The Synchronization for Time-Delay of Linearly Bidirectional Coupled Chaotic System Q

Chaos, Solitons and Fractals 33 (2007) 1197–1203 www.elsevier.com/locate/chaos

The synchronization for time-delay of linearly bidirectional coupled chaotic system q

Yongguang Yu

Department of Mathematics, Beijing Jiaotong University, Beijing 100044, PR China

Accepted 17 January 2006

Communicated by Prof. Ji-Huan He

Abstract

Based on Lyapunov stability theory and matrix measure, this paper presents a new generic criterion of global chaotic synchronization for bidirectional coupled chaotic system with time-delay. A new chaotic system with four-scroll attractor is chosen as an example to verify the eﬀectiveness of the criterion. Numerical simulation are shown for demonstration. 2006 Published by Elsevier Ltd.

1. Introduction

Chaos, a very interesting complex nonlinear phenomenon, has been intensively studied in the last four decades [1–3]. It is found to be useful and it has great potential in many technological fields, such as in information and computer sciences, biomedical systems analysis, flow dynamics and liquid mixing, power systems protection, encryption and communications, and so on. Therefore many researchers have been involved into how to control and utilize it [1–24]. Since the discovery of chaos synchronization [4], it has been a focus of intensive research [4–24]. Many new types of chaos synchronization have appeared in the literatures on this subject, such as phase synchronization [5,6], adaptive synchronization [7], stochastic synchronization [8], lag synchronization [9], projective synchronization [10–12] and generalized synchronization in mismatched systems or even distinct class [14,15]. Chaos synchronization can be applied to many fields including physical, chemical and ecological systems and secure communications, etc. Especially, chaos synchronization of linearly coupled chaotic systems is investigated by some researchers [16–24].So far, there have been many specific results about determining the coupling parameters for chaotic systems. In engineering applications, time-delay always exists, and the value of the delay is often unknown in advance. In practice, It is not reasonable to require the response system to synchronize the drive system at the same time due to the signal propaga- tion delays. So, chaotic synchronization for time-delay of linearly bidirectional coupled chaos systems is investigated in this paper. How to synchronize between the state of response system at time t + s and the drive system at time t? q Supported by the National Nature Science Foundation of China (Grant No. 10461002) and Beijing Jiaotong University Science Foundation of China (No. 2005sm063). E-mail address: [email protected]

0960-0779/$ - see front matter 2006 Published by Elsevier Ltd. doi:10.1016/j.chaos.2006.01.119 1198 Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203 i.e. limt!þ1k~xðt þ sÞxðtÞk ¼ 0, where x and ~x are the states of the drive and response systems, respectively, s(>0) is the delay time. The organization of this paper is as follows. In Section 2, at ﬁrst, the problem statement is presented. Then based on the Lyapunov stability theory and some matrix theory, a generic condition of global chaos synchronization for time- delay of the general chaotic systems with bidirectional linear coupling is obtained. In Section 3, a new chaotic system with four-scroll attractor is taken as examples to demonstrate the eﬀectiveness of the proposed scheme. In Section 4, numerical simulations for given four-scroll chaotic system are presented. Finally, the conclusion of this paper is given in Section 5.

2. Chaos synchronization of linearly bidirectional coupled systems with time-delay

Considering a chaotic system in the form of

X_ ðtÞ¼AX ðtÞþGðX ðtÞÞ ð1Þ where X(t) 2 Rn is the state vector, A 2 Rn·n and G(X(t)) is a continuous nonlinear function. Then a bidirectional coupling scheme between identical chaotic systems with time-delay can be constructed as follows: e X_ ðtÞ¼AX ðtÞþGðX ðtÞÞ þ D1ðX ðt þ sÞX ðtÞÞ ð2Þ and e_ e e X ðt þ sÞ¼AX ðt þ sÞþGðX ðt þ sÞÞ þ D2ðX ðtÞX ðt þ sÞÞ ð3Þ where Xe ðtÞ is the n-dimensional state vector of the response system, s is a ﬁnite time-delay which is an unknown con- n n stant, G : R ! R is a continuous function, D1 and D2 are diagonal matrices which rule the dissipative coupling. Assume that

