Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme

Commun. Theor. Phys. (Beijing, China) 45 (2006) pp. 1049 1056 c International Academic Publishers Vol. 45, No. 6, June 15, 2006 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme WANG Qi Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, China Key Laboratory of Mathematics Mechanization, the Chinese Academy of Sciences, Beijing 100080, China (Received September 19, 2005; Revised November 14, 2005) Abstract In this paper, a bidirectional partial generalized (lag, complete, and anticipated) synchronization of a class of continuous-time systems is defined. Then based on the active control idea, a new systematic and concrete scheme is developed to achieve bidirectional partial generalized (lag, complete, and anticipated) synchronization between two chaotic systems or between chaotic and hyperchaotic systems. With the help of symbolic-numerical computation, we choose the modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius Namajunas Cenys system to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. It is interesting that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. The proposed scheme can also be extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems. PACS numbers: 05.45.Xt Key words: bidirectional partial generalized (lag, complete, and anticipated) synchronization, modified Chua system, Lorenz system, hyperchaotic Tamasevicius Namajunas Cenys system, numerical simulation 1 Introduction Since the pioneering work of Pecora and Carroll, [1] chaos synchronization has become an active research subject in nonlinear science and has attracted much attention due to its potential applications such as physics, secure communication, chemical reaction, control theory, biological networks, artificial neural networks, telecommunications, etc. Up to now, many types of synchronization have been presented. [1 12] At the same time, many powerful methods have been applied to synchronize two identical or different chaotic (hyperchaotic) systems such as coupling control, [13] feedback control, [14] fuzzy control, [15] adaptive control, [16] impulsive control, [17,18] active control theory, [19 21] the scalar signal method, [22] etc. More recent developments in this area were given in Refs. [6] and [23]. Recently, in Ref. [24], M. Hasler et al. presented the definition of partial synchronization, which means that some of variables synchronize but not all. Many real dynamical systems are composed of two or more synchronized systems, giving rise to highly complex dynamics. The synchronized systems usually do not require all variables to synchronize, i.e., partial synchronization takes place. This phenomenon will have many applications in engineering, particularly in signal processing. [25] Most recently, Yan [12] developed a new scheme to investigate the generalized (lag, complete, and anticipated) synchronization. In this paper, we will directly extend the definition by M. Hasler [24] and present a type of bidirectional partial generalized (lag, complete, and anticipated) synchronization, which is defined as the presence of certain relationship between the states of the drive and response systems, i.e., there exists a smooth vector function H such that partial variables y s (t) = H(x s (t τ)) with τ R. Then, based on the theory that all eigenvalues of the error system have negative real parts and the scheme presented by Yan, [12] a systematic and powerful scheme is presented to investigate the bidirectional partial generalized (lag, complete, and anticipated) synchronization between the drive system and response system. Finally, the scheme is applied to investigate bidirectional partial generalized (lag, anticipated, and complete) synchronization between modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius Namajunas Cenys system. It is interesting that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic systems. Numerical simulations are used to verify the effectiveness of the proposed scheme. 2 Bidirectional Partial Generalized (Lag, Complete, and Anticipated) Synchronization Scheme In this section, we consider two chaotic dynamical systems, one with state variable x R p, and the other with E-mail: wangqi dlut@yahoo.com.cn

