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doi:10.1016/j.ph

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Physica A 386 (2007) 513–530

www.elsevier.com/locate/physa

Synchronization in complex delayed dynamical networkswith nonsymmetric coupling

Jianshe Wu�, Licheng Jiao

Institute of Intelligent Information Processing, 132# Xidian University, 2# Taibai South Road, Xi’an 710071, China

Received 4 December 2006; received in revised form 25 June 2007

Available online 2 August 2007

Abstract

A new general complex delayed dynamical network model with nonsymmetric coupling is introduced, and then we

investigate its synchronization phenomena. Several synchronization criteria for delay-independent and delay-

dependent synchronization are provided which generalize some previous results. The matrix Jordan canonical

formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the

coupling configuration matrix of the network does not required to be diagonalizable and may have complex eigenvalues.

Especially, we show clearly that the synchronizability of a delayed dynamical network is not always characterized by the

second-largest eigenvalue even though all the eigenvalues of the coupling configuration matrix are real. Furthermore, the

effects of time-delay on synchronizability of networks with unidirectional coupling are studied under some typical network

structures. The results are illustrated by delayed networks in which each node is a two-dimensional limit cycle oscillator

system consisting of a two-cell cellular neural network, numerical simulations show that these networks can realize

synchronization with smaller time-delay, and will lose synchronization when the time-delay increase larger than a

threshold.

r 2007 Elsevier B.V. All rights reserved.

PACS: 84.35.+i; 05.45.+b

Keywords: Complex network; Synchronization; Time-delay; Nonsymmetric coupling; Exponentially stability

1. Introduction

Complex network models are used to describe various interconnected systems of real world, such as theWorld Wide Web, food webs, electronic power grids, internet, etc. [1–4]. Since the complexity of real worldnetwork, there are various complex network models used to study the dynamics of coupled systems.Synchronization is a basic motion in coupled dynamical networks which has been carefully studied for a longtime [4–16]. Wang and Chen presented a uniform complex network model and investigated its synchronization

e front matter r 2007 Elsevier B.V. All rights reserved.

ysa.2007.07.052

ing author. Tel./fax: +86 29 88201023.

esses: jshwu@mail.xidian.edu.cn (J. Wu), lchjiao@mail.xidian.edu.cn (L. Jiao).

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530514

in small-world and scale-free networks [7–10]. Latterly, Lu and Chen improved the model of Wang and Chenand characterized the synchronizability of small-world networks based on their model [11]. Li and Chenderived a sufficient condition for the global synchronization and asymptotical stability by introducing areference state with the Lyapunov stability theorem [12]. Time-delay in signal transition is a veryfamiliar phenomenon in nature, and the effects of time-delay on the dynamics of various coupled dynamicalsystems have been an interest for research in science and technology. Li and Chen extended the modelof Wang and Chen to a uniform model with the coupling delay, and derived some synchronization criteriafor both delay-independent and delay-dependent exponential stability of the synchronization state [13]. Zhouand Chen investigated the synchronization dynamics of a general model of complex delayed dynamicalnetworks as well as the effects of time-delays, and some criteria which ensure the networks to be delay-independent and delay-dependent exponentially synchronized are derived [14]. Lu and Chen introduced ageneral time-varying complex network model and derived some synchronization criteria for time-varyingcomplex network [15,16].

A commonly used approach to analyzing the synchronization phenomena of a complex dynamical networksis to linearize it at the synchronization state, thus the criteria for synchronization presented so far in mostliteratures require that the coupling configuration matrix of the network is symmetric or diagonalizable. Inthis special case, the coupling configuration matrix has only real eigenvalues. In fact, the coupling betweennodes of many real-world networks is nonsymmetric, such as the food web [17], metabolic network [18],World-Wide-Web [19], epidemic networks [20], document citation networks [21]. Furthermore, the coupling insome real-world networks may be unidirectional, such as the AIDS spreading network. Most of thenonsymmetric matrices have complex conjugate eigenvalues, even though all the eigenvalues of anonsymmetric matrix are real, it may not be diagonalizable [22]. Recently, there are some researches onsynchronization of complex networks with nonsymmetric coupling: without assuming symmetry andirreducibility, some criteria for synchronization in networks and delayed networks were presented in Refs.[23,24], respectively; and some adaptive synchronization criteria for networks with uncertain coupling arereported in Ref. [25].

In this paper, a new general delayed dynamical complex network model with nonsymmetric coupling isintroduced, and then we investigate its synchronization phenomena. The matrix canonical formalizationmethod is used instead of the matrix diagonalization method. Criteria for delay-independent and delay-dependent synchronization are derived, and the synchronization criteria obtained in this paper do not requirethe coupling configuration matrix to be diagonalizable, and made some previous results a special case.Especially, since the effect of time-delay, we show clearly that the synchronizability of a delayed dynamicalnetwork is not always characterized by the second-largest eigenvalue of the coupling configuration matrix eventhough all its eigenvalues are real.

This paper is organized as follows. In Section 2, a general delayed dynamical complex network model withnonsymmetric coupling is introduced, and some mathematical definitions and lemmas are given. The mainresults of this paper are given in Section 3, which contains some general criterion for synchronization, andsome criteria for delay-independent and delay-dependent synchronization in the delayed networks. In Section4, the effects of time-delay on synchronizability of networks with unidirectional coupling under some differenttypical topological structures are analyzed. Illustrating networks and some simulation results are also given inSection 4. Section 5 is the conclusion of this paper.