GðX ðt þ sÞÞ GðX ðtÞÞ ¼ M X ;eX ðX ðt þ sÞX ðtÞÞ where M X ;eX is a matrix with elements depending on X(t) and X(t + s). Obviously it is bounded. Let e(t)=X(t + s) X(t), then by subtracting (2) from (3), we can obtain the error dynamical system:

e_ðtÞ¼ðA þ M X ;eX ðD1 þ D2ÞÞe ð4Þ In order to realize the chaos synchronization between two linearly bidirectional systems with time-delay, we should choose the suitable coupled parameter matrices D1,D2 which can make lim eðtÞ¼0 n!þ1 In the following, a generic criterion is given for judging the global chaos synchronization between two coupled systems with time-delay. Theorem 1. If there exists a positive deﬁnite symmetric matrix P and a constant > 0, such that T 6 ðA þ M X ;eX ðD1 þ D2ÞÞ P þ PðA þ M X ;eX ðD1 þ D2ÞÞ I uniformly for any X(t) and Xe ðt þ sÞ, where I is an identity matrix, then the error dynamics system (4) is globally stable at zero, i.e. the two systems (2) and (3) are realized to globally asymptotically synchronize.

Proof 1. Choose the following Lyapunov function: V ðtÞ¼eTPeðtÞ where P is a positive deﬁnite symmetric matrix, i.e. V(t) is a positive deﬁnite function. Calculating its derivative along the trajectory of the system (4), one have T dV ðtÞ T T T T ¼ e_ ðtÞPeðtÞþe ðtÞPe_ðtÞ¼e A þ M X ;eX ðD1 þ D2Þ PeðtÞþe ðtÞPAþ M X;eX ðD1 þ D2Þ eðtÞ dt T T ¼ e ðtÞ A þ M X ;eX ðD1 þ D2Þ P þ PAþ M X ;eX ðD1 þ D2Þ eðtÞ 6 eTðtÞeðtÞ < 0 Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203 1199 for all e(t) 5 0. Based on the Lyapunov stability theorem, the error dynamics system (4) is globally asymptotically stable at origin. Hence the two coupled systems with time-delay are globally asymptotically synchronized. So, the theorem is proved. h

Remark. It can be seen from Theorem 1, when the coupled parameter matrix D1 =0orD2 = 0, the results can be sim- pliﬁed as the general synchronization criterion of unidirectional coupled system with time-delay [16].

3. Synchronization of a four-scroll chaotic attractor

To verify the result of Theorem 1, a four-scroll chaotic attractor is chosen as an example. One can easily determine the coupling parameters for realizing to synchronize two bidirectional coupled systems with time-delay. A new four-scroll chaotic system can be described by the following three-dimensional quadratic autonomous ODE [22–24]. 8 <> x_ ¼ ax yz y_ ¼by þ xz ð5Þ :> z_ ¼cz þ xy where a >0,b >0,c > 0 and b + c > a. The system is found to be chaotic in a wide parameter range and has many complex dynamic behaviors. Diﬀering from other known chaotic systems, it has ﬁve equilibria, and does not have Hopf bifurcation. At the same time, the system (5) not only can display a two-scroll chaotic attractor when parameters a = 4.5, b = 12 and c = 5 (see Fig. 1), but also can display a four-scroll chaotic attractor when a = 0.4, b = 12 and c = 5 (see Fig. 2). According to Eqs. (2) and (3), two linearly bidirectionally coupled systems with time-delay can be constructed for system (5) 8 <> x_ 1ðtÞ¼ax1ðtÞy1ðtÞz1ðtÞþd11ðx2ðt þ sÞx1ðtÞÞ y_ ðtÞ¼by ðtÞþx ðtÞz ðtÞþd ðy ðt þ sÞy ðtÞÞ ð6Þ :> 1 1 1 1 12 2 1 z_1ðtÞ¼cz1ðtÞþx1ðtÞy1ðtÞþd13ðz2ðt þ sÞz1ðtÞÞ and 8 <> x_ 2ðt þ sÞ¼ax2ðt þ sÞy2ðt þ sÞz2ðt þ sÞþd21ðx1ðtÞx2ðt þ sÞÞ y_ ðt þ sÞ¼by ðt þ sÞþx ðt þ sÞz ðt þ sÞþd ðy ðtÞy ðt þ sÞÞ ð7Þ :> 2 2 2 2 22 1 2 z_2ðt þ sÞ¼cz2ðt þ sÞþx2ðt þ sÞy2ðt þ sÞþd23ðz1ðtÞz2ðt þ sÞÞ where dij(i =1,2;j = 1,2,3) are coupling parameters. T T Let e ¼ðe1; e2; e3Þ ¼ðx1ðtÞ~x2ðt þ sÞ; y1ðtÞy2ðt þ sÞ; z1ðtÞz2ðt þ sÞÞ , the corresponding error dynamics system is

