Neurocomputing 74 (2011) 563–567

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Neurocomputing

0925-23

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/neucom

Global impulsive exponential anti-synchronization of delayed chaoticneural networks

Hongyong Zhao �, Qi Zhang

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

a r t i c l e i n f o

Article history:

Received 20 January 2010

Received in revised form

17 July 2010

Accepted 13 September 2010

Communicated by N. Ozcan& 2010 Elsevier B.V. All rights reserved.

Available online 20 October 2010

Keywords:

Chaotic neural networks

Anti-synchronization

Impulsive perturbation

Inequality method

Feedback control

12/$ - see front matter & 2010 Elsevier B.V. A

016/j.neucom.2010.09.016

esponding author. Tel.: +86 25 52113704.

ail address: hongyongz@126.com (H. Zhao).

a b s t r a c t

In this paper, the impulsive exponential anti-synchronization for chaotic delayed neural networks is

investigated. By establishing an integral delay inequality and using the inequality method, some sufficient

conditions ensuring impulsive exponential anti-synchronization of two chaotic delayed networks are

derived. To illustrate the effectiveness of the new scheme, a numerical example is given.

1. Introduction

Chaos is an interesting phenomenon of nonlinear systems.A chaotic system has some remarkable dynamic characteristics [2].When we use a chaotic signal to drive two identical systems, the twosystems or certain parts of them will have the synchronous behavior.During the last two decades, synchronization of chaotic dynamicsystems has received a great deal of interest among scientists fromvarious research fields [1,3,7,9–11,13,15] since Pecora and Carroll, intheir pioneering work [9], proposed the drive-response concept forconstructing synchronization of coupled chaotic systems.

In recent years, various synchronization phenomena are beingreported for coupled chaotic oscillators, such as generalized synchro-nization [17], lag synchronization [16], and even anti-synchronization[5,6]. Anti-synchronization appears when the sum of two signalsconverges to zero. It is a noticeable phenomenon whereby the statevectors of the synchronized systems have the same amplitude butopposite signs as those of the driving system and has importantapplication significance ([8] and the references therein). Using anti-synchronization to lasers, one may generate not only drop-outs of theintensity as with ordinary low frequency fluctuations but also shortpulses of high intensity, which offers new ways for generating pulsesof special shapes. Using anti-synchronization to communicationsystems, one may transmit digital signals by the transform betweensynchronization and anti-synchronization continuously, which willstrengthen the security and secrecy.

ll rights reserved.

In the same time, in optimal control of signal processing systems,computer networks, automatic control systems, flying object motions,telecommunications, and many systems are characterized by abruptchanges at certain moments due to instantaneous perturbations,which lead to impulsive effects. Mathematically, these systemswith impulsive effects are described by the impulsive differentialequations. Impulsive control synchronization, as a type of synchro-nization, has been developed [4,12,14,18,19].

To the best of our knowledge, for the adaptive anti-synchroni-zation of time-delayed chaotic systems with impulsive effects,there is no result in the literature so far, which still remains openand challenging.

Motivated by the above discussion, this work addresses theexponential anti-synchronization of a class of impulsive perturbedchaotic delayed neural networks. An new integral delay inequalityis established. By employing inequality method [20–24] and theobtained inequality, some simple conditions of impulsive expo-nential anti-synchronization of two coupling chaotic systems arederived. An illustrative example is given to demonstrate theeffectiveness of the presented anti-synchronization scheme.

2. Problem formulation and preliminaries

For convenience, we first introduce some notations used inthis paper.

Rn is the space of n-dimensional real column vectors, Rn�m

denotes the set of n�m real matrices, and Rþ ¼ ½0,þ1Þ.x¼ ðx1, . . . ,xnÞ

T ARn denotes a column vector. JxJ denotes a vectornorm defined by JxJ¼ ð

Pni ¼ 1 jxij

2Þ1=2.

H. Zhao, Q. Zhang / Neurocomputing 74 (2011) 563–567564

Let C(X,Y) denote the space of continuous mappings from thetopological space X to the topological space Y.

PCðJ,RnÞ ¼ fc : J- Rn

jcðsÞ is continuous for all but at most afinite number of points sA J and at these points sA J, cðsþ Þ andcðs�Þ exist, cðsþ Þ ¼cðsÞg, where J�R is a bounded interval, cðsþ Þandcðs�Þdenote the right-hand and left-hand limits of the functioncðsÞ, respectively. In particular, PC9PCð½�t,0�,Rn

Þ, where t40.8j¼ ðj1, . . . ,jnÞ

T APC, JjJt denotes the norm defined byJjJt ¼ sup�tr sr0JjðsÞJ.

