Synchronization of chaotic systems with delay using intermittent linear statefeedbackTingwen Huang, Chuandong Li, and Xinzhi Liu Citation: Chaos: An Interdisciplinary Journal of Nonlinear Science 18, 033122 (2008); doi: 10.1063/1.2967848 View online: http://dx.doi.org/10.1063/1.2967848 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/18/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Amplitude death and synchronized states in nonlinear time-delay systems coupled through mean-fielddiffusion Chaos 23, 043101 (2013); 10.1063/1.4823599 Anticipating synchronization of chaotic systems with time delay and parameter mismatch Chaos 19, 013104 (2009); 10.1063/1.3013600 Response to “Comment on ‘Synchronization of chaotic systems with delay using intermittent linear statefeedback’ ” [Chaos18, 048101 (2008)] Chaos 18, 048102 (2008); 10.1063/1.3046536 Comment on “Synchronization of chaotic systems with delay using intermittent linear state feedback”[Chaos18, 033122 (2008)] Chaos 18, 048101 (2008); 10.1063/1.3046535 Synchronization in networks of chaotic systems with time-delay coupling Chaos 18, 037108 (2008); 10.1063/1.2952450

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

Synchronization of chaotic systems with delay using intermittent linearstate feedback

Tingwen Huang,1 Chuandong Li,2 and Xinzhi Liu3

1Texas A&M University at Qatar, c/o Qatar Foundation, P.O. Box 5825, Doha, Qatar2College of Computer Science, Chongqing University, 400030, China3Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

�Received 10 April 2008; accepted 14 July 2008; published online 25 August 2008�

This paper investigates the synchronization of coupled chaotic systems with time delay by usingintermittent linear state feedback control. An exponential synchronization criterion is obtained bymeans of Lyapunov function and differential inequality method. Numerical simulations on thechaotic Ikeda and Lu systems are given to demonstrate the effectiveness of the theoreticalresults. © 2008 American Institute of Physics. �DOI: 10.1063/1.2967848�

Chaos synchronization is a contemporary topic in nonlin-ear science with applications to diverse areas such as se-cure communications, biological chemical reactions, etc.It has attracted considerable attention from researchersin different areas. Many new effective methods have beenproposed to synchronize chaotic systems. A commonlyused approach is feedback control which includes con-tinuous feedback and discontinuous feedback. Discon-tinuous control dynamical systems are governed by com-plicated mathematical models displaying irregulardynamical behaviors with interesting challenges. Re-cently, the discontinuous feedback control approach, suchas impulsive control, has been explored extensively. How-ever, the intermittent controls which are activated onsome intervals should deserve more investigation on syn-chronization of chaotic systems, especially, the chaoticsystems with delays.

I. INTRODUCTION

As a basic feature in nonlinear science and its applica-tions to diverse areas such as secure communications andbiological chemical reactions, etc., chaos synchronizationhas been known for a rather long time, starting with Fujisakaet al.1 Since then, it has been developed and extensivelyinvestigated. �For a more comprehensive reference, pleaserefer to the review article by Boccaletti et al.2� Chaotic sys-tems are difficult to control due to their sensitiveness; i.e.,sensitive dependence on initial conditions. Accordingly, syn-chronization between two chaotic systems was mistakenlyconsidered to be impossible. However, up to now, a lot ofinvestigations on this subject have been done, and many im-portant and fundamental results have been reported on thesynchronization and control of chaotic systems by research-ers from physics, engineering, biology, mathematics, etc.Lots of control approaches have been developed to synchro-nize chaotic systems such as drive-response,3 couplingcontrol,4 adaptive control,5,6 feedback control,7–9 fuzzycontrol,10 observer-based control,11 manifold-based method12

and impulsive control,13–16 intermittent control,17,18 etc. Re-cently, the discontinuous feedback control approaches, such

as impulsive13–16 and switched,19 have received much inter-est because they are practical and easily implemented in en-gineering such as transportation and communication.

