ARTICLE IN PRESS

0925-2312/$ - se

doi:10.1016/j.ne

$This work

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Neurocomputing 70 (2007) 2477–2485

www.elsevier.com/locate/neucom

Exponential synchronization of stochastic perturbed chaoticdelayed neural networks$

Yonghui Suna, Jinde Caoa,b,�, Zidong Wangb

aDepartment of Mathematics, Southeast University, Nanjing 210096, ChinabDepartment of Information Systems and Computing, Brunel University, Uxbridge, Middlesex, UB8 3PH, UK

Received 9 January 2006; received in revised form 5 September 2006; accepted 10 September 2006

Communicated by S. Arik

Available online 25 October 2006

Abstract

In this paper, we deal with the exponential synchronization problem for a class of stochastic perturbed chaotic delayed neural

networks. Based on the Lyapunov stability theory, by virtue of stochastic analysis, Halanay inequality for stochastic differential

equations, drive-response concept and time-delay feedback control techniques, several sufficient conditions are proposed to guarantee

the exponential synchronization of two identical chaotic delayed neural networks with stochastic perturbation. These conditions, which

are expressed in terms of linear matrix inequalities, rely on the connection matrix in the drive networks as well as the suitable designed

feedback gains in the response networks. Finally, a numerical example with its simulations are provided to illustrate the effectiveness of

the presented synchronization scheme.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Exponential synchronization; Stochastic perturbation; Mean square exponential stability; Halanay inequality; Chaotic delayed neural

networks; Time-delay feedback control

1. Introduction

In the past decade, since the drive-response concept forconstructing synchronization of coupled chaotic systemswas proposed in [27], the control and synchronizationproblems of chaotic systems have been intensively inves-tigated due to their potential applications in various fieldsincluding general complex dynamical networks [24], securecommunication [35], etc. Several different approaches havebeen proposed for the synchronization of chaotic systems,such as linear and nonlinear feedback control [32,19,18,17,6,10,12,11,36], adaptive control [23,7], and so on.Recently, it has been revealed that, if the network’sparameters and time delays are appropriately chosen, thedelayed neural networks (DNNs), such as delayed Hopfield

e front matter r 2006 Elsevier B.V. All rights reserved.

ucom.2006.09.006

was supported in part by the National Natural Science

China under Grant 60574043, the Royal Society of the

m, and the Natural Science Foundation of Jiangsu

ina under Grant BK2006093.

ing author. Tel. +86 25 83792315; Fax:+ 86 25 83792316.

esses: jdcao@seu.edu.cn, jdcaoseu@gmail.com (J. Cao).

neural network and delayed cellular neural network, canexhibit some complicated dynamics and even chaoticbehaviors [14,22,9], in addition to the stability and periodicoscillations considered previously in [15,4,5,8,34]. Hence,the problem of synchronization of chaotic DNNs hasreceived extensive consideration.On the other hand, it is worth pointing out that, like time

delays and parameter uncertainties, noises are ubiquitousin both nature and man-made systems, and the stochasticeffects on neural networks have drawn particular attention.Some results concerning stochastic neural networks havealready been presented. In [20], Liao and Mao studied themean square exponential stability and instability ofcellular neural networks. In [2], the authors discussed thealmost sure exponential stability for a class of stochasticcellular neural networks with discrete delays by using thenonnegative semimartingale convergence theorem. In [30],mean square exponential stability of stochastic delayedHopfield neural network was investigated.Generally, a chaotic system is a nonlinear deterministic

or clean system that possesses complex and unpredictable

ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–24852478

behavior. However, there are some experimental andnumerical results showing that noises can affect thesynchronization between chaotic systems. In [16], it wasreported that the external noise added to two uncoupledchaotic circuits does induce synchronization. In [28], theauthors studied the synchronization problem experimen-tally, and concluded that the noise-induced synchroniza-tion can be induced by a nonzero mean of the signaland not by its stochastic character. In [31], the internalnoise-enhanced phase synchronization of coupled chemicalchaotic oscillators was investigated numerically. Itwas found in [31] that the internal noise can enhance thephase synchronization, and there exists an optimal internalnoise level such that the best phase synchronization isachieved. In [33], the stochastic response phenomenonwas analyzed, and it was claimed that the stochasticresponse enhanced the general interest in noise-inducedphenomena. It was also outlined via some closely relatedexamples that noises play an constructive role in nonlinearsystems.

Since noises can induce stability and instability oscilla-tion to the systems, by virtue of the stability theory forstochastic differential equations and the works mentionedabove, there has been an increasing interest in the study ofsynchronization of chaotic systems with stochastic pertur-bations. Some initial theoretical results have just appeared.For example, in [29], it was shown that two identical fullyconnected chaotic neural networks can achieve a stochasticsynchronization state when linked with a sufficiently largecommon noise. In [21], the complete synchronizationproblem between unidirectionally coupled Chua’s circuitswithin stochastic perturbation was discussed. However, tothe best of our knowledge, so far, the problem ofexponential synchronization of DNNs with stochasticperturbation is still remaining as a challenging openproblem.

