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Chaos, Solitons and Fractals 38 (2008) 461–464

www.elsevier.com/locate/chaos

Coexistence of anti-phase and complete synchronizationin coupled chen system via a single variable q

Chuandong Li a,*, Qian Chen b, Tingwen Huang c

a College of Computer Science and Engineering, Chongqing University 400030, Chinab College of Economics and Business Administration, Chongqing University 400030, China

c Texas A&M University at Qatar, c/o Qatar Foundation, P.O. Box 5825, Doha, Qatar

Accepted 10 November 2006

Abstract

This paper investigates a class of hybrid synchronization phenomenon in coupled identical Chen systems by linearcontrol. Theoretical analysis and numerical simulation show that part of the states of the coupled Chen system are anti-phase synchronized, and part complete-synchronized under certain parameter region.� 2006 Elsevier Ltd. All rights reserved.

1. Introduction

During the last decade, synchronization of chaotic systems has attracted increasing attention and has been exploredintensively. Several typical synchronizations have been identified as complete synchronization (CS) [1–4], phase syn-chronization (PS) [5,6], lag synchronization (LS) [4,7–10], generalized synchronization (GS) [11–13], and so on.Recently, anti-phase synchronization (AS), which is defined as vanishing of the sum of relevant variables, has been sep-arated from GS. AS phenomena have been observed experimentally and numerically in the coupled chaotic systems[14–16].

Many theoretical studies of chaos synchronization have been carried out for unidirectionally coupled identical sys-tems, usually via feedback approach. In this case, linear feedback via a single variable may be a better choice because ofits simplicity for the theoretical analysis and circuit implementation. In most cases of drive-response synchronization,all the states of the response system synchronize to the corresponding states of drive system in terms of the same syn-chronization regime. For example, when we say that two systems are complete-synchronized (or lag-synchronized orphase-synchronized, or something else) with each other, it means that each pair of the states between the interactivesystems is complete-synchronized (or lag-synchronized or phase-synchronized, or something else). Does the phenome-non that part states of the interactive systems are synchronized in terms of one type of synchronization regime, and part

0960-0779/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.chaos.2006.11.028

q Paper presented at the 2005 International Symposium on Nonlinear Dynamics in celebration of M.S. El Naschie’s 60 anniversaryDecember 20–21 (2005) Shanghai, PR China.

* Corresponding author.E-mail address: licd@cqu.edu.cn (C. Li).

462 C. Li et al. / Chaos, Solitons and Fractals 38 (2008) 461–464

synchronized in terms of another type exist in unidirectionally and linearly coupled chaotic systems? There is no doubtthat it is an interesting problem. But to the best of our knowledge, there is few (if any) investigation results reported inthe literature. In this paper, we show that the phenomenon, which is called hybrid synchronization, does exist. By linearcontrol and adaptive control via a single variable, the unidirectionally coupled identical Chen systems may exhibit aclass of hybrid synchronization behavior, i.e., one part of the states is anti-phase synchronized and the others com-plete-synchronized.

2. Coexistence of two synchronization regimes

In this paper, the drive system is chosen as Chen system [17]:

_x1 ¼ aðx2 � x1Þ;_x2 ¼ ðc� aÞx1 � x1x3 þ cx2;

_x3 ¼ x1x2 � bx3;

8><>:

ð1Þ

where a, b and c are the parameters. Chen system exhibits chaotic behavior when we choose a = 35, b = 3, c = 28.Consider the following response system:

_y1 ¼ aðy2 � y1Þ;_y2 ¼ ðc� aÞy1 � y1y3 þ cy2 � kðx2 þ y2Þ;_y3 ¼ y1y2 � by3;

8><>:

ð2Þ

where k > 0 is control gain. Our target is to find a feasible region X of control gain k such that when k 2 X the first twostates variable y1 and y2 in response system are anti-phase synchronized, respectively, to the states x1 and x2, simulta-neously the third state variable y3 is complete-synchronized to x3. For this purpose, let

e1 ¼ x1 þ y1;

e2 ¼ x2 þ y2;

e3 ¼ x3 � y3:

8><>:

ð3Þ

Then, our target is shifted to make error (3) approach asymptotically to zero when t approaches to infinite. It followsfrom (1) and (2) that the error (3) is governed by the following dynamical system:

_e1 ¼ aðe2 � e1Þ;_e2 ¼ ðc� aÞe1 � y1y3 � x1x3 þ ce2 � ke2;

_e3 ¼ x1x2 � y1y2 � be3:

8><>:

ð4Þ

Obviously, the left work is to find an appropriate region X such that the origin of (4) is asymptotically stable whenk 2 X.