Now, we list some mathematical notations used in this paper. We denote a matrix A is an n� n complex(real) matrix by A 2MnðCÞ ðA 2MnðRÞÞ. We denote the positive (nonnegative) definiteness of A by A �

0 ðAk0Þ and the negative (nonpositive) definiteness of A by A � 0 ðA%0Þ. A � B means A� B is a positivedefinite matrix. The vector norm used will be kxk ¼ ðx�xÞ1=2. lR

i and lIi denote the real part and imaginary part

of a complex number li, respectively. _wRil ðtÞ ¼ dwR

il ðtÞ=dt, and _wIilðtÞ ¼ dwI

ilðtÞ=dt, where wilðtÞ is a complexvariable. The Hermitian part of a square matrix P is denoted as PH ¼ 1

2ðP� þ PÞ, where P 2MnðCÞ; and the

symmetric part of a square matrix is denoted as DS ¼ 12ðDT þDÞ, where D 2MnðRÞ. The n� n identity matrix

is denoted by In. The set of all l 2 C that are eigenvalues of G 2MnðCÞ is called the spectrum of G and isdenoted as sðGÞ. The spectral radius of G is the nonnegative real numberrðGÞ ¼ maxfjlj : l 2 sðGÞg. For amatrix P 2MnðCÞ, lPmin and lPmax are the minimum and maximum eigenvalues of P, respectively, ifsðPÞ � R.

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 515

2. Network model and mathematic preliminaries

2.1. Delayed dynamical network model with nonsymmetric coupling

We consider a delayed dynamical network consisting of N linearly and diffusively coupled identical nodes.The state equations of the entire network are described by

_xiðtÞ ¼ f ðxiðtÞÞ þXN

j¼1

gijAxjðt� tÞ; i ¼ 1; . . . ;N, (1)

where xiðtÞ ¼ ðxi1ðtÞ;xi2ðtÞ; . . . ;xinðtÞÞT2 Rn is the state variable of the ith node. A 2MnðRÞ is the

inner-coupling matrix which describe the individual coupling between two connected nodes of thenetwork, tX0is the coupling time-delay, f : Rn ! Rn is a vector-valued function describe the dynamicsof an individual node, gij 2 R describe the time-invariant coupling strength from node j to node iðjaiÞ, gijX0,and

gii ¼ �XN

j¼1jai

gij ; i ¼ 1; . . . ;N, (2)

which imply that the row sum of matrix G ¼ ðgijÞ 2MN ðRÞ are all zero. Obviously, the coupling configurationof the matrix G contains both coupling topological structure and strength information in the network. Notethat the coupling configuration matrix G in our network model is not assumed to be symmetric or irreducible,and it may have complex eigenvalues. So the network (1) is a generalization of the model discussed inRefs. [13,14].

Network (1) can be equivalently expressed as (3) by using Kronecker product [26–28],

_X ðtÞ ¼ f ðX ðtÞÞ þ ðG � AÞX ðt� tÞ, (3)

where X ðtÞ ¼ ðx1ðtÞ; . . . ;xN ðtÞÞT2 RnN , X ðt� tÞ ¼ ðx1ðt� tÞ; . . . ;xN ðt� tÞÞT 2 RnN , f ðX ðtÞÞ ¼ ðf ðx1ðtÞÞ; . . . ;

f ðxNðtÞÞÞT2 RnN , G � A 2MnN ðRÞ denotes the Kronecker product of matrices G and A. It is convenient to

use (3) than (1) sometimes. From the Gersgorin discs theorem [22], the real parts of all the eigenvalues of G areless than or equal to zero. Since the row sum of G are all zero, zero is always an eigenvalue of G correspondingto eigenvector ð1; 1; . . . ; 1ÞT. In this paper, we always assume that zero is an eigenvalue of G with multiplicityone. From Lemma 2 in Ref. [26], zero is an eigenvalue of G with multiplicity one if it is irreducible. Forsimplicity, we always assume l1 ¼ 0, if the coupling configuration matrix have k different eigenvalues and allof them taken to be real, we denote them as l24l34 4lk.

2.2. Mathematic preliminaries

Definition 1 (Synchronization). Let D0 denotes an open set in the state space. If from any initial pointT 0

X ðt0Þ ¼ ðx1ðt0Þ; . . . ;xNðt0ÞÞ 2 D , there is kxiðt0Þ � sðtÞk ! 0 as t!1, i ¼ 1; . . . ;N, the delayed autono-

mous network (1) is said to realize synchronization. The open set D0 is called the region of synchrony, andSðtÞ ¼ ðsðtÞ; . . . ; sðtÞÞT is called the synchronization state of the network.

Obviously, if network (1) realized synchronization, then x1ðtÞ ¼ x2ðtÞ ¼ xN ðtÞ ¼ sðtÞ, from (2) one hasPNj¼1gijAsðtÞ ¼ 0, in turn from (1) to get

_sðtÞ ¼ f ðsðtÞÞ, (4)

which means sðtÞ is a solution of system _xðtÞ ¼ f ðxðtÞÞ. The authors in Ref. [13] include Eq. (4) intheir definition of synchronization, while the definition of synchronization in Ref. [14] does not requirethat sðtÞ is a solution of system _xðtÞ ¼ f ðxðtÞÞ, in fact, they are equivalent. sðtÞ can be an equilibriumpoint, a limit cycle, an aperiodic orbit, or a chaotic orbit, such as the chaotic Lorenz, Chen, andLu attractors [29].

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530516

When the network (1) realized synchronization, the synchronization state SðtÞ is asymptotically stable in thestate space. On the other hand, if the synchronization state SðtÞ is asymptotically stable, then the delayednetwork (1) will realize synchronization by Definition 1.

Definition 2. The synchronization state SðtÞ of the delayed network (1) is said to be delay-independentasymptotically stable, if the synchronization state SðtÞ is asymptotically stable for any fixed time-delayt 2 ½0;1Þ. And the delayed network is then called a delay-independent network.

Definition 3. The synchronization state SðtÞ of the delayed network (1) is said to be delay-dependentasymptotically stable, if the synchronization state SðtÞ is asymptotically stable for any fixed time-delayt 2 ½0; h, h40. And the delayed network is then called a delay-dependent network.

Lemma 1 (Horn and Johnson [22]). Let G 2MnðCÞ be a given complex matrix. There is a nonsingular matrix

F 2MnðCÞ such that

G ¼ F

Jn1ðl1Þ 0

Jn2ðl2Þ

. ..