Motivated by the above discussion, our main aim in thispaper is to shorten this gap by investigating the exponentialsynchronization problem for chaotic DNNs with stochasticperturbation based on Lyapunov stability theory. By virtueof the Halanay inequality for stochastic differentialequations and the drive-response concept, a time-delayfeedback controller is designed to ensure the exponentialsynchronization of chaotic DNNs with stochastic pertur-bation, where the derived criteria are expressed in terms oflinear matrix inequalities (LMIs).

The rest of the paper is organized as follows. InSection 2, the chaotic DNN model with stochasticperturbation considered in this paper is presented, andthe exponential synchronization problem of the drive-response chaotic DNNs is then defined. Our main resultsand the realization of exponential synchronization aredescribed in Section 3. In Section 4, an illustrated exampleand its simulation are provided to demonstrate theeffectiveness of the derived criteria. Finally, in Section 5,the paper is concluded with a discussion on futureinvestigation directions.

2. Notations and preliminaries

Throughout this paper, the following notations will beused: for any symmetric matrix A, A40 means A is apositive definite matrix; k � k denotes a vector or a matrixnorm and kxk2 is used to denote a vector norm defined bykxk2 ¼

Pni¼1 jxij

2; Ef�g stands for the mathematical ex-pectation operator; T represents the transpose of thematrix; I is an identical matrix; DþyðtÞ denotes the Dini-derivative; _oðtÞ is used to denote the Gaussian white noise.In this paper, we consider the following chaotic DNN

model:

_xiðtÞ ¼ �cixiðtÞ þXn

j¼1

aijf jðxjðtÞÞ þXn

j¼1

bijf jðxjðt� tÞÞ,

i ¼ 1; 2; . . . ; n, ð1Þ

or, in a compact form,

_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxtðtÞÞ, (2)

where xðtÞ ¼ ðx1ðtÞ;x2ðtÞ; . . . ; xnðtÞÞT2 Rn is the state vector

associated with the neurons; C ¼ diagðc1; c2; . . . ; cnÞ40represents the rate with which the ith unit will reset itspotential to the resting state in isolation when disconnectedfrom the network and the external inputs; A ¼ ðaijÞn�n andB ¼ ðbijÞn�n represent the connection weight matrix, andthe delayed connection weight matrix, respectively; f and g

are activation functions, f ðxðtÞÞ ¼ ðf 1ðx1ðtÞÞ; f 2ðx2ðtÞÞ; . . . ;f nðxnðtÞÞÞ

T2 Rn, f ðxtðtÞÞ ¼ ðf 1ðx1ðt� tÞÞ; f 2ðx2ðt� tÞÞ; . . . ;

f nðxnðt� tÞÞÞT 2 Rn, where t40 is the transmission delay.

Remark 1. Comparing to the existing literatures such as[15,4,5,8], we have dropped out the external input term I

from the model (2) for the sake of simplicity. Note thatsuch a dropout does not cause any loss of generality in thesense of stability and synchronization analysis.

Based on the drive-response concept for synchronizationof coupled chaotic systems, which was initially proposed byPecora and Carroll in [27], the corresponding responsesystem of (2) is given in the following form:

_yiðtÞ ¼ � ciyiðtÞ þXn

j¼1

aijf jðyjðtÞÞ þXn

j¼1

bijf jðyjðt� tÞÞ

þXn

j¼1

sijðt; zjðtÞ; zjðt� tÞÞ _ojðtÞ þ ui; i ¼ 1; 2; . . . ; n,

ð3Þ

or

_yðtÞ ¼ � CyðtÞ þ Af ðyðtÞÞ þ Bf ðytðtÞÞ

þ sðt; zðtÞ; ztðtÞÞ _oðtÞ þU , ð4Þ

where U ¼ ½u1; u2; . . . ; un�T is the state feedback controller

given to achieve the exponential synchronization between

ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–2485 2479

drive-response system, which is of the form,

U ¼ K

z1ðtÞ

..

.

znðtÞ

2664

3775þM

z1ðt� tÞ

..

.

znðt� tÞ

2664

3775 ¼ KzðtÞ þMztðtÞ. (5)

Here, K and M are feedback gain parameters to bescheduled. Moreover oðtÞ ¼ ðo1ðtÞ;o2ðtÞ; . . . ;onðtÞÞ

T is an-dimensional Brown motion defined on a completeprobability space ðO;F;PÞ with a natural filtrationfFgtX0 ði.e. Ft ¼ sfoðsÞ : 0psptgÞ, and s : Rþ � Rn �

Rn ! Rn�n, s ¼ ðsijÞn�n is the diffusion coefficient matrix.