The discontinuous control dynamical systems are subjectto the complicated mathematical models with more chal-lenges. The intermittent control is different from the impul-sive control since impulsive control is activated only at someisolated instants, while intermittent control has a nonzerocontrol width. In this scheme, the synchronization signals areused in the slave system at periodic time intervals �controlwidth� when the slave system tracks the orbit of the drivingsystem. Recently, the intermittent control has been intro-duced to control of the nonlinear dynamical systems.18 Theauthor of Ref. 18 studied numerically synchronization ofchaotic systems without delays coupled intermittently. To thebest of our knowledge, there is little theoretical analysis ofintermittent control dynamical systems with or without de-lays in the literature. On the other hand, the occurrence oftime-delayed systems is unavoidable in nature, technology,and society due to finite signal transmission times, switchingspeeds, memory effects, etc. Some delayed chaotic systemsare easy to implement and have promising applicationssuch as secure communications using chaoticsynchronization.20–28

Synchronization of chaotic systems with delays is stud-ied in this paper by using intermittent linear state feedbackcontrol. A criterion of synchronization is rigorously derivedby Lyapunov function approach, where the relationship ofduration of the feedback control and that of the delay is ofcrucial importance. In this paper, we let the control width belonger than the time delay of the system. In addition, numeri-cal simulations are given to show the effectiveness of theproposed chaos synchronization scheme.

The paper is organized as follows. In the next section,synchronization scheme by means of intermittent linear statefeedback control is presented. In addition, some necessarypreliminaries are given there. In Sec. III, the criterion ofsynchronization is rigorously derived. In Sec. IV, a numericalexample is given to illustrate the effectiveness of the pro-posed synchronization scheme.

CHAOS 18, 033122 �2008�

1054-1500/2008/18�3�/033122/8/$23.00 © 2008 American Institute of Physics18, 033122-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

II. PROBLEM FORMULATION AND PRELIMINARIES

Consider a class of chaotic systems where a chaotic mas-ter �driver� system with delay:

dx�t�dt

= Ax�t� + Bf�x�t�� + Cg�x�t − ���, t � 0,

�1�x�t� = ��t�, − � � t � 0.

To synchronize system �1� using intermittent feedback con-trol, the slave �response� system is designed as

dy�t�dt

= Ay�t� + Bf�y�t�� + Cg�x�t − ��� + k�t��x�t� − y�t��,

t � 0,

�2��y�t�� + Cgy�t� = ��t�, − � � t � 0,

where x ,y�Rm are the state vectors of systems �1� and �2�,respectively, A ,B ,C�Rm�m, f : Rm→Rm, and g : Rm→Rm

are both continuous nonlinear functions satisfying f�0�=0and g�0�=0, � is the time delay, and k�t� is the intermittentlinear state feedback control gain defined as the following:

k�t� = �K, n� � t � n� + � ,

0, n� + � t � �n + 1�� ,

where K�Rm�m is a constant control gain, ��0 is the con-trol period, and ��0 is called the control width �controlduration�. In this paper, our goal is to design suitable �, �,and K such that the system �2� synchronizes to system �1�.

Let e�t�=y�t�−x�t� be the synchronization error betweenthe states of drive system �1� and response system �2�. Then,for t� �0, �, we have the following error system:

de�t�dt

= Ae�t� + B�f�y�t�� − f�x�t��� + C�f�y�t − ���

− f�x�t − ���� − Ke�t� ,

n� � t � n� + � ,

�3�de�t�

dt= Ae�t� + B�f�y�t�� − f�x�t��� + C�f�y�t − ���

− f�x�t − ���� ,

n� + � t � �n + 1��

As far as we know, the synchronization analysis of delayedsystems �1� and �2� has received little attention so far. How-ever, Senthilkumar et al.29,30 have investigated the phase syn-chronization and generalized synchronization in time-delaysystems, and obtained very nice results. In this paper, we willpresent a criterion of the synchronization.

In this paper, we assume that f and g are Lipschitz con-tinuous functions: there exist positive constants Lf , Lg suchthat, for all x ,y�Rm,

�f�x� − f�y�� � Lf�x − y�, �g�x� − g�y�� � Lg�x − y� . �4�

We also assume that ��� in this paper.

To prove the criterion of synchronization in the next sec-tion, we need the following lemmas.