Fig. 1. The graph of chaotic system (6) with two-scroll attractor when the parameters a = 4.5, b = 12 and c =5. 1200 Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203

Fig. 2. The graph of chaotic system (6) with four-scroll attractor when the parameter a = 0.4, b = 12 and c =5.

e_ðtÞ¼ðA þ M X 1;X 2 ðD1 þ D2ÞÞe ð8Þ where 0 1 0 1 a 00 0 z1ðtÞy ðt þ sÞ B C B 2 C @ A @ A A ¼ 0 b 0 ; M X 1;X 2 ¼ z1ðtÞ 0 x2ðt þ sÞ ; D1 ¼ diagfd11; d12; d13g

00c y2ðt þ sÞ x1ðtÞ 0

and D2 ¼ diagfd21; d22; d23g

When coupling parameter matrices D1 = D2 = 0, and chosen a very small constant s = 0.1 and same initial condition for system (6) and (7),inFig. 3 it can show the two systems are chaotic and do not synchronize.

Fig. 3. The graph of the error dynamical system between systems (6) and (7) when the coupling parameter is zero. Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203 1201

Choose the positive definite symmetric constant matrix P = diag{p1,p2,p3}, (pi >0,i = 1,2,3) and any constant > 0, then ðA þ M ðD þ D ÞÞTP þ PðA þ M ðD þ D ÞÞ þ I 0X 1;X 2 1 2 X 1;X 2 1 2 1 2p1 a þðd11 þ d21Þ ðp3 p1Þy2ðt þ sÞ B 2p1 C B C B C ¼ B ðp2 p1Þz1ðtÞ2p2 b þðd12 þ d22Þ P 2x2ðt þ sÞþp3x1ðtÞ C ð9Þ @ 2p2 A ðp p Þy ðt þ sÞ p x2ðt þ sÞþp x1ðtÞ2p c þðd13 þ d23Þ 3 1 2 2 3 3 2p3 Based on basic matrix theory, the matrix (9) is negative definite if and only if its sub-determinants satisfy DI ¼2p1 a þðd11 þ d21Þ < 0; 2p1 2p a þðd11 þ d21Þ ðp p Þz1ðtÞ 1 2p1 2 1 D2 ¼ > 0 ðp p Þz1ðtÞ2p b þðd12 þ d22Þ 2 1 2 2p2 and T D3 ¼ det ðA þ M X ;X ðD1 þ D2ÞÞ P þ PðA þ M X ;X ðD1 þ D2ÞÞ þ I 1 2 1 2 ¼2p3 c þðd13 þ d23Þ D2 þ d < 0 2p2 2 2 2 where d ¼ðp3 p2Þ y2ðt þ sÞð þ 2p2ðb þ d12 þ d22ÞÞ þ ðp2x2ðt þ sÞþp3x1ðtÞÞ ð a þ d11 þ d21Þþ2ðp2 p1Þðp3 p1Þ y2ðt þ sÞz1ðtÞðp2x2ðt þ sÞþp3x1ðtÞÞ. Through calculations, it can be simplified as

Fig. 4. The graph of the errors between two coupled two-scroll chaotic systems. 1202 Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203 8 > d11 þ d21 > a þ <> 2p1 2 2 ðp2p1Þ z1ðtÞ d12 þ d22 > b þ ð10Þ > 4p1p2ðaþd11þd212p Þ 2p2 > 1 : d d13 þ d23 > c þ 2p3D3 2p3 Due to the fact that the trajectory of the coupled chaotic systems (6) and (7) is bounded, the inequalities (10) can easily be hold by choosing appropriately coupled parameters. For convenience, let the matrix P = I, k > 0, i.e. p = p = p = 1 > 0, one have 8 1 2 3 > d þ d > a þ <> 11 21 2 d d > b 12 þ 22 þ 2 ð11Þ > 2 : ðx2ðtþsÞþx1ðtÞÞ d13 þ d23 > c þ 4ðbþd12þd222Þ 2

4. Numerical simulation

In this section, to demonstrate the eﬀectiveness of the proposed scheme, the following numerical simulations are given. The fourth/ﬁfth-order Runge–Kutta method is used to solve the chaotic systems during the simulation. Let = 0.1. First, the parameters are chosen a = 4.5, b = 12 and c = 5 for system (5), the time-delay s = 0.1, we assume that the initial states of system (6) are (x1(0),y1(0),z1(0)) = (5,3,8), when the coupled parameters are chosen to satisfy the inequalities (13), for instance (d11,d12,d13) = (1,1,20),(d21,d22,d23) = (4,1,20), the simulations in Fig. 4 are shown that the error system (8) converge to zero very quickly, i.e. the synchronization of the coupled systems (6) and (7) is realized. In Fig. 5, we choose the parameters a = 0.4, b = 12 and c = 5, the time-delay s = 0.1, the initial states of system (6) are (x1(0),y1(0),z1(0)) = (5,3,8), the coupled parameters (d11,d12,d13) = (1,1,20),(d21,d22,d23) = (4,1,20), it can display the systems (6) and (7) also synchronized rapidly.