Remark 2. It is well known that time delays alwaysinfluence the dynamic properties of the chaotic DNNs.Hence, in (5), we add the time-delay feedback control termMztðtÞ. As discussed in [32] and [19], such a time-delayedfeedback term could be employed to tackle more generalsystems and achieve less conservative analysis results thanthose results without using the delayed feedback term.

Throughout this paper, we always make the followingassumptions:

ðA1Þ

f iðxÞ satisfies the Lipschitz condition. That is, for eachi ¼ 1; 2; . . . ; n, there is a constant ai40 such that

jf iðxÞ � f iðyÞjpaijx� yj 8x; y 2 R.

ðA2Þ

sðt; x; yÞ satisfies the Lipschitz condition. Moreover,there exist constant matrices of appropriate dimen-sions G1;G2 such that

trace½sTðt;x; yÞsðt;x; yÞ�pkG1xk2 þ kG2yk2

8ðt; x; yÞ 2 Rþ � Rn � Rn.

ðA3Þ

f ð0Þ � 0, sðt; 0; 0Þ � 0:

In order to investigate the problem of exponentialsynchronization for the chaotic DNNs with stochasticperturbation, ziðtÞ ¼ xiðtÞ � yiðtÞ is defined as the synchro-nization error, where xiðtÞ and yiðtÞ are the ith statevariables of drive system (2) and response system (4),respectively. Therefore, the error dynamical system be-tween (2) and (4) is given as follows:

_zðtÞ ¼ � CzðtÞ þ AgðzðtÞÞ þ BgðztðtÞÞ � sðt; zðtÞ; ztðtÞÞ _oðtÞ

� KzðtÞ �MztðtÞ, ð6Þ

where zðtÞ ¼ ½z1ðtÞ; z2ðtÞ; . . . ; znðtÞ�T 2 Rn, and

gðzðtÞÞ ¼ ½f 1ðx1ðtÞÞ � f 1ðy1ðtÞÞ; f 2ðx2ðtÞÞ � f 2ðy2ðtÞÞ; . . . ,

f nðxnðtÞÞ � f nðynðtÞÞ�T,

gðztðtÞÞ ¼ ½f 1ðx1ðt� tÞÞ � f 1ðy1ðt� tÞÞ; f 2ðx2ðt� tÞÞ

� f 2ðy2ðt� tÞÞ; . . . ; f nðxnðt� tÞÞ � f nðynðt� tÞÞ�T.

The initial condition associated with the system (6) isgiven in the following form:

zðsÞ ¼ xðsÞ; �tpsp0,

for any x 2 L2F0ð½�t; 0�;RnÞ, where L2

F0ð½�t; 0�;RnÞ is the

family of all F0-measurable Cð½�t; 0�;RnÞ-valued randomvariables satisfying that sup�tpsp0 EjxðsÞj2o1, andCð½�t; 0�;RnÞ denotes the family of all continuousRn-valued functions xðsÞ on ½�t; 0� with the normkxk ¼ sup�tpsp0 jxðsÞj.Under the Assumptions ðA1Þ and ðA3Þ, we can easily

have that

jgiðziðtÞÞj ¼ jf iðziðtÞ þ yiðtÞÞ � f iðyiðtÞÞjpaijziðtÞj, (7)

and gð0Þ ¼ 0. Hence, together with ðA2Þ, it follows from[13] that the system (6) admits a trivial solution zð0Þ � 0.

Definition 1. The drive system (2) and the response system(4) are said to be exponentially synchronized if, for asuitably designed feedback controller, there exist constantsgX1 and l40 such that EkxðtÞ � yðtÞk2pgEkxð0Þ�yð0Þk2e�lt, for any tX0, and the constant l is defined asthe exponential synchronization rate.

In fact, the exponential synchronization problem con-sidered in this paper is to determine the control input U

associated with the state feedback to synchronize the twoidentical stochastic perturbed chaotic DNNs exponentiallywith the same system parameters, but different initialconditions. It is clear that, if the trivial solution of thecontrolled error dynamical system (6) is exponentiallystable in mean square, then the exponential synchroniza-tion between the drive system (2) and the response system(4) can be realized.

3. Main results

In this section, we will establish several criteria toimplement the exponential synchronization of the chaoticDNNs with stochastic perturbation.Before stating our main results, we need a few more

notations. Let C1;2ðRþ � Rn;RþÞ denote the family of allnonnegative functions V ðt;xÞ on Rþ � Rn which arecontinuously twice differentiable in x and once differentiablein t. For each V 2 C1;2ðRþ � Rn;RþÞ, define an operator Lassociated with the error system (6) acting on V by

LV ðt;xÞ ¼ V tðt;xÞ þ V xðt;xÞ½�CzðtÞ þ AgðzðtÞÞ

þ BgðztðtÞÞ � KzðtÞ �MztðtÞ�

þ 12trace½sTðt; x; yÞV xxðt;xÞsðt;x; yÞ�, ð8Þ

where

V tðt;xÞ ¼qV ðt; xÞ

qt; V xðt;xÞ ¼

qV ðt;xÞ

qx1; . . . ;

qV ðt;xÞ

qxn

� �,

V xxðt;xÞ ¼q2V ðt; xÞqxiqxj

� �n�n

.

ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–24852480

Theorem 1. Let ðA1Þ–ðA3Þ hold. Assume that there exist

constants a4b40, a positive definite matrix Q, and the

control gain matrices K and M in error dynamical system (6)such that the following matrix inequality:

O:¼

2C þ 2K � 2Q� GT1 G1 � aI M AS 0

MT bI � GT2 G2 0 BS

STAT 0 Q 0

0 STBT 0 Q

2666664

377777540

ð9Þ

holds, where S:¼diagða1; a2; . . . ; anÞ, then the drive system

(2) and the response system (4) can be exponentially

synchronized.

In order to prove this theorem, let us present two basicfacts and a lemma.

Fact 1. For any vectors X ;Y 2 Rn and a positive definite

matrix Q 2 Rn�n, the following inequality holds

2XTYpXTQX þ YTQ�1Y . (10)

Fact 2 (Schur complement, Boyd et al. [3]). Let Q and R be

two symmetric matrices, then

Q S

ST R

� �40 (11)

is equivalent to R40 and Q� SR�1ST40.

Lemma 1 (Baker and Buckwar [1]). Under assumptions

ðA1Þ–ðA3Þ, let xðtÞ be a solution of system (6) and assume

that there exists a positive, continuous function V ðt; xðtÞÞ (for

tXt0 � t and x 2 R) for which there exist positive constants

c1; c2, and p41, such that c1jxjppV ðt;xÞpc2jxj

p, when tXt0and for some constants 0pboa,

DþEðV ðt;xðtÞÞÞp� aEðV ðt;xðtÞÞÞ þ bEðV ðt� t;xðt� tÞÞÞ,

when tXt0. Then

Eðjxðt; t0;fÞjpÞpc2

c1E sup

s2½t0�t;t0�jfðsÞjp

!expð�vþðt� t0ÞÞ,

where vþ 2 ð0; a� b� is the unique positive solution of the

equation v ¼ a� bevt. Furthermore, the trivial solution of

system (6) is globally exponentially stable in pth mean.

Remark 3. The global exponential stability in pth mean isequivalent to the definition of pth moment exponentialstability in [25]. In this paper, we only consider the case ofp ¼ 2 which means that the trivial solution of the errorsystem (6) is exponentially stable in mean square.

Remark 4. The function V in Lemma 1 is positive andcontinuous on tXt0 � t, while a similar function V in thegeneral Halanay inequality used in [26] is required to bepositive and continuous only on t0 � tptpt0. The reasonfor our tighter condition is that we study the pth momentexponential stability of stochastic differential equations,where a precondition is c1jxj

ppV ðt;xÞpc2jxjp. Note that

such a difference does not contradict with the conditionneeded in [26].

Let us now begin to prove our main results.

Proof of Theorem 1. Construct the following Lyapunovfunctional:

V ðt; zðtÞÞ ¼ zTðtÞzðtÞ ¼ kzðtÞk2.

It can be easily verified that V ðt; zðtÞÞ is a nonnegativefunction over ½t0 � t;þ1Þ. Evaluating the time derivativeof V along the trajectory of system (6), we have

_V ðt; zðtÞÞ ¼LV ðt; zðtÞÞ þ Vzðt; zðtÞÞsðt; zðtÞ; ztðtÞÞ _oðtÞ,(12)