For any d¼ ðd1, . . . ,dnÞT ARn, we denote by d ¼maxiA f1,...,ngfdig,

d ¼miniA f1,...,ngfdig.Consider a class of impulsive neural networks model described

by

_xiðtÞ ¼�cixiðtÞþXn

j ¼ 1

aijfjðxjðtÞÞ

þXn

j ¼ 1

bijfjðxjðt�tijðtÞÞÞ, tatk, tZ0,

DxiðtkÞ ¼ xiðtkÞ�xiðt�k Þ ¼ hikxiðt

�k Þ, t¼ tk, k¼ 1,2, . . . ,

xiðtÞ ¼fiðtÞ, tA ½�t,0�, i¼ 1, . . . ,n,

8>>>>>>>>>><>>>>>>>>>>:

ð2:1Þ

where xi denotes the state variable associated with ith neuron andfj(xj (t)) is the output of the jth neuron of the network. ci40represents the rate with which the ith unit will reset its potential tothe resting state in isolation when disconnection from the networkand the external inputs. aij, bij denote the strengths of connectivitybetween the cell i and j at time t and at time t�tijðtÞ, respectively.tijðtÞ are time-varying delays of the neural network satisfying0rtijðtÞrt. hik is the impulsive controller. The impulsive time tk,k¼1,2,yare constants and satisfy 0¼ t0ot1o � � �otkotkþ1o � � �with tk-1 as k-1. fðtÞ ¼ ðf1ðtÞ, . . . ,fnðtÞÞ

T APC and JfJtoþ1.System (2.1) is considered as drive system. And the response

system for (2.1) is constructed as follows

_yiðtÞ ¼ �ciyiðtÞþXn

j ¼ 1

aijfjðyjðtÞÞþXn

j ¼ 1

bijfjðyjðt�tijðtÞÞÞ

þXn

j ¼ 1

kð1Þij ðxjðtÞþyjðtÞÞ

þXn

j ¼ 1

kð2Þij ðxjðt�tijðtÞÞþyjðt�tijðtÞÞÞ, tatk, tZ0,

DyiðtkÞ ¼ yiðtkÞ�yiðt�k Þ ¼ hikyiðt

�k Þ, t¼ tk, k¼ 1,2, . . . ,

yiðtÞ ¼ciðtÞ, tA ½�t,0�, i¼ 1, . . . ,n,

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð2:2Þ

where K ð1Þ ¼ ðkð1Þij Þn�n and K ð2Þ ¼ ðkð2Þij Þn�n are real controller gainmatrices. cðtÞ ¼ ðc1ðtÞ, . . . ,cnðtÞÞ

T APC and JcJtoþ1.Letting eiðtÞ ¼ xiðtÞþyiðtÞ be the anti-synchronization error,

where eðtÞ ¼ ðe1ðtÞ, . . . ,enðtÞÞT , xi(t) and yi(t) are the state variables

of drive system (2.1) and response system (2.2). Thus, we can derivethe error dynamical system as follows:

_eiðtÞ ¼�cieiðtÞþXn

j ¼ 1

aijgjðejðtÞÞþXn

j ¼ 1

bijgjðejðt�tijðtÞÞÞ

þXn

j ¼ 1

kð1Þij ejðtÞþXn

j ¼ 1

kð2Þij ejðt�tijðtÞÞ, tatk, tZ0,

DeiðtkÞ ¼ eiðtkÞ�eiðt�k Þ ¼ hikeiðt

�k Þ, t¼ tk, k¼ 1,2, . . . ,

eiðtÞ ¼fiðtÞþciðtÞ, tA ½�t,0�,

8>>>>>>>>>><>>>>>>>>>>:

ð2:3Þ

where gjðejðtÞÞ ¼ fjðxjðtÞÞþ fjðyjðtÞÞ, gjðejðt�tijðtÞÞÞ ¼ fjðxjðt�tijðtÞÞÞ

þ fjðyjðt�tijðtÞÞÞ,i,j¼ 1, . . . ,n.

Definition 2.1 (Meng and Wang [8], Ren and Cao [11]). The drivesystem (2.1) and the response system (2.2) are said to be globallyexponentially anti-synchronized if for a suitably designed feedbackcontroller, for any f,cAPC, there exist constants MZ1 and l40

such that

JeðtÞJ2rMJfþcJ2te�lt , tZ0

the constant l is defined as the globally exponential anti-synchronization rate.