Lemma 1: �Sanchez and Perez31� For any vectors x ,y�Rm and positive-definite matrix Q�Rm�m, the followingmatrix inequality holds:

2xTy � xTQx + yTQ−1y .

The following lemma is the well known result in differentialequation with delays.

Lemma 2: �Halanay Inequality1� Suppose that functiony�t� is non-negative when t� �−� , � and satisfies the fol-lowing:

dy�t�dt

� − k1y�t� + k2y�t − ��, t � 0, �5�

where k1 ,k2 are positive constants, and k1�k2. We then havethe following inequality:

y�t� � �y�0���e−rt, t � 0, �6�

where �y�0���= max−��s�0

�y�s�� and r is the unique positive so-

lution of

− r = − k1 + k2er�.

Lemma 3: Let u�t� is continuous and non-negative func-tions on I= �a−� ,b� ���0�, for which the following inequal-ity,

u��t� �1u�t� + �2u�t − ��, t � �a,b� ,

holds. Then,

u�t� � �u�a��� exp��1 + �2�t, t � �a,b� ,

where �u�a���=maxa−��t�a �u�t� � .Proof: Let z�t�= �u�a���e��1+�2�t for t� �a ,b�. We need to

prove u�t��z�t�, or u�t�−z�t��0 for t� �a ,b�. Suppose it isnot correct; there is a t0� �a ,b� such that

u�t0� − z�t0� = 0, u��t0� − z��t0� � 0,

�1��u�t� − z�t� � 0, t � �a,t0� .

We always have u�t0−���z�t0�, since if

t0 − � � �a − �,a�, u�t0 − �� � �u�a��� = z�a� � z�t0� ,

or

t0 − � � �a,t0�, u�t0 − �� � �u�t0 − ���� � z�t0 − �� � z�t0� .

Evaluate and estimate u��t0�−z��t0� as

u��t0� − z��t0� �1u�t0� + �2u�t0 − �� − ��1 + �2�z�t0�

� �1z�t0� + �2z�t0� − ��1 + �2�z�t0� = 0.

This is contradicted by Eq. �1��. Thus, u�t��z�t� for t� �a ,b�. We have completed the proof. �

Throughout this paper, P�0�0, �0, �0� denotes asymmetrical positive �negative, semi-negative, semi-positive� definite matrix P; also, PT , �M�m��P� are the trans-pose and the maximum �minimum� eigenvalue of a squarematrix P, respectively. The vector �or matrix� norm is takento be Euclidian, denoted by � · �.

033122-2 Huang, Li, and Liu Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

III. CRITERIA FOR SYNCHRONIZATION

In this section, we rigorously derive some criteria forsynchronization through intermittent linear state feedbackusing a Lyapunov function and differential inequality ap-proach. The main result is stated as follows.

Theorem 1: Suppose there exist positive definite matrixP�0 and positive constants , � , � , � , �2 , �2 , �1 , �1

such that the following conditions hold:

�a� PA+ATP+ −1Lf2I+ PBBTP− PK−KTP+�1P�0,

�b� �PCCTP+�−1Lg2I−�2P�0,

�c� PA+ATP+�−1Lf2I+�PBBTP−�1P�0,

�d� �PCCTP+�−1Lg2I−�2P0,

�e� �=r��−��− ��1+�2���−���0,

where r is the unique positive solution to −r=−�1+�2er�.Then system �1� and system �2� achieve exponential syn-chronization.

Proof: Consider the following Lyapunov function:

V�t� = e�t�TPe�t� . �5��

We shall calculate the derivative V�t� defined by Eq. �5� withrespect to time t along the trajectory of error system �3�, andestimate it.