where

LV ðt; zðtÞÞ

¼ Vtðt; zðtÞÞ þ V zðt; zðtÞÞ½�CzðtÞ þ AgðzðtÞÞ

þ BgðztðtÞÞ � KzðtÞ �MztðtÞ�

þ 12trace½sTðt; zðtÞ; ztðtÞÞV zzðt; zðtÞÞsðt; zðtÞ; ztðtÞÞ�

¼ 2zTðtÞ½�CzðtÞ þ AgðzðtÞÞ þ BgðztðtÞÞ � KzðtÞ �MztðtÞ�

þ trace½sTðt; zðtÞ; ztðtÞÞsðt; zðtÞ; ztðtÞÞ�

¼ �2zTðtÞCzðtÞ þ 2zTðtÞAgðzðtÞÞ

þ 2zTðtÞBgðztðtÞÞ � 2zTðtÞKzðtÞ

� 2zTðtÞMztðtÞ þ trace½sTðt; zðtÞ; ztðtÞÞsðt; zðtÞ; ztðtÞÞ�.ð13Þ

It follows from Fact 1 that

2zTðtÞAgðzðtÞÞpzTðtÞQzðtÞ þ gTðzðtÞÞATQ�1AgðzðtÞÞ

pzTðtÞQzðtÞ þ zTðtÞSTATQ�1ASzðtÞ, ð14Þ

2zTðtÞBgðztðtÞÞ

pzTðtÞQzðtÞ þ gTðztðtÞÞBTQ�1BgðztðtÞÞ

pzTðtÞQzðtÞ þ zTðt� tÞSTBTQ�1BSzðt� tÞ. ð15Þ

From Assumption ðA2Þ, we also have

trace½sTðt; zðtÞ; ztðtÞÞsðt; zðtÞ; ztðtÞÞ�

pzTðtÞGT1 G1zðtÞ þ zTðt� tÞGT

2 G2zðt� tÞ. ð16Þ

Substituting (14)–(16) into (13), it can be derived that

LV ðt; zðtÞÞp� 2zTðtÞCzðtÞ þ 2zTðtÞQzðtÞ

þ zTðtÞSTATQ�1ASzðtÞ

þ zTðt� tÞSTBTQ�1BSzðt� tÞ

� 2zTðtÞKzðtÞ � 2zTðtÞMzðt� tÞ

þ zTðtÞGT1 G1zðtÞ þ zTðt� tÞGT

2 G2zðt� tÞ

ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–2485 2481

pzTðtÞð�2C � 2K þ 2Qþ STATQ�1AS

þ GT1 G1 þ aIÞzðtÞ

þ zTðt� tÞðGT2 G2 þ STBTQ�1BS� bIÞzðt� tÞ

� 2zTðtÞMzðt� tÞ � azTðtÞzðtÞ

þ bzTðt� tÞzðt� tÞ

p� ZTðtÞPZðtÞ � azTðtÞzðtÞ

þ bzTðt� tÞzðt� tÞ, ð17Þ

where

ZðtÞ ¼zðtÞ

ztðtÞ

" #,

and

P:¼2C þ 2K � 2Q� STATQ�1AS� GT

1 G1 � aI M

MT bI � GT2 G2 � STBTQ�1BS

" #. (18)

By Fact 2, the condition (9) in Theorem 1 is equivalent tothe inequality P40, and therefore, we have

LV ðt; zðtÞÞp� azTðtÞzðtÞ þ bzTðt� tÞzðt� tÞ. (19)

To this end, we obtain

_V ðt; zðtÞÞ ¼LV ðt; zðtÞÞ þ V zðt; zðtÞÞsðt; zðtÞ; ztðtÞÞ _oðtÞ

p� azTðtÞzðtÞ þ bzTðt� tÞzðt� tÞ

þ Vzðt; zðtÞÞsðt; zðtÞ; ztðtÞÞ _oðtÞ. ð20Þ

Then, take the expectations on both sides of (20), it holds

E _V ðt; zðtÞÞpELV ðt; zðtÞÞ þ EV zðt; zðtÞÞsðt; zðtÞ; ztðtÞÞ _oðtÞ

p� aEV ðt; zðtÞÞ þ b maxt�tpspt

EV ðs; zðsÞÞ. ð21Þ

It follows by Lemma 1 that

EV ðt; zðtÞÞp maxt0�tpspt0

EV ðs; zðsÞÞe�lðt�t0Þ. (22)

In other words, we arrive at the conclusion that

EkzðtÞk2pe�lðt�t0ÞEkxk2, (23)

where l is the unique positive solution of the equationl ¼ a� belt. This completes the proof. &

Remark 5. Based on the Lyapunov stability theory forstochastic systems and the drive-response concept, we haveestablished theoretical results in Theorem 1 on exponentialsynchronization of stochastic chaotic DNNs. If the feed-back gain matrices K and M in response networks aresuitably designed, the drive system (2) and the responsesystem (4) can be exponentially synchronized with a certainexponential synchronization rate. So, the problem ofsynchronization of chaotic systems with stochastic pertur-bation can be easily solved by solving a LMI, which can beconducted readily using Matlab LMI Toolbox.

Remark 6. In this paper, noise perturbation was taken intoaccount when studying the exponential synchronization

problem, which reflects a more realistic dynamics than theresults in [12,11], where the noise perturbation was ignored.Meanwhile, the time-delay feedback controllers adoptedhere can tackle more general systems and achieve lessconservative results than those in [10,12,11]. Moreover,the simulation results provided later will demonstratethe noise-perturbed synchronization phenomena success-fully, and they are found to be coincident with thetheoretical results well. At the same time, those novelchaotic phenomena which are reflected by the simulationresults can enlighten us to do some other interestinginvestigations.

If the time-delay feedback controller U is designedwithout the time-delay term, we can also have the followingresults.