Theorem 1. Suppose that M and N are the upper bounds of the absolute values of variables x2 and x3, respectively. Then,

the origin of system (4) is asymptotically stable if there exists a positive constant p > 0 such that the following conditions

hold:

(i) 4abp �M2 > 0;(ii) k > cþ bðapþc�aþNÞ2

4abp�M2 .

Proof. Notice that

x1x3 þ y1y3 ¼ x1x3 þ y1x3 � y1x3 þ y1y3 ¼ x3e1 � y1e3;

x1x2 � y1y2 ¼ ðx1x2 þ y1x2Þ � ðy1x2 þ y1y2Þ ¼ x2e1 � y1e2:ð5Þ

Consider the following Lyapunov function:

V ¼ 1

2pe2

1 þ e22 þ e2

3

� �; ð6Þ

where p > 0 is constant. The time derivative of V along the trajectories of (4) is

Fig. 1. The convergence dynamics of the error system (4) when k = 160.

C. Li et al. / Chaos, Solitons and Fractals 38 (2008) 461–464 463

_V ¼ pe1 _e1 þ e2 _e2 þ e3 _e3 ¼�ape21 � ðk� cÞe2

2 � be23 þ ðapþ c� a� x3Þe1e2 þ x2e1e3 ¼�½ e1 e2 e3 �Q½ e1 e2 e3 �T;

where

Q ¼ap � 1

2ðap þ c� a� x3Þ � 1

2x2

� 12ðap þ c� a� x3Þ k � c 0

� 12x2 0 b

264

375:

Obviously, to ensure that the origin of the error system (4) is asymptotically stable, we let the matrix Q be positive def-inite. This is the case if and only if the following three conditions hold:

(A) ap > 0;(B) 4apðk � cÞ > ðap þ c� a� x3Þ2;(C) 4abp � x2

2

� �ðk � cÞ > bðap þ c� a� x3Þ2.

It is easy to show that conditions (A)–(C) hold if conditions (i) 4abp �M2 > 0 and (ii) k > cþ bðapþc�aþNÞ24abp�M2 are satisfied.

Then the matrix Q be positive definite, and therefore, _V < 0 (if ½e1; e2; e3� 6¼ 0Þ, which implies that the origin of errorsystem (4) is asymptotically stable. This concludes the proof. h

The conditions (i) and (ii) in Theorem 1 give an estimate for the range of the control gain k. Specifically, we canobtain a critical gain k* such that aforementioned hybrid synchronization can be achieved if k > k*. In fact, it is easyto calculate k* � 159.33 when p � 6.19 (M = 32, N = 53).

3. Numerical simulation

We have investigated analytically the existence of hybrid synchronization phenomenon in coupled identical Chensystems. In what follows, we will verify our theoretical results by computer simulations. Notice that the coupled Chensystems in this paper are solved numerically through the program ODE45 in MATLAB. The initial conditions of driveand response systems are ((�15,20,10)) and ((10,5,5)), respectively. Fig. 1 shows the convergence dynamics of the errorsystem (4) when the control gain k = 160. From Fig. 1, we can see that x1 + y1! 0, x2 + y2! 0 and x3! y3 as time t

approaches to infinite.

4. Conclusions

The paper presented a class of hybrid synchronization phenomenon, i.e., two pairs of states are anti-phase synchro-nized, and the third pair is complete-synchronized, of coupled Chen system by linear control via a single variable. Suf-ficient conditions for the hybrid synchronization were obtained in terms of asymptotical stability.

Acknowledgement

The work described in this paper was partially supported by the National Natural Science Foundation of China(Grant No.60574024), Natural Science Foundation Project of CQ CSTC (Grant No. 2006BB2228), and Program forNew Century Excellent Talents in University.

464 C. Li et al. / Chaos, Solitons and Fractals 38 (2008) 461–464

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