For k�� t�k�+�, using condition �a� and condition �b�in addition to Lemma 1 and assumption �4�, we have thefollowing:

V�t� = 2e�t�TPe�t�

= 2e�t�TP�Ae�t� + B�f�y�t�� − f�x�t��� + C�g�y�t − ���

− g�x�t − ���� − Ke�t��

� e�t�T�PA + ATP − PK − KTP�e�t� + e�t�TPBBTPe�t�

+ −1�f�y�t�� − f�x�t���2 + �e�t − ��TPCCTPe�t − ��

+ �−1�g�y�t − ��� − g�x�t − ����2

� e�t�T�PA + ATP + −1Lf2I + PBBTP − PK − KTP�e�t�

+ e�t − ��T��PCCTP + �−1Lg2I�e�t − ��

= e�t�T�PA + ATP + −1Lf2I + PBBTP − PK − KTP

+ �1P�e�t� − �1e�t�TPe�t�

+ e�t − ��T��PCCTP + �−1Lg2I − �2P�e�t − ��

+ �2e�t − ��TPe�t − ��

� − �1e�t�TPe�t� + �2e�t − ��TPe�t − ��

= − �1V�t� + �2V�t − �� .

Namely, we have

V�t� � − �1V�t� + �2V�t − ��, for n� � t � n� + � . �6��

For k�+�� t� �k+1��, using condition �c� and condition�d� in addition to Lemma 1 and assumption �4�, we have thefollowing:

V�t� = 2e�t�TPe�t�

= 2e�t�TP�Ae�t� + B�f�y�t�� − f�x�t��� + C�g�y�t − ���

− g�x�t − �����

� e�t�T�PA + ATP�e�t� + �e�t�TPBBTPe�t�

+ �−1�f�y�t�� − f�x�t���2 + �e�t − ��TPCCTPe�t − ��

+ �−1�g�y�t − ��� − g�x�t − ����2

� e�t�T�PA + ATP + �−1Lf2I + �PBBTP�e�t�

+ e�t − ��T��PCCTP + �−1Lg2I�e�t − ��

= e�t�T�PA + ATP + �−1Lf2I + �PBBTP − �1P�e�t�

+ �1e�t�TPe�t� + e�t − ��T��PCCTP + �−1Lg2I − �2P�

�e�t − �� + �2e�t − ��TPe�t − ��

�1e�t�TPe�t� + �1e�t − ��TPe�t − ��

= �1V�t� + �2V�t − �� .

Namely, we have the following:

V�t� �1V�t� + �2V�t − ��, for n� + � � t � �n + 1�� . �7�

In the following, we will prove the error �e�t� � →0, in otherwords, system �1� and system �2� are synchronizing undercondition �e� in addition to conditions �a�–�d�. By HalanayInequality �Lemma 2� and differential inequality �6�, wehave

V�t� � �V�0���e−rt for 0 � t � � , �8�

where r is the unique positive solution to −r=−�1+�2er�.For �� t��, using Lemma 3, we have the following:

V�t� � �V�����e��1+�2�t. �9�

By Eq. �8�, we have for t� �0,��,

V�t� � �V�0���e−rt.

From the above inequality, we are able to obtain the follow-ing:

�V����� = max�−��t��

�V�t�� � �V�0���e−r��−��.

By the above inequality and Eq. �9�, for �� t��, we have

V�t� � �V�����e��1+�2��t−�� � �V�0���e

−r��−��e��1+�2��t−��.

Especially,

V��� � �V�0���e−r��−��+��1+�2���−�� = �V�0���e

−�,

where �=r��−��− ��1+�2���−��.It is clear that

�V����� = max�−��t��

�V�t��

� max�−��t��

��V�0���e−r��−��e��1+�2��t−��

� max�−��t��

�V�0���e−r��−��e��1+�2���−��

and

033122-3 Synchronization of chaotic systems Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

�V����� � �V�0���e−��1−�2���−��+��1+�2���−�� = �V�0���e

−�. �10�

Using mathematical induction, we can prove, for positiveinteger k,

V�k�� � �V�0���e−k�. �11�

Assume Eq. �11� holds when k� l. Now, we prove Eq. �11� istrue when k= l+1. Similar to the process to obtain Eq. �10�,we have

�V�l���� � �V�0���e−l�.

When t� �l� , l�+��,

V�t� � �V�l����e−r�t−l�� � �V�0���e

−l�e−r�t−l��.

Thus, we have

�V�l� + ���� � �V�0���e−l�e−r�.

For t� �l�+� , �l+1���,

V�t� � �V�l� + ����e��1+�2��t−l�−��

� �V�0���e−l�e−r�e��1+�2��t−l�−��.