Corollary 1. Let ðA1Þ–ðA3Þ hold. If there exist constants

a4b40, a positive definite matrix Q, and the feedback gain

matrix K in the response system (4) satisfying the following

LMI:

O:¼

2C þ 2K � 2Q� GT1 G1 � aI 0 AS 0

0 bI � GT2 G2 0 BS

STAT 0 Q 0

0 STBT 0 Q

2666664

377777540,

ð24Þ

where S:¼diagða1; a2; . . . ; anÞ, then the drive and response

systems can be exponentially synchronized.

If there is no noise perturbation in response system (4),then the considered model degenerate to the generalchaotic DNNs, and the synchronization problem becomesthe exponential synchronization of two identical chaoticDNNs. In this case, we have the following results.

Corollary 2. Let ðA1Þ–ðA3Þ hold. If there exist constants

a4b40, a positive definite matrix Q, and the control gain

matrices K and M in system (4) satisfying the following

LMI:

O:¼

2C þ 2K � 2Q� aI M AS 0

MT bI 0 BS

STAT 0 Q 0

0 STBT 0 Q

266664

37777540, (25)

where S:¼diagða1; a2; . . . ; anÞ, then the drive and response

systems can be exponentially synchronized.

Remark 7. In [10], by virtue of Halanay inequalitytechnique, the authors proposed several exponentialsynchronization conditions for two identical chaoticDNNs, where the criteria derived depend on the matrix

ARTICLE IN PRESS

0

0.2

0.4

0.6

0.8

1

x2

Y. Sun et al. / Neurocomputing 70 (2007) 2477–24852482

norm, and the feedback gain matrix in their results must bepositive definite. The conditions in Corollary 2, however,do not have these restrictions. In particular, the connectionmatrix in the drive networks and the desired controllergain matrix in the response networks are the parametersof a LMI, which can be solved directly. Hence, Corollary 2improves and further generalizes the results derivedin [10], and the presented synchronization scheme is lessconservative.

Remark 8. In [12], the authors presented an exponentialsynchronization scheme for a class of DNNs, the conditionof which can be verified based on a certain Hamiltonianmatrix. Though it can avoid solving an algebraic Riccatiequation, it is difficult to verify this condition and notapplicable in the real-life application.

Remark 9. In [21], by means of the so-called LaSalle-typeinvariance principle for stochastic differential equations,the authors discussed the complete synchronization be-tween unidirectionally coupled Chua’s circuits withinstochastic perturbation. Unfortunately, the derived cou-pling strengths in [21] are positive, that is, the feedbackgain matrices are required to be diagonal positive definitematrices. On the other hand, our results can remove theserestrictions, and the obtained criteria are less conservativethan those derived in [21]. Moreover, the techniques usedin our paper cannot only deduce the exponential synchro-nization of the chaotic DNNs with stochastic perturba-tions, but also calculate the exponential synchronizationrate.

Remark 10. It is worth pointing out that the topic ofsynchronization with noise perturbation or synchroniza-tion based on white-noise coupling has attracted a greatdeal of attention. The novel dynamic behaviors with noiseperturbation is interesting, and we do believe that thetheoretical results derived here could stimulate increasinginterests on the synchronization of chaotic systems withstochastic perturbation, which can also enrich the topics ofchaos synchronization. Moreover, there are many otherinteresting experimental observations with noise perturba-tion, such as stochastic resonance, noise-induced synchro-nization and noise-enhanced synchronization, etc. Hence,developing the corresponding analytical results to supportthese experimental observations leave some interestingtopics to the future investigations.

-15 -10 -5 0 5 10 15-1

-0.8

-0.6

-0.4

-0.2

x1

Fig. 1. Chaotic behaviors of the DNNs.

4. Illustrative example

In this section, we employ an example to illustrate theeffectiveness of the obtained results.

Example. Consider the following chaotic DNNs [14]:

_xðtÞ ¼ �CxðtÞ þ Af ðxðtÞÞ þ Bf ðxðt� tÞÞ, (26)

where f ðxÞ ¼ 12ðjxþ 1j � jx� 1jÞ;

C ¼1 0

0 1

" #; A ¼

1þ p4

20

0:1 1þ p4

" #,

B ¼�

ffiffiffi2p

p41:3 0:1

0:1 �ffiffiffi2p

p41:3

24

35.

The response system is designed as follows:

_yðtÞ ¼ � CyðtÞ þ Af ðyðtÞÞ þ Bf ðytðtÞÞ þ sðt; zðtÞ; ztðtÞÞ _oðtÞ

þ KzðtÞ þMztðtÞ, ð27Þ

where

sðt; zðtÞ; ztðtÞÞ ¼

ffiffiffi2p

2z1ðtÞ 0

0

ffiffiffi2p

2z2ðtÞ

26664

37775

þ

ffiffiffi2p

2z1ðt� tÞ 0

0

ffiffiffi2p

2z2ðt� tÞ

26664

37775.