1050 WANG Qi Vol. 45 state variable y R q and there is no cross coupling between the two sets of state variables. Here we define a coordinate subset of x, x s R s, and similarly y s R s, which represent the coordinates which require synchronization to achieve coupling between the two systems. In general, we just consider the case that the two systems can be expressed as ẋ m = f 11 (x m, x s, t), ẋ s = Ax s + f 12 (x m, x s, t) + C(y n, y s, t), (1) ẏ n = f 22 (y n, y s, t), ẏ s = By s + f 21 (y n, y s, t) + U(x, y, t), (2) where f 11 : R p R 0,+ R m, f 12 : R p R 0,+ R s, f 21 : R q R 0,+ R s and f 22 : R q R 0,+ R n (R 0,+ = [0, + )) are all continuous vector functions including nonlinear terms and can contain the explicit time t as well as the constant term, A, B R s s, x m = {x i x : x i / x s } and x i denotes the i-th element of x, and likewise y n = {y i y : y i / y s }. Here U(x, y, t) is an unknown column vector controller and C(y n, y s, t) is a column vector which represents the bidirectional influence on the behavior of system (1) from system (2). Let the vector error state be e s (t) = y s (t) H(x s (t τ)), where τ R and H(x s (t τ)) = [H 1 (x s (t τ)), H 2 (x s (t τ)),..., H s (x s (t τ))] T is a smooth column vector function. Thus the error dynamical system between the drive system (1) and the response system (2) is ė s (t) = ẏ s (t) Ḣ(x s(t τ)) = By s + f 21 (y n (t), y s (t), t) + U(x(t τ), y(t), t) DH(x s (t τ))(ax s (t τ) + f 12 (x m (t τ), x s (t τ), t τ) + C(y n (t), y s (t), t)), (3) where DH(x s (t τ)) is the Jacobian matrix of the vector function H(x s (t τ)), DH(x s (t τ)) H 1(x s(t τ)) x s1(t τ) H 2(x s(t τ)) x s1(t τ) H 1(x s(t τ)) x s2(t τ) H 1(x s(t τ)) x ss(t τ) H 2(x s(t τ)) x ss(t τ) H 2(x s(t τ)) x s2(t τ) =.. (4).. H s(x s(t τ)) H s(x s(t τ)) H x s1(t τ) x s2(t τ) s(x s(t τ)) x ss(t τ) In the following we would give the definition of the bidirectional partial generalized synchronization. Definition For the drive system (1) and response system (2), it is said that the drive system (1) and response system (2) are bidirectional partial (i) generalized lag synchronous (τ > 0, τ is called the generalized synchronization lag), (ii) generalized (complete) synchronous (τ = 0), or (iii) generalized anticipated synchronous (τ < 0, τ is called the generalized synchronization anticipation) with respect to the vector transformation H, if there exist a vector controller U(x, y, t) and a smooth vector function H : R s R s such that all trajectories (x s (t τ), y s (t)) in Eqs. (1) and (2) with any initial conditions (x s (0), y s (0)) in P = Rx s s Ry s s R p R q approach the manifold M = {(x s (t τ), y s (t)) : y s (t) = H(x s (t τ))} with M P as time t goes to infinity, that is to say, lim t e s (t) = lim t y s (t) H(x s (t τ)) = 0, which implies that the error dynamical system (3) between the drive system and response system is globally asymptotically stable. Remark 1 The definition is extension of the partial synchronization [24] and generalized (lag, anticipated, and complete) synchronization [12] in the sense of the same dimensional spaces. When τ = 0 and H is an identity mapping, the synchronization mentioned above is bidirectional partial synchronization; [26] When C = τ = 0 and H is an identity mapping, the synchronization mentioned above is unidirectional (master-slave) system; [24,25] When p = q = s, C = τ = 0, the synchronization mentioned above is just generalized synchronization; [3 6] When p = q = s, C = τ = 0 and H is a linear function, the synchronization mentioned above becomes the