0 JnkðlkÞ

2666664

3777775F�1 ¼ FJF�1 (5)

and n1 þ n2 þ þ nk ¼ N. The Jordan matrix J of G is unique up to permutation of the diagonal Jordan blocks.The eigenvalues li, i ¼ 1; . . . ; k, are not necessarily distinct. If G is a real matrix with only real eigenvalues

(sðGÞ � RÞ, then the similarity matrix F can be taken to be real. JniðliÞ is the Jordan blocks:

JniðliÞ ¼

li 1 0

li. ..

. ..

1

0 li

0BBBBB@

1CCCCCA; i ¼ 1; . . . ; k. (6)

Obviously, if li is an eigenvalue with multiplicity one ðni ¼ 1Þ, then the corresponding Jordan block J1ðliÞ ¼ li.

Lemma 2. For any two n-dimensional complex vectors Y and Z, and a positive definite matrix Q 2MnðCÞ,

Y �Z þ Z�YpY �QY þ Z�Q�1Z. (7)

In particular, if Q ¼ In, Lemma 2 reduced to

Y �Z þ Z�YpY �Y þ Z�Z, (8)

else if x and y are real vectors and Q 2MnðRÞ, then

xTyp12xTQyþ 1

2xTQ�1y. (9)

Proof. Since a positive definite matrix is Hermitian, from the spectral theorem for Hermitian matrices [22],there is a unitary matrix U, such that Q ¼ U�GU , G ¼ diagðl1; . . . ; lnÞ is a diagonal matrix with its entries areeigenvalues of Q. Then,

0pðffiffiffiffiGp

UY �ffiffiffiffiffiffiffiffiG�1

pUZÞ�ð

ffiffiffiffiGp

UY �ffiffiffiffiffiffiffiffiG�1

pUZÞ

¼ Y �U�GUY þ Z�U�G�1UZ � Y �U�ffiffiffiffiGp ffiffiffiffiffiffiffiffi

G�1p

UZ � Z�U�ffiffiffiffiffiffiffiffiG�1

p ffiffiffiffiGp

UY

¼ Y �QY þ Z�Q�1Z � Y �Z � Z�Y , ð10Þ

thus, obtain the result. &

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 517

Definition 4 (Zhou and Chen [14]). Function kðz1; z2Þ is defined as

kðz1; z2Þ ¼

1 if z140 or z1 ¼ 0 and z240;

0 if z1 ¼ z2 ¼ 0;

�1 if z1o0 or z1 ¼ 0 and z2o0;

8><>: (11)

then, kðuðtÞ; _uðtÞÞuðtÞ ¼ juðtÞj, djuðtÞj=dt ¼ kðuðtÞ; _uðtÞÞ _uðtÞ.

Lemma 3 (Zhou and Chen [14]). Let V ðtÞ40 for t 2 R; t 2 ½0;1Þ and t0 2 R. Suppose that

_V ðtÞp� aV ðtÞ þ b supt�tpapt

V ðaÞ (12)

for all t4t0. If a4b40, there exist two constants g40 and k40, such that

V ðtÞpke�gðt�t0Þ; t4t0.

Lemma 4. Suppose P 2MnðCÞ is a positive definite matrix, and w 2 Cn, then

lPminw�wpw�PwplPmaxw�w. (13)

Proof. Since a positive definite matrix is Hermitian, from the spectral theorem for Hermitian matrices [22],there is a unitary matrix U and a real diagonal matrix L, such thatP ¼ U�LU , where the entries of L areeigenvalues of P, then

w�Pw ¼ ðUwÞ�LUw

plPmaxw�U�Uw ¼ lPmaxw�w.

Similarly, one has w�PwXlPminw�w. The proof is completed. &

3. Synchronization in delayed dynamical networks

Since the coupling configuration matrix G in our network model is not symmetric and may have complexeigenvalues, it should be treated as a complex matrix in the process of transformation.

3.1. General synchronization analysis

Theorem 1. Suppose f ðxðtÞÞ is continuously differentiable at sðtÞ 2 Rn, then the synchronization state SðtÞ of the

delayed network (1) is exponentially stable if the following k � 1 ni-node time-varying systems are exponentially

stable about their zero solutions:

_W ðtÞ ¼ ðIni�Df ðsðtÞÞÞW ðtÞ þ ðJni

ðliÞ � AÞW ðt� tÞ; i ¼ 2; . . . ; k, (14)

where 2pkpN, n2 þ n3 þ þ nk ¼ N � 1, W ¼ ðw1; . . . ;wj ; . . . ;wniÞT2 Cnni , wj 2 Cn, Df ðsðtÞÞ is the

Jacobian matrix of f ðxðtÞÞ at sðtÞ, Ini�Df ðsðtÞÞ denotes the Kronecker product of matrices Ini

and Df ðsðtÞÞ.JniðliÞ are the Jordan blocks of the coupling configuration matrix G.

Proof. To investigate the stability of the synchronous solution SðtÞ, let

ZiðtÞ ¼ xiðtÞ � sðtÞ; i ¼ 1; 2; . . . ;N, (15)

ZiðtÞ is the error between xiðtÞ and sðtÞ. Substituting (15) into (1) and considering (4), one has

_ZiðtÞ ¼ f ðsðtÞ þ ZiðtÞÞ � f ðsðtÞÞ þXN

j¼1

gijAZjðt� tÞ

¼ Df ðsðtÞÞZjðtÞ þXN

j¼1

gijAZjðt� tÞ; i ¼ 1; 2; . . . ;N. ð16Þ

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530518

By using Kronecker product, one can rewrite (16) as

_ZðtÞ ¼ ðIN �Df ðsðtÞÞ þ G � AÞZðt� tÞ, (17)

where ZðtÞ ¼ ðZ1ðtÞ; . . . ; ZnðtÞÞTis the error variable between X ðtÞ and SðtÞ. From Lemma 1 there is a non-

singular matrix F 2 CN�N such that

G ¼ FJF�1. (18)

Let

ZðtÞ ¼ ðF� InÞyðtÞ, (19)

then

_ZðtÞ ¼ ðF� InÞ_yðtÞ. (20)

Substituting (18), (19), and (20) into (17), by some algebra, one has

_yðtÞ ¼ ðIN �Df ðsðtÞÞ þ J � AÞyðt� tÞ, (21)

where yðtÞ ¼ ðy1ðtÞ; . . . ; yNðtÞÞT2 CnN , yiðtÞ 2 Cn, i ¼ 1; 2; . . . ;N. From the Lyapunov linearization method for

non-autonomous system [30], the synchronization state SðtÞ of network (1) is exponentially stable if system(21) is exponentially stable about its zero solution. Considering the special form of J given in (5) and (6),system (21) can be easily transformed to (14) equivalently. The proof is thus completed. &

Remark 1. If the Jordan matrix J of the coupling configuration matrix G happen to be diagonal, and all theeigenvalues of G taken to be real, it is a special case that has been discussed in Refs. [13,14].