Hence we can easily have G1 ¼10

01

� �; G2 ¼

10

01

� �, and

obviously, the drive system and the response system satisfyðA1Þ–ðA3Þ with S ¼ 1

001

� �. Here we let t ¼ 1.

We can find that, when a ¼ 11, b ¼ 4, Q ¼ 13�6

�613

� �, the

control gain matrices are K ¼ 20�3

�338

� �, M ¼ 2

002

� �, which

satisfy the condition in Theorem 1, and therefore thesystems (26) and (27) can be exponentially synchronizedwith the exponential synchronization rate l ¼ 0:9239.In the simulations, we use the Euler–Maruyama numer-

ical scheme to simulate the drive-response systems (26) and(27). Some initial parameters are given as follows: T ¼ 200and time step size is dt ¼ 0:04. Fig. 1 shows the chaoticbehavior of system (26) in phase space with the initial

ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–2485 2483

condition ½x1ðtÞ;x2ðtÞ�T ¼ ½0:1; 0:1�T, for �1ptp0. Fig. 2

depicts the exponential synchronization of the chaoticDNNs with stochastic perturbation in phase space. Fig. 3shows the states of response system (27) with the feedbackgain matrices K and M. Fig. 4 depicts the state response ofthe error dynamical system with the initial conditions½x1ðtÞ; x2ðtÞ�

T ¼ ½0:1; 0:1�T and ½y1ðtÞ; y2ðtÞ�T ¼ ½0:2;�0:2�T

for �1ptp0, respectively. Fig. 5 depicts the exponentialsynchronization rate of the error dynamical system .

If we take K1 ¼13�6

�613

� �, M1 ¼

10

01

� �, one can easily

check that the condition in Theorem 1 is not satisfied, so

-15 -10 -5 0 5 10 15-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y1

y2

Fig. 2. Synchronization of DNNs with stochastic perturbation.

0 50 100 150 200-15

-10

-5

0

5

10

15

Times (sec)

sta

te y

1

Fig. 3. The state of th

system (26) and the system (27) cannot be exponentiallysynchronized with the same parameters and the same initialconditions as in Figs. 1–5. Fig. 6 depicts the states of theunsynchronized systems with the feedback gain matricesK1;M1.

5. Conclusions

In this paper, we have investigated the exponentialsynchronization of stochastic perturbed chaotic DNNs.

0 50 100 150 200-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Times (sec)

sta

te y

2

e response system.

0 1 2 3 4 5 6-0.1

0

0.1

0.2

0.3synchronization error |z1|=|x1-y1|

Times (sec)|z

1|

0 1 2 3 4 5 6-0.1

0

0.1

0.2

0.3synchronization error |z2|=|x2-y2|

Times (sec)

|z2

|

Fig. 4. The state response of error dynamical system.

ARTICLE IN PRESS

0 5 10 15 20 25 30-25

-20

-15

-10

-5

0

Times (sec)

ln (

|z1|)

/t

0 5 10 15 20 25 30-25

-20

-15

-10

-5

0

Times (sec)

ln (

|z2|)

/t

Fig. 5. Exponential synchronization rate of error dynamical system.

-15 -10 -5 0 5 10 15-1.5

-1

-0.5

0

0.5

1

1.5

y1

y2

Fig. 6. Unsynchronized dynamics in phase space.

Y. Sun et al. / Neurocomputing 70 (2007) 2477–24852484

Utilizing time-delay feedback control and LMI techniques,we have proposed several criteria to guarantee exponentialsynchronization of chaotic DNNs with stochastic pertur-bation. With the suitably designed feedback gain matrix inthe response networks, the drive system and the responsesystem with stochastic perturbation can be exponentiallysynchronized, and the exponential synchronization rate canalso be derived at the same time. Finally, an illustratedexample with its simulations have been utilized todemonstrate the effectiveness of the provided results.

In fact, there are many fruitful stability theories forstochastic systems, for example, Razumikihin-type theo-rems on exponential stability for stochastic functionaldifferential equations in [25], and so on. We can utilizethese results to investigate other synchronization problemsbetween chaotic systems with stochastic perturbation, for

example, lag synchronization, phase synchronization,general synchronization, etc., which will be our futureresearch topics.

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Yonghui Sun was born in Henan Province, China.

He received the B.S. degree in mathematics from

Henan University, Kaifeng, China, in 2004. Now,

he is working toward the M.S. degree in

mathematics from Southeast University, Nanj-

ing, China. His current research interests include

stochastic theory and applications, chaos syn-

chronization, and neural networks.