Throughout this paper, we make the following assumption:(H1) The neurons activation functions fið�Þ are Lipschitz-con-

tinuous, that is, there exist constants Li40 such that

jfiðxÞ�fiðyÞjrLijx�yj for all x,yAR, i¼ 1, . . . ,n: ð2:4Þ

Remark 2.1. Since the activation functions fi of the Hopfield neuralnetworks and the cellular neural networks are odd functions [8],according to the assumption (H1), gið�Þ possesses the followingproperties:

jgiðeiðtÞÞjrLijeiðtÞj ð2:5Þ

and

gið0Þ ¼ fiðxiðtÞÞþ fið�xiðtÞÞ ¼ 0, i¼ 1, . . . ,n: ð2:6Þ

Therefore, system (2.3) admits a zero solution eiðtÞ � 0,i¼ 1, . . . ,n.

Remark 2.2. From (H1), we know the problem of anti-synchroni-zation between the drive system (2.1) and the response system(2.2) is shifted into the stability of the anti-synchronization errorsystem (2.3).

In fact, from the analysis above, we can see that (2.1) and (2.2) areexponentially anti-synchronized if and only if the equilibriumpoint of (2.3) is exponentially stable for any bounded initialcondition, eðtÞ ¼fðtÞþcðtÞ, tA ½�t,0� with JfþcJtoþ1. So theglobal impulsive exponential anti-synchronization problemcan be solved if the controller gain matrices K(1) and K(2) aresuitably designed such that the zero solution of (2.3) is globallyexponentially stable.

3. Impulsive anti-synchronization of chaotic system

The following integral delay inequality is necessary to developthe main result in this paper.

Lemma 3.1. For �1osobrþ1, let ui(t) ði¼ 1, . . . ,nÞACð½s,bÞ,RÞ satisfy the following integral delay inequality:

uiðtÞre�biðt�sÞuiðsÞþR ts e�biðt�sÞ

Xn

j ¼ 1

mijujðsÞ

24

þXn

j ¼ 1

Zijujðs�tijðsÞÞ

35ds, tA ½s,bÞ,

uiðsþsÞ ¼jiðsÞ, sA ½�t,0�,

8>>>>>>>>><>>>>>>>>>:

ð3:1Þ

where 0rtijðsÞrt,jðtÞ ¼ ðj1ðtÞ, . . . ,jnðtÞÞT APC, JjJtoþ1,bi,mij,

and Zij (i,j¼1,y,n) are positive constants. We assume that

�biþPn

j ¼ 1ðmijþZijÞo0 and

uiðtÞrM1JjJ2t , tA ½s�t,s�: ð3:2Þ

Then

uiðtÞrM1JjJ2t , tAðs,bÞ, ð3:3Þ

where i¼1,y,n, M1 is a positive constant.

Proof. To prove (3.3), we first prove for any given constant h41,

uiðtÞohM1JjJ2t for tA ½s,bÞ, i¼ 1, . . . ,n: ð3:4Þ

H. Zhao, Q. Zhang / Neurocomputing 74 (2011) 563–567 565

If (3.4) is not true, then there must be t�Aðs,bÞ and some integer

mAf1, . . . ,ng such that

umðt�Þ ¼ hM1JjJ2

t , umðtÞohM1JjJ2t , tA ½s�t,t�Þ ð3:5Þ

and

uiðtÞrhM1JjJ2t , tA ½s�t,t��, i¼ 1, . . . ,n: ð3:6Þ

By using (3.1), (3.2), (3.5), and (3.6), we have

umðt�Þre�bmðt

��sÞumðsÞþZ t�

se�bmðt

��tÞ

�Xn

j ¼ 1

mmjujðtÞþXn

j ¼ 1

Zmjujðt�tÞ

24

35dtoe�bmðt

��sÞhM1JjJ2t

þ

Z t�

se�bmðt

��tÞXn

j ¼ 1

ðmmjþZmjÞhM1JjJ2t dt

r e�bmðt��sÞ þ

1

bm

Xn

j ¼ 1

ðmmjþZmjÞð1�e�bmðt��sÞÞ

8<:

9=;hM1JjJ2

t

ohM1JjJ2t , ð3:7Þ

which implies umðt�ÞohM1JjJ2t , this contradicts the first equality in

(3.5). So (3.4) holds. Let h-1, then (3.3) holds. The proof is

complete. &

We make the following assumption.(H2) There exist positive constants pi, i¼1, y, n, such that

�2ðci�kð1Þii ÞþXn

j ¼ 1

ðjaijjLjþjbijjLjþjkð2Þij jþjk

�ijjÞ

þXn

j ¼ 1

ðjaijjLjþjbijjLjþjkð2Þij jþjk

�ijjÞpip

�1j o0, ð3:8Þ

where k�ij ¼ kð1Þij as ia j, and kij*¼0 as i¼ j.From Lemma 3.1, we have the following result.