Theorem 2. Suppose f ðxðtÞÞ is continuously differentiable at sðtÞ 2 Rn, and the coupling configuration matrix G

has k different eigenvalues, then the synchronization state SðtÞ of the delayed network (1) is exponentially stable if

the following k � 1 individual time-varying systems are exponentially stable about their zero solutions:

_wðtÞ ¼ Df ðsðtÞÞwðtÞ þ liAwðt� tÞ; i ¼ 2; . . . ; k, (22)

where 2pkpN, wðtÞ 2 Cn, and li 2 sðGÞ.

Proof. We will show that the exponential stability of the zero solution of the ith ni-node time-varying systemin (14) is equivalent to that of the corresponding ith individual time-varying system in (22). Suppose ni ¼ 2,then the ith two-node system in (14) can be described by the following two equations with W ¼ ðw1;w2Þ

T:

_w1ðtÞ ¼ Df ðsðtÞÞw1ðtÞ þ liAw1ðt� tÞ þ Aw2ðt� tÞ, (23)

_w2ðtÞ ¼ Df ðsðtÞÞw2ðtÞ þ liAw2ðt� tÞ. (24)

Firstly, it is obvious that, if the zero solution of the time-varying system (24) is exponentially stable, then thezero solution of the ith individual time-varying system in (22) is exponentially stable. Secondly, if the zerosolution of the ith equation in (22) is exponentially stable, then the zero solution of (24) is exponentially stable,in turn it follows that the zero solution of (23) is exponentially stable. The case for ni42 can be proved thesame way and omitted here. From Theorem 1, the proof is thus completed. &

Remark 2. By Theorem 2, to estimate whether it can realize synchronization exponentially in the delayeddynamical network (1), one only need to find out all the different eigenvalues of the coupling configurationmatrix G no matter whether it is symmetric or diagonalizable, and estimate the exponential stability of thek � 1 individual time-varying systems in (22). So, Theorem 2 is a generalization of the results in Refs. [13,14].

3.2. Delay-independent synchronization analysis

By defining a Lyapunov–Krasovskii function and using Lemma 2, a sufficient condition ensuring theexponential stability of the synchronization state SðtÞ of the delayed network (1) is obtained.

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 519

Theorem 3. Suppose f ðxðtÞÞ is continuously differentiable at sðtÞ 2 Rn. If there exist two n� n positive definite

matrix, P;Q � 0, such that

PDf ðsðtÞÞ þDf ðsðtÞÞTPþQþ r2ðGÞPAQ�1ATP%� eIn, (25)

where rðGÞ is the spectral radius of G, e is a positive number, then the synchronization state SðtÞ of the delayed

network (1) is delay-independent exponentially stable.

Proof. Define a Lyapunov–Krasovskii function as follows:

V ðtÞ ¼ w�ðtÞPwðtÞ þ

Z t

t�tw�ðaÞQwðaÞda, (26)

where wðtÞ 2 Cn. By using Lemmas 2 and 4, the derivative of V ðtÞ along the trajectory of the ith individualtime-varying system in (22) is

_V ðtÞ ¼ w�ðtÞðPDf ðsðtÞÞ þDf ðsðtÞÞTPÞwðtÞ þ w�ðtÞPliAwðt� tÞ þ wðt� tÞ�l�i ATPwðtÞ

þ w�ðtÞQwðtÞ � w�ðt� tÞQwðt� tÞ

pw�ðtÞðPDf ðsðtÞÞ þDf ðsðtÞÞTPþQþ jlij2PAQ�1ATPÞwðtÞ

p� ekwðtÞk2,

p�e

lPmax þ tlQmaxV ðtÞ; i ¼ 2; . . . ; k. ð27Þ

From the Lyapunov’s theorem for local stability [30] and Theorem 2, the proof is thus completed. &

In particular, if P ¼ Q ¼ In, Theorem 3 reduced to the following simple and useful corollary.

Corollary 1. For the delayed dynamical network (1), if

Df ðsðtÞÞ þDf ðsðtÞÞT þ r2ðGÞAAT%� ðeþ 1ÞIn, (28)

where e is a positive number, then the synchronization state SðtÞ is delay-independent exponentially stable. From

Lemma 4, sufficient condition (28) can be further written as

Df ðsðtÞÞS þ1

2r2ðGÞrðAATÞ%�

ðeþ 1Þ

2In.

Remark 3. Corollary 1 means that smaller spectral radius of G and AAT easily implies delay-independentsynchronization.

3.3. Delay-dependent synchronization analysis

Compared to the delay-independent networks, there are networks which can realize synchronization onlywhen the time-delays less than a threshold.

Theorem 4. Suppose f ðxðtÞÞ is continuously differentiable at sðtÞ 2 Rn, P;Q 2MnðCÞ are positive definite

matrices. If there exist a positive constant b, such that

h2lPmax

lPminr2ðGÞl

ðPAÞ�Q�1PAmaxðlDf ðsðtÞÞTDf ðsðtÞÞmax þ r2ðGÞlATAmaxÞ þlQmax

2ob, (29)

and

ðPDf ðsðtÞÞ þ liPAÞH%� bIn; i ¼ 2; . . . ; k, (30)

where li 2 sðGÞ, then the synchronization state of the delayed network (1) is delay-dependent exponentially stable

for any fixed time-delay t 2 ½0; h.