Jinde Cao received the B.S. degree from Anhui

Normal University, Wuhu, China, the M.S.

degree from Yunnan University, Kunming,

China, and the Ph.D. degree from Sichuan

University, Chengdu, China, all in mathematics/

applied mathematics, in 1986, 1989, and 1998,

respectively. From March 1989 to May 2000, he

was with Yunnan University. In May 2000, he

joined the Department of Mathematics, South-

east University, Nanjing, China. From July 2001

to June 2002, he was a Post doctoral Research Fellow in the Department

of Automation and Computer-aided Engineering, Chinese University of

Hong Kong, Hong Kong. From August 2002 to October 2002, he was a

Senior Visiting Scholar at the Institute of Mathematics, Fudan University,

Shanghai, China. From February 2003 to May 2003, he was a Senior

Research Associate in the Department of Mathematics, City University of

Hong Kong, Hong Kong. From July 2003 to September 2003, he was a

Senior Visiting Scholar in the Institute of Intelligent Machines, Chinese

Academy of Sciences, Hefei, China. From January 2004 to April 2004, he

was a Research Fellow in the Department of Manufacturing Engineering

and Engineering Management, City University of Hong Kong, Hong

Kong. From January 2005 to April 2005, he was a Research Fellow in the

Department of Mathematics, City University of Hong Kong, Hong Kong.

From January 2006 to April 2006, he was a Research Fellow in the

Department of Electronics Engineering, City University of Hong Kong,

Hong Kong. From July 2006 to September 2006, he was a Visiting

Research Fellow of Royal Society in the School of Information Systems,

Computing and Mathematics, Brunel University, UK.

He is currently a Professor and Doctoral Advisor at the Southeast

University. Prior to this, he was a Professor at Yunnan University from

1996 to 2000. He is the author or coauthor of more than 130 journal

papers and five edited books and a reviewer of Mathematical Reviews and

Zentralblatt-Math. His research interests include nonlinear systems,

neural networks, complex systems and complex networks, stability theory,

and applied mathematics.

Professor Cao is an Associate Editor of the IEEE Transaction on

Neural Networks, Mathematics and Computers in Simulation, and

Neurocomputing.

Zidong Wang was born in Jiangsu, China, in

1966. He received the B.Sc. degree in mathe-

matics in 1986 from Suzhou University, Suzhou,

China, and the M.Sc. degree in applied mathe-

matics in 1990 and the Ph.D. degree in electrical

engineering in 1994, both from Nanjing Uni-

versity of Science and Technology, Nanjing,

China.

Dr. Wang was appointed as Lecturer in 1990

and Associate Professor in 1994 at Nanjing

University of Science and Technology. From January 1997 to December

1998, he was an Alexander von Humboldt research fellow with the

Control Engineering Laboratory, Ruhr-University Bochum, Germany.

From January 1999 to February 2001, he was a Lecturer with Department

of Mathematics, University of Kaiserslautern, Germany. From March

2001 to July 2002, he was a University Senior Research Fellow with the

School of Mathematical and Information Sciences, Coventry University,

UK. In August 2002, he joined the Department of Information Systems

and Computing, Brunel University, UK, as a Lecturer, and was promoted

to Reader in September 2003. He is also holding an Adjunct TePin

Professorship at Donghua University, Shanghai, a Visiting Professorship

at Fuzhou University, Fuzhou, and a Guest Professorship at Nanjing

Normal University, Nanjing, China.

Dr. Wang’s research interests include dynamical systems, signal

processing, bioinformatics, control theory and applications. He has

published more than 60 papers in refereed international journals. He

was awarded the Humboldt research fellowship in 1996 from Alexander

von Humboldt Foundation, the JSPS Research Fellowship in 1998 from

Japan Society for the Promotion of Science, and the William Mong

Visiting Research Fellowship in 2002 from the University of Hong Kong.

He was a recipient of the Outstanding Science and Technology

Development Awards, once in 2005 and twice in 1997 from the National

Education Committee of China, once in 1997 from the Military Industry

General Company of China, once in 1997 from Jiangsu Province

Government of China, and once in 1999 from the National Education

Ministry of China.

Dr. Wang is currently serving as an Associate Editor for IEEE

Transactions on Signal Processing, IEEE Transactions on Systems, Man,

and Cybernetics—Part C, IEEE Transactions on Control Systems

Technology, Circuits, Systems and Signal Processing, an Editorial Board

Member for International Journal of Systems Science, Neurocomputing,

International Journal of Computer Mathematics, and an Associate Editor

on the Conference Editorial Board for the IEEE Control Systems Society.

He is a Senior Member of the IEEE, a Fellow of the Royal Statistical

Society, a member of program committee for many international

conferences, and a very active reviewer for many international journals.

He was nominated as an outstanding reviewer for IEEE Transactions on

Automatic Control in 2004 and for the journal Automatica in 2000. He

received the Standing Membership of the Technical Committee on

Control of IASTED (International Association of Science and Technol-

ogy for Development) in 2000. From 2001 to 2006 he served as a member

of technical program committee for 28 international conferences in the

general area of systems theory and computing.