Theorem 3.1. If (H1) and (H2) hold. Assume furthermore that system

(2.3) satisfies the following condition:

(H3) Let dkZ1 satisfies the following inequality:

ð1þhikÞ2rdk, i¼ 1, . . . ,n, k¼ 1,2, . . . ð3:9Þ

and there exists a constant aZ0 such that

lndk

tk�tk�1raoe,

where the positive constant e is determined by the following inequality:

�biþXn

j ¼ 1

ðmijþZijÞo0 ð3:10Þ

with bi ¼ 2ðci�kð1Þii Þ�e�Pn

j ¼ 1ðjaijjLjþjbijjLjþjkð2Þij jþjk

�ijjÞ40, mij ¼

ðjaijjLjþjk�ijjÞpip

�1j , Zij ¼ ðjbijjLjþjk

ð2Þij jÞpip

�1j eet, i,j¼1,y,n.

Then the zero solution of (2.3) is globally exponentially stable,

implying that the two systems (2.1) and (2.2) are globally impulsively

exponentially anti-synchronized.

Proof. From the condition (H2) and by using continuity, we knowthat (3.10) has at least one positive solution e. Let

uiðtÞ ¼ eetpie2i ðtÞ, i¼ 1, . . . ,n: ð3:11Þ

Calculating the upper right derivative DþuiðtÞ along the solutions

of (2.3). From the condition (H1), we know for tAðtk�1,tkÞ,

i¼ 1, . . . ,n, k¼ 1, . . .,

DþuiðtÞ ¼ eeetpie2i ðtÞþ2eetpieiðtÞ_eiðtÞ

¼ euiðtÞþ2eetpieiðtÞ �cieiðtÞþXn

j ¼ 1

aijgjðejðtÞÞ

8<:

þXn

j ¼ 1

bijgjðejðt�tijðtÞÞÞþXn

j ¼ 1

kð1Þij ejðtÞþXn

j ¼ 1

kð2Þij ejðt�tijðtÞÞ

9=;

reuiðtÞþ2eetpieiðtÞ �ðci�kð1Þii ÞeiðtÞþXn

j ¼ 1

jaijjLjjejðtÞj

8<:

þXn

j ¼ 1

jbijjLjjejðt�tijðtÞÞjþXn

j ¼ 1,ja i

jkð1Þij jjejðtÞj

þXn

j ¼ 1

jkð2Þij jjejðt�tijðtÞÞj

9=;

reuiðtÞþ �2ðci�kð1Þii ÞuiðtÞþXn

j ¼ 1

jaijjLjeetpiðe

2i ðtÞþe2

j ðtÞÞ

8<:

þXn

j ¼ 1

jbijjLjeetpiðe

2i ðtÞþe2

j ðt�tijðtÞÞÞþXn

j ¼ 1,ja i

jkð1Þij jeetpiðe

2i ðtÞ

þe2j ðtÞÞþ

Xn

j ¼ 1

jkð2Þij jeetpiðe

2i ðtÞþe2

j ðt�tijðtÞÞÞ

9=;

r �2ðci�kð1Þii ÞþeþXn

j ¼ 1

ðjaijjLjþjbijjLjþjkð2Þij jþjk

�ijjÞ

24

35uiðtÞ

þXn

j ¼ 1

½jaijjLjpip�1j þjk

�ijjpip

�1j �ujðtÞ

þXn

j ¼ 1

ðjbijjLjpip�1j þjk

ð2Þij jpip

�1j Þe

etujðt�tijðtÞÞ

¼�biuiðtÞþXn

j ¼ 1

mijujðtÞþXn

j ¼ 1

Zijujðt�tijðtÞÞ: ð3:12Þ

Thus we have

uiðtÞre�bituið0Þþ

Z t

0e�biðt�sÞ

Xn

j ¼ 1

mijujðsÞþXn

j ¼ 1

Zijujðs�tijðsÞÞ

24

35ds

with the initial condition uiðtÞ ¼jiðtÞ,tA ½�t,0�, jiðtÞ ¼fiðtÞ

þciðtÞ, i¼ 1, . . . ,n.