Proof. Define a Lyapunov function as

ViðtÞ ¼ w�i ðtÞPwiðtÞ; i ¼ 2; . . . ; k. (31)

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530520

Rewrite the ith equation of (22) as

_wiðtÞ ¼ ½Df ðsðtÞÞ þ liAwiðtÞ � liA

Z t

t�t_wiðaÞda; i ¼ 2; . . . ; k. (32)

Thus, by using Lemmas 2 and 4, the derivative of ViðtÞ along trajectory of (32) is

_ViðtÞ ¼ w�i ðtÞ ðDf ðsðtÞÞ þ liAÞ�Pþ PðDf ðsðtÞÞ þ liAÞð ÞwiðtÞ

� PliA

Z t

t�t_wiðaÞda

� ��wiðtÞ � w�i ðtÞ PliA

Z t

t�t_wiðaÞda

� �p2w�i ðtÞðPðDf ðsðtÞÞ þ liAÞÞ

HwiðtÞ

þ liPA

Z t

t�t_wiðaÞda

� ��Q�1 liPA

Z t

t�t_wiðaÞda

� �þ w�i ðtÞQwiðtÞ

pð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ jlij

2

Z t

t�t_wiðaÞda

� ��ðPAÞ�Q�1PA

Z t

t�t_wiðaÞda

� �

pð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ lij j

2lðPAÞ�Q�1PAmax

Z t

t�t_wiðaÞda

� �� Z t

t�t_wiðaÞda

� �

¼ ð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ jlij

2lðPAÞ�Q�1PAmax

Z t

t�t

Z t

t�tð _wiðaÞÞ

� _wiðgÞda dg� �

pð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ jlij

2lðPAÞ�Q�1PAmaxt

2 supt�tpapt

ðð _wiðaÞÞ� _wiðaÞÞ

pð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ r2ðGÞl

ðPAÞ�Q�1PAmaxh2

� supt�tpapt

ððDf ðsðaÞÞwðaÞ þ liAwða� tÞÞ�ðDf ðsðaÞÞwðaÞ þ liAwða� tÞÞÞ

pð�2bþ lQmaxÞw�i ðtÞwiðtÞ þ r2ðGÞl

ðPAÞ�Q�1PAmaxh2

�ð2lDf ðsðtÞÞTDf ðsðtÞÞmax þ 2r2ðGÞlATAmaxÞ sup

t�2tpapt

ðw�i ðaÞwiðaÞÞ

p1

lPmaxð�2bþ lQmaxÞViðtÞ

þ2h2

lPminr2ðGÞl

ðPAÞ�Q�1PAmaxðlDf ðsðtÞÞTDf ðsðtÞÞmax þ r2ðGÞlATAmaxÞ supt�2tpapt

ðViðaÞÞ; i ¼ 2; . . . ; k.

From Lemma 3, ViðtÞ ði ¼ 2; . . . ; kÞ are exponentially approach to zero, therefore, the k � 1 individual time-varying systems in (22) are exponentially stable about their zero solutions. From Theorem 2, the proof iscompleted. &

In the following of this paper, sometimes we denote the Jacobian matrix of f ðxðtÞÞ at sðtÞ as

Df ðsðtÞÞ ¼ ðsmlðtÞÞn�n

where smlðtÞ are continuous or piecewise continuous functions.

Theorem 5. Suppose f ðxðtÞÞ is continuously differentiable at sðtÞ 2 Rn. If there exist a positive matrix

P ¼ ðpilÞðk�1Þ�n, and a positive vector b ¼ ðb2; b3; . . . ;bkÞT, such that

pilðsllðtÞ þ lRi allÞ þ

Xn

m¼1mal

pimðsmlðtÞ þ lRi amlÞ

��������������

þXn

m¼1

pim lIi aml

�� ��þ hðjlRi j þ jl

Ii jÞXn

m¼1

pimjaml jXn

m¼1

jslmðtÞjp� bi; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n, ð33Þ

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 521

and

hpimaxðjlRi j þ jl

Ii jÞ

2

pimin

Xn

m¼1

jalmjXn

m¼1

pimjaml jobi; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n, (34)

where pimin ¼ minfpi1; . . . ; ping, pimax ¼ maxfpi1; . . . ; ping, and li 2 sðGÞ, then the synchronization state SðtÞ of

the delayed dynamical network (1) is delay-dependent exponentially stable for any fixed time-delay t 2 ½0; h.

Proof. For i ¼ 2; . . . ; k, from (32) one has

_wilðtÞ ¼Xn

m¼1

ðslmðtÞ þ lialmÞwimðtÞ � li

Xn

m¼1

Z t

t�talm _wimðaÞda; l ¼ 1; . . . ; n. (35)

Since wilðtÞ are complex variables, Eq. (35) can be equivalently written as (36) and (37)

_wRil ðtÞ ¼

Xn

m¼1

ðslmðtÞ þ lRi almÞw

RimðtÞ �

Xn

m¼1

lIi almwI

imðtÞ

� lRi

Xn

m¼1

Z t

t�talmwR

imðaÞdaþ lIi

Xn

m¼1

Z t

t�talmwI

imðaÞda; l ¼ 1; . . . ; n, ð36Þ

_wIilðtÞ ¼

Xn

m¼1

ðslmðtÞ þ lRi almÞw

IimðtÞ þ

Xn

m¼1

lIi almwR

imðtÞ

� lRi

Xn

m¼1

Z t

t�talmwI

imðaÞda� lIi

Xn

m¼1

Z t

t�talmwR

imðaÞda; l ¼ 1; . . . ; n. ð37Þ

Similarly, from (22) one has

_wRil ðtÞ ¼

Xn

m¼1

slmðtÞwRimðtÞ þ lR

i

Xn

m¼1

almwRimðt� tÞ

� lIi

Xn

m¼1

almwIimðt� tÞ; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n, ð38Þ

_wIilðtÞ ¼

Xn

m¼1

slmðtÞwIimðtÞ þ lR

i

Xn

m¼1

almwIimðt� tÞ

þ lIi

Xn

m¼1

almwRimðt� tÞ; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n. ð39Þ

From (38) and (39), it is easy to obtain

j _wRil ðtÞ � _wI

ilðtÞjpXn

m¼1

slmðtÞðwRimðtÞ � wI

imðtÞÞ

����������þ jlR

i jXn

m¼1

almj j supt�tpapt

jwRimðaÞ � wI

imðaÞj

þ jlIi jXn

m¼1

jalmj supt�tpapt

jwRimðaÞ � wI

imðaÞj

pXn

m¼1

jslmðtÞjðjwRimðtÞj þ jw

IimðtÞjÞ

þ ðjlRi j þ jl

Ii jÞXn

m¼1

jalmj supt�tpapt

ðjwRimðaÞj þ jw

IimðaÞjÞ. ð40Þ

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530522

Define Lyapunov functions as

ViðtÞ ¼Xn

l¼1

pilðjwRil ðtÞj þ jw

IilðtÞjÞ; i ¼ 2; . . . ; k, (41)