Fig. 5. The graph of the errors between two coupled four-scroll chaotic systems. Y. Yu / Chaos, Solitons and Fractals 33 (2007) 1197–1203 1203

5. Discussion

In this paper, we investigate the synchronization for time-delay of linearly bidirectional coupled chaotic systems. A new generic criterion is given for judging the synchronization of two coupled systems. We choose a new chaotic system with four-scroll attractor as an example to show the eﬀectiveness of the method. Finally, numerical simulation is pro- vided for demonstration.

References

[1] Chen G, Dong X. From chaos to order: methodologies, perspectives and applications. Singapore: World Scientific; 1998. [2] Lorenz EN. Deterministic non-periodic flows. J Atmos Sci 1963;20:130–41. [3] Ott E, Grebogi C, Yorke J. Controlling chaos. Phys Rev Lett 1990;64:1196–9. [4] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [5] Rosenblum M, Pikovsky A, Kurth J. Phase synchronization in chaotic oscillators. Phys Rev Lett 1996;76:1804–7. [6] Rosenblum M. A characteristic frequency of chaotic dynamical system. Chaos, Solitons & Fractals 1993;3(6):617–26. [7] Chen S, Lu J. Synchronization of uncertain unified chaotic system via adaptive control. Chaos, Solitons & Fractals 2002;14(4):643–7. [8] Alexeyev A, Shalfeev V. Chaotic synchronization of mutually-coupled generators with frequency-controlled feedback loop. Int J Bifurcat Chaos 1995;5:551–8. [9] Taherion S, Lai Y. Experimental observation of lag synchronization in coupled chaotic systems. Int J Bifurcat Chaos 2000;10:2587–94. [10] Xu D, Li Z. Controlled projective synchronization in nonpartially-linear chaotic systems. Int J Bifurcat Chaos 2002;12(6):1395–402. [11] Xu D, Chee C, Li C. A necessary condition of projective synchronization in discrete-time systems of arbitrary dimensions. Chaos, Solitons & Fractals 2004;22(1):175–80. [12] Li Z, Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 2004;22(2): 477–81. [13] Rulkov N, Sushchik K, Tsimring L, Abarbanel H. Generalized synchronization of chaos in directionally coupled chaotic systems. Phys Rev E 1995;51:980–94. [14] Chen G, Liu S. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 2003;15(2):311–8. [15] Yang X, Chen G. Some observer-based criteria for discrete-time generalized chaos synchronization. Chaos, Solitons & Fractals 2002;13(6):1303–8. [16] Jiang G, Zheng W, Chen G. Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals 2004;20(2):267–75. [17] Jiang GP, Tang W. A global synchronization criterion for coupled chaotic systems via unidirectional linear error feedback approach. Int J Bifurcat Chaos 2002;12(10):2239–53. [18] Zhou T, Lu¨ J, Chen G, Tang Y. Synchronization stability of three chaotic systems with linear coupling. Phys Lett A 2002;301:231–40. [19] Kyprianidis I, Stouboulos I. Chaotic synchronization of three coupled oscillators with ring connection. Chaos, Solitons & Fractals 2003;17:327–36. [20] Yu Y, Zhang S. The synchronization of linearly bidirectional coupled chaotic systems. Chaos, Solitons & Fractals 2004;22:189–97. [21] Yu Y, Zhang S. Global synchronization of three coupled chaotic systems with ring connection. Chaos, Solitons & Fractals 2005;24:1233–42. [22] Elabbasy E, Agiza H. Chaos synchronization criterion and adaptive synchronization for new chaotic system. Chaos, Solitons & Fractals 2005;23:1299–309. [23] Park J. Chaos synchronization of a chaotic system via nonlinear control. Chaos, Solitons & Fractals 2005;25:579–84. [24] Lu J, Chen G, Cheng D. A new chaotic system and beyond: the generalized Lorenz-like system. Int J Bifurcat Chaos 2004;14:1507–37.