u and v in the form

f共u,t兲 − f共v,t兲 =

f1共u,t兲 − f1共v,t兲 ]

fn共u,t兲 − fn共v,t兲

¬

fu共u,t兲 − fu共v,t兲fc共u,t兲 − fc共v,t兲

¬

Q共v,t兲关u − v兴fc共u,t兲 − fc共v,t兲

k+1

n + r共u,v,t兲

, 共B1兲 where fc共·,t兲苸Rk, fu共·,t兲苸Rn−k, and matrix Q共v,t兲 is of the size共n−k兲⫻共n−k兲. Since r共u,v,t兲 could depend on all com-ponents of u and v, such a decomposition in Eq. 共B1兲 can always be achieved.

Proposition 4: Suppose fi共·,t兲, i=1, ... ,k are uniformly Lipschitz, i.e., there exists a positive constant r⬎0 such that

兩fi共u,t兲 − fi共v,t兲兩 ⱕ r储u − v储 共B2兲

for all i = 1 , . . . , k. Then the inequality in 共H3†兲 is satisfied

with b1= r

k cond共C1CT兲. Here C1is given as in Eq.(8)and cond共C1CT兲=储C1CT储储共C1CT兲−1储 is the condition number of 共C1CT兲.

Proof: Note first that C1= C1CTC and C =共C1CT兲−1C1. Now, 储Fc共y,t兲储 =

冩冢

Cf˜1共x˜,t兲 ] Cf˜k共x˜,t兲

冣冩

=

共Ik共C1CT兲−1兲

C1˜f1共x˜,t兲 ] C1˜fk共x˜,t兲

冣冩

ⱕ 储共C1CT兲−1储

冩冢

C1˜f1共x˜,t兲 ] C1˜fk共x˜,t兲

冣冩

. Since

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储C1˜f1共x˜,t兲储2=

冩冢

fi共x1,t兲 − fi共x2,t兲 ] fi共x1,t兲 − fi共xm,t兲

冣冩

2 ⱕ r2

冩冢

x1− x2 ] x1− xm

冣冩

2

for all i = 1 , . . . , k, we have that 储Fc共y,t兲储 ⱕ

kr储共C1CT兲−1储

冩冢

x1− x2 ] x1− xm

冣冩

=

kr储共C1CT兲−1储

共C1丢In兲

x1 ] xm

冣冩

=

kr储共C1CT兲−1储储共C1CTIn兲共CIn兲x储

kr cond共C1CT兲储y储. 共B3兲 The proof of the proposition is completed.

The above Proposition 4 amounts to saying that if fi, i = 1 , . . . , k, the coupled parts of the vector field of the indi-vidual oscillator are uniformly Lipschitz, then hypothesis 共H3†兲 holds.

We next turn our attention to the structure of the vector field of the uncoupled parts.

Proposition 5: (i) Suppose matrix Q共v,t兲 can be written as the block diagonal form

Q共v,t兲 = diag共Q1共v,t兲, ... ,Ql共v,t兲兲,

where the size of matrices Qj共v,t兲 is kj⫻kj,∀j=1, ... ,l and indices l, kjare given as in 共H4†兲. Moreover, there is some

⬎0 such that

␮2共Qj共v,t兲兲 ⱕ −␥. 共B4a兲

Hereis independent ofv , t. Then the inequality in共H4†兲 is

fulfilled. (ii) Denoted by s1= k and sj= k +i=1 j−1k

i, j = 2 , . . . , l,

where ki and l are defined in 共H4†兲. Let C=共ci,j兲共m−1兲⫻m.

Suppose, for any 1ⱕ jⱕl, there is⬎0 such that

储关r共u,v,t兲兴s

j+1

sj+kj储 ⱕ␦储关u − v兴

1

sj储. 共B4b兲

Then the inequality in 共H5†兲 is satisfied with b 2 =␦储C˜储储C1CT储. Here

C

˜ = 共ci,j+1兲 苸 R共m−1兲⫻共m−1兲, 1ⱕ i, j ⱕ m − 1. 共B4c兲

Proof: Write Fu共y,t兲 as 共Fu1共y,t兲, ... ,Ful共y,t兲兲T, which is in consistence with the block diagonal form of U共t兲. Now, for 1ⱕ jⱕl, Fuj共y,t兲 =

Cf˜sj+1共x˜,t兲 ] Cf˜sj+kj共x˜,t兲

=

k=1 m c1,kfsj+1共xk,t兲 

k=1 m cm−1,kfsj+1共xk,t兲 ]

k=1 m c1,kfsj+kj共xk,t兲 

k=1 m cm−1,kfsj+kj共xk,t兲

= P

k=1 m c1,kfsj+1共xk,t兲 

k=1 m c1,kfsj+kj共xk,t兲 ]

k=1 m cm−1,kfsj+1共xk,t兲 

k=1 m cm−1,kfsj+kj共xk,t兲

¬ Ph.

Here P is a permutation matrix. That is, we exchange certain rows of Fuj共y,t兲 to obtain F. Using the fact that the row sums of C are all zeros, we have that for 1ⱕiⱕm−1, sj+ 1ⱕlⱕsj+ kj,

k=1 m ci,kfl共xk,t兲 =

k=2 m ci,k共fl共xk,t兲 − fl共x1,t兲兲. 共B5兲

To save notations,∀i=1, ... ,kj, we denote by关rs

j+i共xl, x1, t兲兴l=2

m

the vector共rs

j+i共x2, x1, t兲,rsj+i共x3, x1, t兲, ... ,rsj+i共xm, x1, t兲兲

T. Applying Eq.共B1兲and Eqs.共B4a兲–共B4c兲, we shall be able to rewrite h as

數據

TABLE I. The table gives the matrix measures of G C i , i = 1 , 2 with various size of G, which is given in Eq
FIG. 1. 共Color online兲 The matrix measures of G共t兲 and G C i 共t兲, i=1,2 with G being given in Eq
FIG. 2. 共Color online兲 The matrix measures of G共t兲 and G C i 共t兲, i=1,2 with G given in Eq
FIG. 3. 共Color online兲 The matrix measures of G共t兲 and G C i 共t兲, i=1,2 with G being given in Eq
+4

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