then from Definition 4 and considering (36), (37), (40), and (41), the derivative of V iðtÞ along the trajectoryof (35) is

dV iðtÞ

dt¼Xn

l¼1

pil

dðjwRil ðtÞj þ jw

IilðtÞjÞ

dt

¼Xn

l¼1

pilkðwRil ðtÞ; _w

Ril ðtÞÞ _w

Ril ðtÞ þ

Xn

l¼1

pilkðwIilðtÞ; _w

IilðtÞÞ _w

IilðtÞ

pXn

l¼1

pilðsllðtÞ þ lRi allÞjw

Ril ðtÞj þ

Xn

m¼1mal

pimðsmlðtÞ þ lRi amlÞ

��������������jw

Ril ðtÞj

8><>:

�Xn

m¼1

pimjlIi amlw

IilðtÞj þ

Xn

m¼1

pimjlIi amlw

Ril ðtÞj

þ pilðsllðtÞ þ lRi allÞjw

IilðtÞj þ

Xn

m¼1mal

pimðsmlðtÞ þ lRi amlÞ

��������������jw

IilðtÞj

�tjlRi jXn

m¼1

pimjaml j supt�tpapt

j _wRil ðaÞ � _wI

ilðaÞj

þ tjlIi jXn

m¼1

pimjaml j supt�tpapt

j _wRil ðaÞ � _wI

ilðaÞj

9>=>;

p�bi

pimax

V iðtÞ þhðjlR

i j þ jlIi jÞ

2

pimin

Xn

m¼1

jalmjXn

m¼1

pimjaml j supt�2tpapt

V iðaÞ; i ¼ 2; . . . ; k.

From Lemma 3, ViðtÞ ! 0 exponentially, thus wilðtÞ ! 0 exponentially for all i ¼ 2; . . . ; k and l ¼ 1; . . . ; n.Then from Theorem 2, the proof is completed. &

Remark 4. Obviously, it is easy to obtain an allowable bound of time-delay for synchronization in term ofexplicit expression from Theorems 4 or 5.

Theorem 5 is somewhat complicated in practical use; one can obtain the following simpler corollary whichcan be used in some special cases.

Corollary 2. If there exist nþ 1 positive numbers p1; p2; . . . ; pn and b, such that

plðsllðtÞ þ lRi allÞ þ

Xn

m¼1mal

pmðsmlðtÞ þ lRi amlÞ

��������������

þXn

m¼1

pmjlIi aml j þ hðjlR

i j þ jlIi jÞXn

m¼1

pmjaml jXn

m¼1

jslmðtÞjp� b; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n, ð42Þ

and

hpmaxðjlRi j þ jl

Ii jÞ

2

pmin

Xn

m¼1

jalmjXn

m¼1

pmjaml job; i ¼ 2; . . . ; k; l ¼ 1; . . . ; n, (43)

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 523

where pmin ¼ minfp1; . . . ; png, pmax ¼ maxfp1; . . . ; png, and li 2 sðGÞ, then the synchronization state SðtÞ of the

delayed network (1) is delay-dependent exponentially stable for any fixed time-delay t 2 ½0; h.

Corollary 2 is really a special case of Theorem 5, where each row of the matrix P are all equal to ðp1; . . . ; pnÞ,and b2 ¼ b3 ¼ ¼ bk ¼ b.

In particular, if G has only real eigenvalues (negative numbers except forl1 ¼ 0Þ; one can obtain thefollowing useful corollary from Corollary 2.

Corollary 3. If li 2 sðGÞ � R, and there exist n positive numbers p1; p2; . . . ; pn, such that

ho mini¼2;...;kl¼1;...;n

Hðli; lÞ, (44)

where

Hðli; lÞ ¼plðsllðtÞ þ liallÞ þ j

Pnm¼1mal

pmðsmlðtÞ þ liamlÞj

li

Pnm¼1pmjaml j

Pnm¼1jslmðtÞj � ðpmaxl

2i =pminÞ

Pnm¼1jalmj

Pnm¼1pmjaml j

, (45)

then the synchronization state SðtÞ of the delayed network (1) is delay-dependent exponentially stable for any

fixed time-delay t 2 ½0; h.

Definition 5. The synchronizability of the delayed dynamical network (1) is said to be determined by lp ifHðlp; lÞ ¼ mini¼2;...;k

l¼1;...;nHðli; lÞ. The definition in case of which the coupling configuration matrix has complex

eigenvalues can be given similarly.

Remark 5. It can be seen clearly from Theorem 5 and all its corollaries that, unlike the non-delayed networkswith symmetric coupling, whose synchronizability is characterized by the second-largest eigenvalue of itscoupling configuration matrix [7–11], for a delayed network with nonsymmetric or symmetric coupling, itssynchronizability is not always characterized by the second-largest eigenvalue of its coupling configurationmatrix even though all the eigenvalues are real.

4. Synchronizability analysis of unidirectional coupled networks with different structures

In this section, we analyze the effect of time-delay on the synchronizability of delayed networks withunidirectional coupling under different topological structures by using the theoretical results above.Simulations are also done to verify the results by using delayed networks in which each node is a two-dimensional oscillator system consisting of a two-cell cellular neural network. The state equations of the N

−2 −1 0 1 2−1.5

−1

−0.5

0

0.5

1

1.5

x (1)

x (

2)

Fig. 1. Limit cycle in the two-cell CNN with a11 ¼ a22 ¼ 1:2, a12 ¼ 0:5, a21 ¼ �0:5.