For the initial condition jðtÞ ¼ ðj1ðtÞ, . . . ,jnðtÞÞT APC, we can get

uiðtÞ ¼ eetpie2i ðtÞrpie

2i ðtÞ ¼ pijfiðtÞþciðtÞj

2

rpiJjJ2t , �trtr0, i¼ 1, . . . ,n: ð3:13Þ

So all the conditions of Lemma 3.1 hold. Then we obtain that

uiðtÞrpiJjJ2t , tA ½0,t1Þ, i¼ 1, . . . ,n: ð3:14Þ

In the following, we will use the mathematical induction to prove

uiðtÞrd0d1 � � � dk�1piJjJ2t , tA ½tk�1,tkÞ, i¼ 1, . . . ,n, k¼ 1,2, . . . ,

ð3:15Þ

where d0 ¼ 1.

When k¼1, we know from (3.14) that (3.15) holds. Suppose that

the inequalities

uiðtÞrd0d1 � � � dk�1piJjJ2t , tA ½tk�1,tkÞ, i¼ 1, . . . ,n, k¼ 1, . . . ,m

ð3:16Þ

hold.

H. Zhao, Q. Zhang / Neurocomputing 74 (2011) 563–567566

From (H3) and (3.11), we have

uiðtmÞ ¼ eetm pie2i ðtmÞ ¼ eetm pið1þhimÞ

2e2i ðt�mÞreetmdmpie

2i ðt�mÞ

¼ dmuiðt�mÞrd0d1 � � �dm�1dmpiJjJ2

t , i¼ 1, . . . ,n:

This, together with (3.16), leads to

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−5

−4

−3

−2

−1

0

1

2

3

4

5

x1(t)

x2(t)

Fig. 1. The chaotic behavior of system (4.1).

_y1ðtÞ ¼�y1ðtÞþ2:1f1ðy1ðtÞÞ�0:12f2ðy2ðtÞÞ�1:6f1ðy1ðt�1ÞÞ�0:1f2ðy2ðt�1ÞÞ

�20ðx1ðtÞþy1ðtÞÞþ0:1ðx2ðtÞþy2ðtÞÞþ8ðx1ðt�1Þþy1ðt�1ÞÞ,

_y2ðtÞ ¼�y2ðtÞ�5:1f1ðy1ðtÞÞþ3:2f2ðy2ðtÞÞ�0:2f1ðy1ðt�1ÞÞ�2:4f2ðy2ðt�1ÞÞ

þ0:1ðx1ðtÞþy1ðtÞÞ�20ðx2ðtÞþy2ðtÞÞþ8ðx2ðt�1Þþy2ðt�1ÞÞ, tatk,

Dy1ðtkÞ ¼ �0:1y1ðt�k Þ, Dy2ðtkÞ ¼ �0:1y2ðt

�k Þ, t¼ tk,tZ0,

y1ðtÞ ¼ 1:5sint, y2ðtÞ ¼ e2t , tA ½�1,0�, k¼ 1, . . . :

8>>>>>>>>><>>>>>>>>>:

ð4:2Þ

0

2

4

6

8

10

t) an

d e2

(t)

e1(t)

e2(t)

uiðtÞrd0d1 � � � dm�1dmpiJjJ2t , tA ½�t,tm�, i¼ 1, . . . ,n: ð3:17Þ

Employing Lemma 3.1 again, we can prove that

uiðtÞrd0d1 � � � dm�1dmpiJjJ2t , tA ½tm,tmþ1Þ, i¼ 1, . . . ,n: ð3:18Þ

By the mathematical induction, we conclude that (3.15) holds.

Furthermore, from (H3) we have

uiðtÞrd0d1 � � � dk�1piJjJ2treatk�1 piJjJ2

t

reatpiJjJ2t ,tA ½tk�1,tkÞ, i¼ 1, . . . ,n, k¼ 1,2, . . . : ð3:19Þ

That is

e2i ðtÞr

p

pe�ðe�aÞtJjJ2

t , i¼ 1, . . . ,n: ð3:20Þ

Therefore

JeðtÞJ2rMJjJ2te�lt , i¼ 1, . . . ,n, ð3:21Þ

where M¼ np=pZ1 and l¼ e�a40. The proof is complete. &

Corollary 3.1. If (H1) holds. Assume furthermore that the conditions

(H2)–(H3) with pi � 1ði¼ 1, . . . ,nÞ are satisfied, then the zero solution

of (2.3) is also globally exponentially stable, implying that the two

systems (2.1) and (2.2) are globally impulsively exponentially anti-

synchronized.