linear generalized synchronization, which was recently studied; [27] When p = q = s, C = 0, H is an identity mapping and τ > 0 (τ = 0, τ < 0), the synchronization mentioned above becomes lag (complete, anticipated) synchronization; [7 10] When p = q = s, C = 0, the synchronization mentioned above becomes generalized (lag, anticipated, and complete) synchronization. [12] In the following we would like to present a systematic, powerful and concrete scheme to study bidirectional partial generalized (lag, complete, and anticipated) synchronization between the drive system and response system simultaneously. Theorem For the drive system (1) and response system (2), let e s (t) = y s (t) H(x s (t τ)), if the vector control U(x, y) in Eq. (2) is given by U(x(t τ), y(t), t) = e(t) BH(x(t τ)) f 21 (y n, y s, t) + DH(x s (t τ))(ax s (t τ) + f 12 (x m (t τ), x s (t τ), t τ) + C(y n, y s, t)), (5) where = (δ ij ) s s, δ ij R are constants satisfying each of the conditions δ ii < b ii, δ ij = b ij, (i > j), i, j = 1, 2,..., s, (6) δ ii < b ii, δ ij = b ij, (i < j), i, j = 1, 2,..., s, (7)

No. 6 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme 1051 then lim t e s (t) = lim t y s (t) H(x s (t τ)) = 0, that is to say, the bidirectional partial generalized lag synchronization (τ > 0), bidirectional partial generalized anticipated synchronization (τ < 0) or bidirectional partial generalized synchronization (τ = 0) alway occurs between the drive system (1) and the response system (2) under the vector controller U(x(t τ), y(t), t). Proof The substitution of the chosen vector controller U(x(t τ), y(t), t) in Eq. (5) into the error dynamical system (3) yields a linear homogeneous error system, ė s (t) = By s (t) + f 21 (y n (t), y s (t), t) + e(t) BH(x(t τ)) f 21 (y n (t), y s (t), t) DH(x s (t τ))(ax s (t τ) + f 12 (x m (t τ), x s (t τ), t τ) + C(y n (t), y s (t), t)) + DH(x s (t τ))(ax s (t τ) + f 12 (x m (t τ), x s (t τ), t τ) + C(y n (t), y s (t), t)) = (B + )e s (t). Since δ ij satisfy Eq. (6) or (7), thus the error system (8) reduces to ė s (t) = (ė s1 (t), ė s2 (t),..., ė ss (t)) T = (B + )(e s1 (t), e s2 (t),..., e ss (t)) T, (9) (8) where b 11 + δ 11 b 12 + δ 12 b 1s + δ 1s 0 b 22 + δ 22 b 2s + δ 2s (B+ ) (6) =... 0 0 b ss + δ ss b 11 + δ 11 0 0 b 21 + δ 21 b 22 + δ 22 0 (B+ ) (7) =... b s1 + δ s1 b s2 + δ s2 b ss + δ ss It is easy to see from the condition (10) or (11) that for any given parameters b ij in Eq. (2), the error system (9) with Eq. (10) or (11) admits n negative eigenvalues λ i = b ii + δ ii < 0, (i = 1, 2,..., s). Thus we know that lim t e s (t) = 0. This completes the proof of the Theorem. Remark 2 It is obvious that for the given b ij, there may have many other choices for δ ij such that the error system (8) is globally asymptotically stable. where F (x 3 ) = bπ 2a (x 3 2ac), if x 3 2ac, b sin ( πx 3 2a + d), if 2ac < x 3 < 2ac, bπ 2a (x 3 + 2ac), if x 3 2ac. (13) Here, in Eqs. (12) and (13), α, β, a, b, c, d are suitable constants, and α > 0, β > 0, a > 0, b > 0. 3 Applications of Above-Mentioned Scheme In this section, we will only use the case of Eq. (6) to investigate the bidirectional partial generalized (lag, complete, and anticipated) synchronization between chaotic and chaotic (hyperchaotic) systems. 3.1 Bidirectional Partial Generalized Synchronization between Modified Chua System and Lorenz System The modified Chua system, [28] unlike the classic Chua s circuit, is governed by a trigonometric function, which is a continuous function. It is reported that n- scroll attractors can be obtained, as shown in Fig. 1(a). The dimensionless state equation is given by ẋ 1 = βx 2, ẋ 2 = x 3 x 2 + x 1, ẋ 3 = α(x 2 F (x 3 )). (12) Fig. 1 (a) Five-scroll attractor of modified Chua system; (b) The attractor of Lorenz system. An n-scroll attractor is generated with the following relationship: n = c + 1, (14) { π, if n is odd, d = (15) 0, if n is even.