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530524

nodes are described by (46) and (47) [31,32], where the parameters are a11 ¼ a22 ¼ 1:2, a12 ¼ 0:5, anda21 ¼ �0:5. Fig. 1 shows that there is a limit cycle in the state space of the two-cell cellular neural network(CNN):

_xi1 ¼ �xi1 þ a11yi1 þ a12yi2;

_xi2 ¼ �xi2 þ a21yi1 þ a22yi2;

(i ¼ 1; . . . ;N, (46)

yilðxilÞ ¼12ðjxil þ 1j � jxil þ 1jÞ; l ¼ 1; 2. (47)

Obviously, the Jacobian matrix of f ðxðtÞÞ at sðtÞ in this system satisfy

Df ðsðtÞÞ ¼s11 ¼ �1 or 0:2 s12 ¼ 0:5

s21 ¼ �0:5 s22 ¼ �1 or 0:2

" #.

As analyzed above, the synchronizability of the delayed network (1) has relation with the inner-couplingmatrix A. In order to compare the synchronizability of delayed networks with different structures, we alwaysassume A ¼ In, p1 ¼ p2 ¼ ¼ pn ¼ 1 in Corollaries 2 and 3, and all the coupling strength from node j tonode i are 1 (iajÞ in this section. All simulations are done using networks of 50 nodes with different structures.

4.1. Global unidirectional coupling

The coupling configuration matrix of the globally unidirectional coupled network with N nodes is

Ggu ¼

�ðN � 1Þ 1 1

0 �ðN � 2Þ 1 1

..

.0 . .

. . .. ..

.

..

. ... . .

.�1 1

0 0 0

266666664

377777775, (48)

which has only real eigenvalues: l1 ¼ 0, l2 ¼ �1, l3 ¼ �2; . . . ; lN ¼ �ðN � 1Þ. From Corollary 3, one canobtain a sufficient condition for synchronization as follows:

tpho minl¼1;...;ni¼2;...;N

H1ðli; lÞ ¼ðsllðtÞ þ liÞ þ j

Pnm¼1mal

smlðtÞj

li

Pnm¼1jslmðtÞj � l2i

8<:

9=;. (49)

Let H 01ðl; lÞ denotes the derivative of H1ðl; lÞ with respect to the variable l, then, we have the following results:

(i)

The synchronizability of the delayed network with global unidirectional coupling is determined by thesecond-largest eigenvalue l2 if H 0ðl; lÞp0 for all l ¼ 1; . . . ; n in interval l 2 ½lN ; l2. In other words, thedelayed network will realize synchronization if

tpho minl¼1;...;n

fH1ð�1; lÞg. (50)

(ii)

The synchronizability of the delayed network with global unidirectional coupling is determined by thesmallest eigenvalue lN if H 0ðl; lÞX0 for all l ¼ 1; . . . ; n in interval l 2 ½lN ; l2. That is, the delayednetwork will realize synchronization if

tpho minl¼1;...;n

fH1ð�ðN � 1Þ; lÞg. (51)

(iii)

Or else, the synchronizability of the delayed network with global unidirectional coupling is determined byeigenvalue li, 2pipN, �ðN � 1Þplip� 1.

ARTICLE IN PRESS

0 5 10 15 20 25 30−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

s

25 26 27 28 29 30−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

sFig. 2. Time responses of the state variables in the delayed network with global unidirectional coupling in time interval ½0 30: (a)

t ¼ 0:0194; (b) t ¼ 0:0320.

J. Wu, L. Jiao / Physica A 386 (2007) 513–530 525

For example, if N ¼ 6, the synchronizability of the delayed network with global unidirectional couplingconsisting of the two-cell CNNs is determined by l2 ¼ �1, it will synchronize when to0:12 from (49); whereasN ¼ 50, it is determined by l50 ¼ �49, and it will synchronize when to0:0195 from (49). The simulationresults are shown in Fig. 2, obviously, it is a delay-dependent network.

For comparing, the coupling configuration matrix of the delayed network with global symmetric coupling is

Ggs ¼

�ðN � 1Þ 1 1

1 �ðN � 1Þ . .. ..

.

..

. . .. . .

.1

1 1 �ðN � 1Þ

26666664

37777775, (52)

which has only two different eigenvalues, l1 ¼ 0 and l2 ¼ �N with multiplicity N � 1. From Corollary 3, asufficient condition for synchronization is easily obtained as follows:

tpho minl¼1;...;n

fH1ð�N; lÞg. (53)

For the delayed network consisting of 50 two-cell CNNs with global symmetric coupling, it will synchronizewhen to0:0191 from (53). Simulation results are shown in Fig. 3.

Remark 6. Comparing (51) and (53), one can see clearly that the synchronizability of the delayed networkwith global unidirectional coupling, if it is determined by the smallest eigenvalue lN , will approach to that ofthe delayed network with global symmetric coupling as the number of nodes increasing.

Remark 7. As an illustrating example of Remark 5, it can be seen clearly that the synchronizability of adelayed network is not always characterized by the second-largest eigenvalue l2 of its coupling configurationmatrix.

4.2. Nearest-neighbor unidirectional coupling

Secondly, consider a nearest-neighbor coupled network consisting of N nodes arranged in a ringand each node only connected to its neighbor nodes with unidirectional coupling. The coupling configuration

ARTICLE IN PRESS

0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

s

19 19.2 19.4 19.6 19.8 20−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

sFig. 3. Time responses of the state variables in the delayed network with global symmetric coupling: (a) t ¼ 0:0190 in time interval ½0 5;

(b) t ¼ 0:0310 in time interval ½0 20.

J. Wu, L. Jiao / Physica A 386 (2007) 513–530526

matrix is

Gnu ¼

�1 1 0 0

0 �1 1 . .. ..

.

..

. . .. . .

. . ..