Remark 3.1. The paper studied exponential anti-synchronizationproblem with impulsive perturbation, which reflects more realisticdynamics than the those in [8,11,25], where the impulsive perturbationwas ignored. Meanwhile, the controller gain matrices adopted here cantackle more general systems than that in [8,25]. In addition, themethods of this paper may extend to study adaptive anti-synchroniza-tion problems for chaotic neural networks with time-varying and forchaotic neural networks without impulsive such as [8,11,25].

4. Illustrative example

−10 0 10 20 30 40 50 60−10

−8

−6

−4

−2

time t

erro

r e1(

Fig. 2. The anti-synchronization error e1(t), e2(t) without the controller gain

matrices in (4.2), respectively.

Example. Consider a chaotic delayed Hopfield neural network asfollows:

_x1ðtÞ ¼�x1ðtÞþ2:1f1ðx1ðtÞÞ�0:12f2ðx2ðtÞÞ

�1:6f1ðx1ðt�1ÞÞ�0:1f2ðx2ðt�1ÞÞ,

_x2ðtÞ ¼�x2ðtÞ�5:1f1ðx1ðtÞÞþ3:2f2ðx2ðtÞÞ

�0:2f1ðx1ðt�1ÞÞ�2:4f2ðx2ðt�1ÞÞ, tatk,

Dx1ðtkÞ ¼�0:1x1ðt�k Þ,Dx2ðtkÞ ¼�0:1x2ðt

�k Þ, t¼ tk,tZ0,

x1ðtÞ ¼ cost, x2ðtÞ ¼ 5sint, tA ½�1,0�, k¼ 1, . . . ,

8>>>>>>>>><>>>>>>>>>:

ð4:1Þ

where fiðxiÞ ¼ tanhðxiÞ, i¼ 1,2, tk ¼ tk�1þk, t¼ 1. Clearly, thecondition (H1) with Li¼1(i¼1,2) holds.

As shown in Fig. 1, system (4.1) possesses a chaotic behavior.

Now the response system is designed as follows:

Note that the controller gain matrices K(1) and K(2) are chosen asfollows:

K ð1Þ ¼�20 0:1

0:1 �20

� �, K ð2Þ ¼

8 0

0 8

� �:

0 5 10 15 20 25 30 35 40 45 50−2

−1.5

−1

−0.5

0

0.5

1

time t

erro

r e1(

t) an

d e2

(t)

e1(t)

e2(t)

Fig. 3. The anti-synchronization error e1(t), e2(t) with the controller gain matrices in

(4.2), respectively.

H. Zhao, Q. Zhang / Neurocomputing 74 (2011) 563–567 567

Let e¼ 0:1, p1¼2, p2¼1. It is easy to check that condition (H2) isalso satisfied.

Choose dk ¼ 1 and a¼ 0, then the condition (H3) is satisfied withl¼ e�a¼ 0:1.

It follows from Theorem 3.1 that systems (4.1) and (4.2) areimpulsively exponentially anti-synchronized.

Figs. 2 and 3 depict the anti-synchronization error of the statevariables between the drive system and the response systemwithout or with the controller gain matrices in (4.2), respectively.

From Figs. 2 and 3, we see that the controller gain matricescontribute to the chaos anti-synchronization.

5. Conclusion

This paper studies two coupling chaotic systems. By establishingan integral delay inequality and using the inequality method, weobtain some new criteria for the impulsive exponential anti-synchronization. An example is given to show the effectiveness ofthe obtained criteria. In addition, the methods of this paper mayextend to discuss other systems [8,11,25].

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Hongyong Zhao received the Ph.D. degree from SichuanUniversity, Chengdu, China, and the Post-DoctoralFellow in the Department of Mathematics at NanjingUniversity, Nanjing, China.

He was with the Department of Mathematics atNanjing University of Aeronautics and Astronautics,Nanjing, China. He is currently a Professor of NanjingUniversity of Aeronautics and Astronautics, Nanjing,China. He also is the author or coauthor of more than 40journal papers. His research interests include nonlineardynamic systems, neural networks, control theory, andapplied mathematics.

Zhang Qi a teacher of Nanjing University of Aeronauticsand Astronautics. She received her M.S. degree in puremathematics from Ningbo University in 2005 and D.S.degree in applied mathematics from Nanjing Universityin 2008. Her research interests include stochasticdifferential equation, impulsive differential equationand neural networks.