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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 propagation 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!þ1 k~xðt þ sÞ xðtÞk 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 timedelay 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_ ðtÞ ¼ AX ðtÞ þ GðX 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 ðt þ sÞ X ðtÞÞ X_ ðtÞ ¼ AX ðtÞ þ GðX ðtÞÞ þ D1 ð X ð2Þ e_ ðt þ sÞ ¼ A X e ðt þ sÞ þ GðX ðt þ sÞÞ þ D2 ðX ðtÞ X e ðt þ sÞÞ X ð3Þ and e ðtÞ is the n-dimensional state vector of the response system, s is a ﬁnite time-delay which is an unknown conwhere X stant, G : Rn ! Rn is a continuous function, D1 and D2 are diagonal matrices which rule the dissipative coupling. Assume that ðX ðt þ sÞ X ðtÞÞ GðX ðt þ sÞÞ GðX ðtÞÞ ¼ M X ;e X where M X ;e is a matrix with elements depending on X(t) and X(t + s). Obviously it is bounded. X Let e(t) = X(t + s) X(t), then by subtracting (2) from (3), we can obtain the error dynamical system: e_ ðtÞ ¼ ðA þ M X ;e ðD1 þ D2 ÞÞe X 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 eðtÞ ¼ 0 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 ðA þ M X ;e ðD1 þ D2 ÞÞT P þ P ðA þ M X ;e ðD1 þ D2 ÞÞ 6 I X X e ðt þ sÞ, where I is an identity matrix, then the error dynamics system (4) is globally stable at uniformly for any X(t) and X zero, i.e. the two systems (2) and (3) are realized to globally asymptotically synchronize. Proof 1. Choose the following Lyapunov function: V ðtÞ ¼ eT PeðtÞ 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 ðD1 þ D2 Þ PeðtÞ þ eT ðtÞP A þ M X ;e ðD1 þ D2 Þ eðtÞ ¼ e_ T ðtÞPeðtÞ þ eT ðtÞP e_ ðtÞ ¼ eT A þ M X ;e X X dt T ðD þ D Þ P þ P A þ M ðD þ D Þ eðtÞ ¼ eT ðtÞ A þ M X ;e 1 2 1 2 X X ;e X 6 eT ðtÞeðtÞ < 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 = 0 or D2 = 0, the results can be simpliﬁed 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_ ¼ by þ xz ð5Þ > : z_ ¼ cz 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_ 1 ðtÞ ¼ ax1 ðtÞ y 1 ðtÞz1 ðtÞ þ d 11 ðx2 ðt þ sÞ x1 ðtÞÞ y_ 1 ðtÞ ¼ by 1 ðtÞ þ x1 ðtÞz1 ðtÞ þ d 12 ðy 2 ðt þ sÞ y 1 ðtÞÞ ð6Þ > : z_ 1 ðtÞ ¼ cz1 ðtÞ þ x1 ðtÞy 1 ðtÞ þ d 13 ðz2 ðt þ sÞ z1 ðtÞÞ and 8 > < x_ 2 ðt þ sÞ ¼ ax2 ðt þ sÞ y 2 ðt þ sÞz2 ðt þ sÞ þ d 21 ðx1 ðtÞ x2 ðt þ sÞÞ y_ 2 ðt þ sÞ ¼ by 2 ðt þ sÞ þ x2 ðt þ sÞz2 ðt þ sÞ þ d 22 ðy 1 ðtÞ y 2 ðt þ sÞÞ > : z_ 2 ðt þ sÞ ¼ cz2 ðt þ sÞ þ x2 ðt þ sÞy 2 ðt þ sÞ þ d 23 ðz1 ðtÞ z2 ðt þ sÞÞ ð7Þ where dij(i = 1, 2; j = 1, 2, 3) are coupling parameters. Let e ¼ ðe1 ; e2 ; e3 ÞT ¼ ðx1 ðtÞ ~x2 ðt þ sÞ; y 1 ðtÞ y 2 ðt þ sÞ; z1 ðtÞ z2 ðt þ sÞÞT , 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_ ðtÞ ¼ ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞe where 0 1 a 0 0 B C A ¼ @ 0 b 0 A; 0 0 c and ð8Þ 0 M X 1 ;X 2 1 0 z1 ðtÞ y 2 ðt þ sÞ B C ¼ @ z1 ðtÞ 0 x2 ðt þ sÞ A; y 2 ðt þ sÞ x1 ðtÞ 0 D1 ¼ diagfd 11 ; d 12 ; d 13 g D2 ¼ diagfd 21 ; d 22 ; d 23 g 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), in Fig. 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 X 1 ;X 2 ðD1 þ D2 ÞÞT P þ P ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞ þ I 0 2p1 a þ ðd 11 þ d 21 Þ 2p ðp3 p1 Þy 2 ðt þ sÞ 1 B B ¼B 2p2 b þ ðd 12 þ d 22 Þ 2p 2 ðp2 p1 Þz1 ðtÞ B @ p2 x2 ðt þ sÞ þ p3 x1 ðtÞ ðp3 p1 Þy 2 ðt þ sÞ 1 C C C P 2 x2 ðt þ sÞ þ p3 x1 ðtÞ C A 2p3 c þ ðd 13 þ d 23 Þ 2p ð9Þ 3 Based on basic matrix theory, the matrix (9) is negative deﬁnite if and only if its sub-determinants satisfy < 0; DI ¼ 2p1 a þ ðd 11 þ d 21 Þ 2p1 2p1 a þ ðd 11 þ d 21 Þ 2p 1 D2 ¼ ðp2 p1 Þz1 ðtÞ and > 0 2p2 b þ ðd 12 þ d 22 Þ 2p 2 ðp2 p1 Þz1 ðtÞ D3 ¼ det ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞT P þ P ðA þ M X 1 ;X 2 ðD1 þ D2 ÞÞ þ I D2 þ d < 0 ¼ 2p3 c þ ðd 13 þ d 23 Þ 2p2 where d ¼ ðp3 p2 Þ2 y 22 ðt þ sÞð þ 2p2 ðb þ d 12 þ d 22 ÞÞ þ ðp2 x2 ðt þ sÞ þ p3 x1 ðtÞÞ2 ð a þ d 11 þ d 21 Þ þ 2ðp2 p1 Þðp3 p1 Þ y 2 ðt þ sÞz1 ðtÞðp2 x2 ðt þ sÞ þ p3 x1 ðtÞÞ. 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 d 11 þ d 21 > a þ 2p 1 > > > < ðp2 p1 Þ2 z21 ðtÞ d 12 þ d 22 > 4p1 p2 ðaþd b þ 2p 2 11 þd 21 2p Þ > 1 > > :d þ d > d c þ 13 23 2p3 D3 2p3 ð10Þ 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. p1 = p2 = p3 = 1 > 0, one have 8 d 11 þ d 21 > a þ 2 > > < d 12 þ d 22 > b þ 2 ð11Þ > > ðx2 ðtþsÞþx1 ðtÞÞ2 : d 13 þ d 23 > 4ðbþd 12 þd 22 Þ c þ 2 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. 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