Especially, when t= l�+�, we have

�V��l + 1���� � �V�0���e−l�e−r�e��1+�2���−��

� �V�0���e−�l+1��.

Thus, Eq. �11� is true for all positive integers.For any t�0, there is a n0�0, such that n0�� t

� �n0+1��.

V�t� � �V�0���e��1+�2��e−n0� � �V�0���e

��1+�2��e� exp−�

�t� .

Let M = �V�0���e��1+�2��e�. We have

V�t� � M exp−�

�t�, for t � 0. �12�

By Eqs. �5� and �12�, we have

�m����e�t��2 � M exp−�

�t� .

Thus, we have obtained the following:

�e�t�� �� M

�m���exp−

�

2�t�, for t � 0,

which means that the two systems synchronize exponentially.The proof is complete. �

Remark 1: The result in Theorem 1 includes the generalchaotic systems. If B ,C are identity matrices, and we choosethe Lyapunov function as V�t�= �x�t��2; in other words, P= I,also, the intermittent control gain K=kI, then we have thefollowing result.

Corollary 1: Suppose that there exist positive constants , � , � , � such that the following conditions hold:

�a� A+AT+ � −1Lf2+ −2k+�1�I�0,

�b� �+�−1Lg2−�2�0,

�c� A+AT+ ��−1Lf2+�−�1�I�0,

�d� �+�−1Lg2−�2�0, and

�e� �=r��−��− ��1+�2���−���0,

where r is the unique positive solution to −r=−�1+�2er�.Then the systems �1� and �2� achieve exponential synchroni-zation.

Remark 2: If scalar conditions �M�A+AT�+ � −1Lf

2+ −2k+�1��0 and �M�A+AT�+ ��−1Lf

2+�−�1��0 hold, then conditions �a� and �c� in theabove Corollary 1 hold. Thus, the conditions for criterion ofthe given chaotic systems are all given by scalar conditions.From the proof of Theorem 1, we know that � is the rate ofconvergence, so the bigger value of �, the faster conver-gence.

Corollary 2: Suppose there exist positive constants , � , � , � , �2 , �2 , �1 , �1 such that the following condi-tions hold:

�a� �M�A+AT�+ � −1Lf2+ −2k+�1��0,

�b� �+�−1Lg2−�2�0,

�c� �M�A+AT�+ ��−1Lf2+�−�1��0,

�d� �+�−1Lg2−�2�0,

�e� �=r��−��− ��1+�2���−���0,

where r is the unique positive solution to −r=−�1+�2er�.Then the systems �1� and �2� achieve exponential synchroni-zation.

Remark 3: Since �+�−1Lg2�2Lg and �+�−1Lg

2�2Lg,we let �2=�2=2Lg and �=�=Lg, then conditions �b� and �d�are satisfied. Furthermore, we chose =�=Lf. Thus, we canfurther reduce the result in the above corollary to the follow-ing.

Corollary 3: Suppose there exist positive constants�1 ,�1 such that the following conditions hold:

�i� �M�A+AT�+2Lf −2k+�1�0,�ii� �M�A+AT�+2Lf −�1�0,�iii� �=r��−��− ��1+2Lf���−���0,

where r is the unique positive solution to −r=−�1+�2er�.Then the systems �1� and �2� achieve exponential synchroni-zation.

To end this section, we show the following claim holdsby means of Corollary 3.

Corollary 4: Given control period � and control dura-tion �. Systems �1� and �2� achieve exponential synchroniza-tion, if the control strength k satisfies

k � ��r*� , �13�

where

��r� =1

2�M�A + AT� + Lf +

r

2+ Lger� �14�

and

r* =� − �

� − ���M�A + AT� + 2Lf + 2Lg� � 0. �15�

033122-4 Huang, Li, and Liu Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

Proof: In Corollary 3, we chose �1=2k−�M�A+AT�−2Lf and v1=�M�A+AT�+2Lf. Obviously, the conditions�i� and �ii� in Corollary 3 hold. Note that the condition �iii� inCorollary 3 is equivalent to r�r*. Substituting �1, �1 chosenabove into the equation −r=−�1+�2er� yields k=��r�.Since ��r� is increasing monotonically with respect to theargument r, the condition k���r*� leads to the condition�iii� in Corollary. This concludes the proof. �

Remark 4: Corollary 4 allows us to determine the con-trol strength in a simple way provided that the control period� and control duration � are given. In addition, in case thatonly the control period is known, one can estimate the fea-

sible region D of control parameter �� ,k�, or alternately, con-trol parameter �� ,k�, where �=� /�; i.e.,

D = ��,k��k �12

�M�A + AT� + Lf + r*

2+ Lger*�,

r* =��1 − ���� − �

��M�A + AT� + 2Lf + 2Lg� � 0� .

�16�

IV. NUMERICAL EXAMPLES

In this section, we present two examples to show theeffectiveness of the proposed results.

Example 1: Consider the Ikeda-type oscillator of theform

x�t� = − ax�t� + b sin�x�t − ��� . �17�

This system exhibits chaotic behavior when �=0.5, a=1, andb=8, as shown in Fig. 1. We assume that the response systemassociated with Eq. �17� is of the form

y�t� = − ay�t� + b sin�y�t − ��� + u�t��x�t� − y�t�� , �18�

where

u�t� = �k, t0 + n� � t � t0 + n� + � ,

0, t0 + n� + � t � t0 + �n + 1�� .�19�

Note that the parameters involved in Corollary 4 are as fol-lows: A=−1, Lf =0, and Lg=8. Given �, one can plot thefeasible region D of the control parameters �� ,k�, as shownin Fig. 2, where the region above the curve for every � is justthe feasible region D. For numerical simulation, we chosethe parameters �� ,k�= �0.92,16� that are included in D, and

FIG. 1. �Color online� The chaotic attractor of the time-delay system �17�.

FIG. 2. �Color online� The feasible re-gions of the control parameters �� ,k�for different control periods in Ex-ample 1.

033122-5 Synchronization of chaotic systems Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

FIG. 3. �Color online� Synchroniza-tion error in absolute form in Example1. The initial values for drive and re-sponse systems are, respectively, x���=−2 and y���=3, �� �−0.5,0�.

FIG. 4. �Color online� The chaotic at-tractor of Lu oscillator �Eq. �20��.

033122-6 Huang, Li, and Liu Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

plot the error curve in the absolute form, as shown in Fig. 3.Example 2: Consider Lu oscillator32

x�t� = − Ax�t� + f�x�t�� + g�x�t − ��� , �20�

where �=1,

A = 1 0

0 1�, f�x� = 3.0 5.0

0.1 2.0�tanh�x1�

tanh�x2�� ,

and

FIG. 5. �Color online� The feasible re-gions of the control parameters �� ,k�for different control periods in Ex-ample 2.

FIG. 6. �Color online� Synchroniza-tion error in absolute form in Example2. The initial values for drive and re-sponse systems are, respectively, x���= �−0.2 0.3�T and y���= �0.2 −0.3�T;�� �−1, 0�.

033122-7 Synchronization of chaotic systems Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58

g�y� = − 2.5 0.2

0.1 − 1.5�tanh�y1�

tanh�y2�� .

This model was investigated by Lu in Ref. 32, where itis shown to be chaotic, as shown in Fig. 4. For numericalsimulation, we assume that the slave system associated withEq. �20� is of the form

y�t� = − Ay�t� + f�y�t�� + g�y�t − ��� + u�t��x�t − �� − y�t�� ,

�21�

where u�t� is defined by Eq. �19�.Note that the parameters involved in Corollary 4 are as

follows: Lf =6.0989 and Lg=2.5226. Given �, one can plotthe feasible region D of the control parameters �� ,k�, asshown in Fig. 5, where the region above the curve for every� is just the feasible region D. For numerical simulation, wechose the parameters �� ,k�= �0.96,11� that are included inD, and plot the error curve in the absolute form, as shown inFig. 6.

ACKNOWLEDGMENTS

The authors highly appreciate reviewers’ detailed com-ments and suggestions. Also, the first author appreciates Mr.Greg Klein at the Department of Mathematics of Texas A&MUniversity for his polishing English. The work described inthis paper was partially supported by the National NaturalScience Foundation of China �Grant No. 60574024 and60774051� and Program for New Century Excellent Talentsin University of China �Grant No. NCET-06-0764�.

