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ARTICLE IN PRESS
0925-2312/$ - se
doi:10.1016/j.ne
$This work
Foundation of
United Kingdo
Province of Ch�CorrespondE-mail addr
Neurocomputing 70 (2007) 2477–2485
www.elsevier.com/locate/neucom
Exponential synchronization of stochastic perturbed chaoticdelayed neural networks$
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: [email protected], [email protected] (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 thatjf 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 thattrace½sTðt;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.
References
[1] C. Baker, E. Buckwar, Exponential stability in pth mean of solutions,
and of convergent Euler-type solutions, of stochastic delay differ-
ential equations, J. Comput. Appl. Math. 184 (2005) 404–427.
[2] S. Blythea, X. Mao, X. Liao, Stability of stochastic delay neural
networks, J. Franklin Inst. 338 (2001) 481–495.
[3] S. Boyd, L. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix
Inequalities in System and Control Theory, SIAM, Philadelphia, PA,
1994.
[4] J. Cao, Periodic oscillation and exponential stability of delayed
CNNs, Phys. Lett. A 270 (2000) 157–163.
[5] J. Cao, New results concerning exponential stability and periodic
solutions of delayed cellular neural networks, Phys. Lett. A 307
(2003) 136–147.
[6] J. Cao, H. Li, D.W.C. Ho, Synchronization criteria of Lur’e systems
with time-delay feedback control, Chaos Solitons Fractals 23 (2005)
1285–1298.
[7] J. Cao, J. Lu, Adaptive synchronization of neural networks with or
without time-varying delay, Chaos 16 (2006) 013133.
[8] J. Cao, J. Wang, Global asymptotic and robust stability of recurrent
neural networks with time delays, IEEE Trans. Circuits Syst. I 52
(2005) 417–426.
[9] G. Chen, J. Zhou, Z. Liu, Global synchronization of coupled delayed
neural networks and applications to chaotic CNN model, Int.
J. Bifur. Chaos 14 (2004) 2229–2240.
[10] C. Cheng, T. Liao, C. Hwang, Exponential synchronization of a class
of chaotic neural networks, Chaos Solitons Fractals 15 (2005)
197–206.
[11] C. Cheng, T. Liao, J. Yan, C. Hwang, Synchronization of neural
networks by decentralized feedback control, Phys. Lett. A 338 (2005)
28–35.
[12] C. Cheng, T. Liao, J. Yan, C. Hwang, Exponential synchronization
of a class of neural networks with time-varying delays, IEEE Trans.
Syst. Man Cybern. B 36 (2006) 209–215.
[13] A. Friedman, Stochastic Differential Equations and Applications,
Academic Press, New York, 1976.
[14] M. Gilli, Strange attractors in delayed cellular neural networks, IEEE
Trans. Circuits Syst. I 40 (1993) 849–853.
[15] J. Hopfield, Neurons with graded response have collective computa-
tional properties like those of two-state neurons, Proc. Natl. Acad.
Sci. USA, Biophys. 81 (1984) 3088–3092.
[16] G. Hu, L. Pivka, A. Zheleznyak, Synchronization of a one-
dimensional array of Chua’s circuits by feedback control and noise,
IEEE Trans. Circuits Syst. I 42 (1995) 736–740.
[17] X. Huang, J. Cao, D.W.C. Ho, Synchronization criteria for Lur’e
systems by dynamic output feedback with time-delay, Int. J. Bifur.
Chaos, 16 (2006) 2293–2307.
[18] G. Jiang, G. Chen, K. Tang, A new criterion for chaos synchroniza-
tion using linear state feedback control, Int. J. Bifur. Chaos 13 (2003)
2343–2351.
[19] X. Liao, G. Chen, Chaos synchronization of general Lur’e systems
via time-delay feedback control, Int. J. Bifur. Chaos 13 (2003)
207–213.
[20] X. Liao, X. Mao, Exponential stability and instability of stochastic
neural networks, Stochast. Anal. Appl. 14 (1996) 165–185.
[21] W. Lin, Y. He, Complete synchronization of the noise-perturbed
Chua’s circuits, Chaos 15 (2005) 023705.
[22] H. Lu, Chaotic attractors in delayed neural networks, Phys. Lett. A
298 (2002) 109–116.
[23] J. Lu, J. Cao, Adaptive complete synchronization of two identical or
different chaotic (hyperchaotic) systems with fully unknown para-
meters, Chaos 15 (2005) 043901.
ARTICLE IN PRESSY. Sun et al. / Neurocomputing 70 (2007) 2477–2485 2485
[24] J. Lu, X. Yu, G. Chen, Chaos synchronization of general complex
dynamical networks, Physica A 334 (2004) 281–302.
[25] X. Mao, Exponential Stability of Stochastic Differential Equations,
Marcel Dekker, New York, 1994.
[26] S. Mohamad, K. Gopalsamy, Continuous and discrete Halanay-type
inequalities, Bull. Austral. Math. Soc. 61 (2000) 371–385.
[27] L. Pecora, T. Carroll, Synchronization in chaotic systems, Phys. Rev.
Lett. 64 (1990) 821–824.
[28] E. Sachez, M. Matıas, V. Munuzuri, Analysis of synchronization of
chaotic systems by noise: an experimental study, Phys. Rev. E 56
(1997) 4068–4071.
[29] J. Shuai, K. Wong, Noise and synchronization in chaotic neural
networks, Phys. Rev. E 57 (1998) 7002–7007.
[30] L. Wan, J. Sun, Mean square exponential stability of stochastic
delayed Hopfield neural networks, Phys. Lett. A 343 (2005) 306–318.
[31] M. Wang, Z. Hou, H. Xin, Internal noise-enhanced phase
synchronization of coupled chemical chaotic oscillators, J. Phys. A
38 (2005) 145–152.
[32] X. Wang, G. Zhong, K. Tang, K. Man, Z. Liu, Generating chaos in
Chua’s circuit via time-delay feedback, IEEE Trans. Circuits Syst. I
48 (2001) 1151–1156.
[33] T. Wellens, V. Shatokhin, A. Buchleitner, Stochastic resonance,
Rep. Prog. Phys. 67 (2004) 45–105.
[34] Q. Zhang, R. Ma, C. Wang, J. Xu, On the global stability of delayed
neural networks, IEEE Trans. Automat. Control 48 (2003) 794–797.
[35] Y. Zhang, Z. He, A secure communication scheme based on cellular
neural networks, Proceedings of the IEEE International Conference
on Intelligent Processing Systems, vol. 1, 1997, pp. 521–524.
[36] X. Huang, J. Cao, Generalized synchronization for delayed chaotic
neural networks: a novel coupling scheme, Nonlinearity 19 (2006)
2797–2811.
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.