0

0 0 �1 1

1 0 0 �1

266666664

377777775, (54)

which has eigenvalues:

liþ1 ¼ e2ipj=N � 1; i ¼ 0; . . . ;N � 1, (55)

where j ¼ffiffiffiffiffiffiffi�1p

. From Corollary 2, a sufficient condition for synchronization in the nearest-neighborunidirectional coupled network is then obtained

to mini¼1;...;N�1

l¼1;...;n

�ðsllðtÞ þ cosð2ip=NÞ � 1þ jPn

m¼1malðsmlðtÞÞj þ j sinð2ip=NÞjÞ

ðj cosð2ip=NÞ � 1j þ j sinð2ip=NÞjÞPn

m¼1jslmðtÞj þ ðj cosð2ip=NÞ � 1j þ j sinð2ip=NÞjÞ2. (56)

For the nearest-neighbor unidirectional coupled network consisting of 50 two-cell CNNs, we cannot find asolution from (56).

The coupling configuration matrix of a nearest-neighbor coupled network with symmetric coupling is asfollows:

Gns ¼

�2 1 0 0 1

1 �2 1 0 . ..

0

0 1 . .. . .

. . .. ..

.

..

. . .. . .

. . .. . .

.0

0 0 1 �2 1

1 0 0 1 �2

266666666664

377777777775, (57)

ARTICLE IN PRESS

100 150 200−3

−2

−1

0

1

2

3

time

sta

te v

aria

ble

s

199 199.2 199.4 199.6 199.8 200−3

−2

−1

0

1

2

3

time

sta

te v

aria

ble

sFig. 4. Time responses of the state variables in the delayed network with nearest-neighbor symmetric coupling in time interval ½0200: (a)

t ¼ 0:0155; (b) t ¼ 0:4000.

J. Wu, L. Jiao / Physica A 386 (2007) 513–530 527

which has only real eigenvalues:

li ¼ �4 sin2 ði � 1Þp

N

� �; i ¼ 1; . . . ;N. (58)

From Corollary 3, a sufficient condition ensuring synchronization is easily obtained as follows:

tpho mini¼2;...;Nl¼1;...;n

H1 �4 sin2 ði � 1Þp

N

� �; l

� �� �. (59)

From (59) with N ¼ 50, and by some algebra, one can obtain that the synchronizability of thedelayed network with nearest-neighbor symmetric coupling consisting of the two-cell CNNs isdetermined by l8 ¼ �0:7252, and it will synchronize when to0:0156. Simulation results are shown inFig. 4.

4.3. Star unidirectional coupling

At last, in a delayed network with star unidirectional coupling, all nodes are connected to one center node,and the coupling configuration matrix is

Gsu ¼

�1 0 0 1

0 �1 0 1

..

. . .. . .

. . .. ..

.

�1 1

0 0 0

26666664

37777775, (60)

which has only two different eigenvalues also, l1 ¼ 0 and l2 ¼ �1 with multiplicityN � 1. Similarly, asufficient condition for synchronization is obtained the same as (50). One can get that the delayed networkconsisting of the two-cell CNNs with star unidirectional coupling will realize synchronization if to0:12.Simulation results are shown in Fig. 5.

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530528

Whereas the coupling configuration matrix of a delayed network with star symmetric coupling is

Gss ¼

�ðN � 1Þ 1 1 1

1 �1 0 0

1 0 . .. . .

. ...

..

. ... . .

.�1 0

1 0 0 �1

266666664

377777775, (61)

0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

s

90 92 94 96 98 100−3

−2

−1

0

1

2

3

time

sta

te v

ariable

s

Fig. 5. Time responses of the state variables in the delayed network with star unidirectional coupling: (a) t ¼ 0:1199 in time interval ½0 20;

(b) t ¼ 1:5 in time interval ½0 100.

0 5 10 15 20−4

−3

−2

−1

0

1

2

3

4

time

sta

te v

ariable

s

18 18.5 19 19.5 20−6

−4

−2

0

2

4

6

time

sta

te v

ariable

s

Fig. 6. Time responses of the state variables in the delayed network with star symmetric coupling in time interval [0 20]: (a) t ¼ 0:0190; (b)t ¼ 0:0315.

ARTICLE IN PRESSJ. Wu, L. Jiao / Physica A 386 (2007) 513–530 529

which has three different real eigenvalues, l1 ¼ 0; l2 ¼ �1 with multiplicity N � 2, and l3 ¼ �N. Then, it iseasy to get the following interesting conclusions from Corollary 3:

(i)

If the synchronizability of the delayed network with star symmetric coupling is determined by the second-largest eigenvalue l2 ¼ �1, sufficient condition for synchronization is the same as that with starunidirectional coupling given in (50), which is also a sufficient condition for synchronization in the delayednetwork with global unidirectional coupling when its synchronizability is determined by the second-largesteigenvalue.

(ii)

Else if the synchronizability of the delayed network with star symmetric coupling is determined by thesmallest eigenvalue l3 ¼ �N, sufficient condition for synchronization is the same as that with globalsymmetric coupling given in (53).

For the network consisting of 50 two-cell CNNs with star symmetric coupling, its synchronizability isdetermined by l3 ¼ �50 and it will synchronize when to0:0191 from (53). Simulation results are shown inFig. 6.

Simulations show that all the above networks consisting of the two-cell CNNs are delay-dependent, that is,they can realize synchronization only when the time-delay less than a threshold, or else, they will losesynchronization.

5. Conclusion

We investigate synchronization phenomena based on a new general delayed complex dynamical networkmodel with nonsymmetric coupling. The matrix canonical formalization method is used instead the matrixdiagonalization method, several criteria for both delay-independent and delay-dependent synchronization innetworks with nonsymmetric coupling are derived, which generalize some previous results. Analysis indicatesthat the synchronizability of a delayed network cannot be characterized by the second-largest eigenvalue ofthe coupling configuration matrix even though it has only real eigenvalues. The effects of time-delay on thesynchronizability of unidirectional coupled networks with several typical topological structures are analyzed,and the results are verified by simulations on networks in which each node is a two-dimensional limit cycleoscillator system consisting of a two-cell cellular neural network.

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