1H. Fujisaka and T. Yamada, Prog. Theor. Phys. 69, 32 �1983�.2S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou,Phys. Rep. 366, 1 �2002�.

3E. Ott, G. Grebogi, and J. A. Yorke, Phys. Rev. Lett. 64, 1196 �1990�.

4J. F. Heagy, T. L. Carroll, and L. M. Pecora, Phys. Rev. E 50, 1874,�1994�.

5E. M. Elabbasy, H. N. Agiza, and M. M. El-Dessoky, Chaos, SolitonsFractals 30, 1133 �2006�.

6M. T. Yassen, Chaos, Solitons Fractals 37, 465 �2008�.7J. H. Park, S. M. Lee, and O. M. Kwon, Phys. Lett. A 371, 263 �2007�.8M. Rafikov and J. M. Balthazar, Commun. Nonlinear Sci. Numer. Simul.

13, 1246 �2008�.9I. Masubuchi and M. Tsutsui, in Proc. 40th IEEE Conf. Dec. Control,2001, pp. 2466–2471.

10K. Tanaka and H. O. Wang, in IEEE World Congress on Fuzzy SystemsProceedings, Vol. 1, pp. 434–439.

11H. Nijmeijer and I. M. Y. Mareels, IEEE Trans. Circuits Syst., I: Fundam.Theory Appl. 44, 882 �1997�.

12J. Q. Fang, Y. Hong, and G. Chen, Phys. Rev. E 59, R2523 �1999�.13T. Yang and L. O. Chua, IEEE Trans. Circuits Syst., I: Fundam. Theory

Appl. 44, 976 �1997�.14C. D. Li, X. F. Liao, X. F. Yang, and T. W. Huang, Chaos 15, 043103

�2005�.15A. Khadra, X. Liu, and X. Shen, IEEE Trans. Circuits Syst., I: Fundam.

Theory Appl. 50, 341 �2003�.16A. Khadra, X. Liu, and X. Shen, Automatica 41, 1491 �2005�.17C. D. Li, X. F. Liao, and T. W. Huang, Chaos 17, 013103 �2007�.18M. Zochowski, Physica D 145, 181 �2000�.19C. D. Li, X. F. Liao, and X. F. Yang, Chaos 16, 033104 �2006�.20D. Yue, Q. L. Han, and C. Peng, IEEE Trans. Circuits Syst., II: Express

Briefs 52, 204 �2005�.21S. S. Yang and C. K. Duan, J. Phys. A: Math Gen. 30, 3273 �1997�.22H. N. Agiza, Nonlinear Anal. Theory, Methods Appl. 58, 11 �2004�.23H. T. Lu and C. Leeuwen, Chaos, Solitons Fractals 30, 166 �2006�.24S. Boyd, L. El Ghaoui, and V. Balakrishnan, Linear matrix Inequalities in

Systems and Control Theory �SIAM, Philadelphia, 1994�.25M. C. Mackey and L. Glass, Science 197, 287 �1977�.26Chaos Control: Theory and Applications, edited by G. Chen and X. Yu

�Springer, New York, 2003�.27H. Zhu and B. Cui, Chaos 17, 043122 �2007�.28K. Ikeda, H. Daido, and O. Akimoto, Phys. Rev. Lett. 45, 709 �1980�.29D. V. Senthilkumar, M. Lakshmanan, and J. Kurths, Chaos 18, 023118

�2008�.30D. V. Senthilkumar, M. Lakshmanan, and J. Kurths, Phys. Rev. E 74,

035205�R� �2006�.31E. N. Sanchez and J. P. Perez, IEEE Trans. Circuits Syst., I: Fundam.

Theory Appl. 46, 1395 �1999�.32H. T. Lu, Phys. Lett. A 298, 109 �2002�.

033122-8 Huang, Li, and Liu Chaos 18, 033122 �2008�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

192.195.95.193 On: Sat, 21 Feb 2015 05:33:58