1052 WANG Qi Vol. 45 The well-known Lorenz system was given as ẏ 1 = 10y 1 + 10y 2, ẏ 2 = y 1 y 3 + 28y 1 y 2, ẏ 3 = y 1 y 2 8 3 y 3, (16) by Lorenz, [29] which led to the discovery of the butterflylike Lorenz attractor as shown in Fig. 1(b). According to the above-mentioned method, we will consider the following two cases. (i) Single Variable Synchronization Now let us consider the case when we wish to synchronize x 3 and y 1. In this case A = 0, B = 10, f 12 (t) = α(x 2 F (x 3 (t))), f 21 (t) = 10y 2 (t). Let the error states e(t) = y 1 (t) H(x 3 (t τ)), where τ R and H(x 3 (t τ)) is a smooth function. Then we have the error dynamical system, ė(t) = Be(t) + BH(x 3 (t τ)) + f 21 (t) + U(x(t τ), y(t), t) DH(x 3 (t τ))(f 12 (t τ) + C(y 1 (t), y 2 (t), y 3 (t), t)). (17) Let U = e (t) BH (x 3 (t τ)) f 21 (t) + DH (x s (t τ)) (f 12 (t τ) + C (y 1 (t), y 2 (t), y 3 (t), t)). (18) Then the error dynamical system (17) becomes ė(t) = ( + B)e(t) = ( + 10)e(t). (19) It can be shown that there exist many types of solutions for < 10, such that all eigenvalues of system (17) all have negative real parts, that is to say, lim t e(t) = lim t y 1 (t) H(x 3 (t τ)) = 0, such that system (17) is global asymptotically stable. In what follows we would like to use the numerical simulations to verify the effectiveness of the above-designed controllers. Let the initial values of systems (12) and (16) be H(x 3 (t τ)) = 5x 3 (t τ), α = 10.814, β = 14, a = 1.3, b = 0.11, c = 4, d = π, = 13, [x 1 (0) = 1, x 2 (0) = 0, x 3 (0) = 0], and [y 1 (0) = 2, y 2 (0) = 2, y 3 (0) = 2]. Remark 3 values. This choice of parameters and initial conditions is arbitrary: control can be applied for any parameter Case i(a) Partial generalized lag synchronization. In the case τ > 0, without loss of generality, we set τ = 1. Thus by calculation, the initial values of the error dynamical system (17) is e(0) = y 1 (0) 2x 3 ( 1) = 30.716 754 07. The dynamical of partial generalized lag synchronization errors for the drive system (12) and the response system (16) are shown in Fig. 2(a) Case i(b) Partial generalized (complete) synchronization. In the case τ = 0. Thus the initial values of the error dynamical system (17) are e(0) = y 1 (0) 2x 3 (0) = 2. Similarly, we also display the dynamical of generalized synchronization errors for the drive system (12) and the response system (16) are shown in Fig. 2(b). Case i(c) Partial generalized anticipated synchronization. In the case τ < 0, without loss of generality, we set τ = 1. Thus the initial values of the error dynamical system (17) are e(0) = y 1 (0) 2x 3 (1) = 7.575 411 550. The dynamics of generalized anticipated synchronization errors for the drive system (12) and the response system (16) is shown in Fig. 2(c). Fig. 2 (a) Partial generalized lag synchronization error; (b) Partial generalized (complete) synchronization error; (c) Partial generalized anticipated synchronization error. (ii) Multivariable Synchronization Now let us consider the case when we wish to synchronize x 2, x 3 and y 1, y 2. Then ( ) ( ) 10 0 x 3 (t) + x 1 (t) B =, f 12 (t) =, x s (t τ) = 0 1 α(x 2 (t) F (x 3 (t))) ( ) x2 (t τ), x 3 (t τ)

No. 6 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme 1053 A = ( ) ( ) 1 0 10y 2 (t), f 21 (t) = 0 0 y 1 (t)y 3 (t) + 28y 1 (t). (20) Let the error states e(t) = (y 1 (t), y 2 (t)) T H(x 2 (t τ), x 3 (t τ)), where H(x 2 (t τ), x 3 (t τ)) is a smooth column vector function and τ R. Then we have the error dynamical system Let ė(t) = (ė 1 (t), ė 2 (t)) T = Be(t) + BH(x 2 (t τ), x 3 (t τ)) + f 21 (t) + U(x(t τ), y(t), t) DH(x 2 (t τ), x 3 (t τ))(ax s (t τ) + f 12 (t τ) + C(y 1 (t), y 2 (t), y 3 (t), t)). (21) U = (u 1, u 2 ) T = e(t) BH(x 2 (t τ), x 3 (t τ)) f 21 (t) + DH(x 2 (t τ), x 3 (t τ))(ax s (t τ) + f 12 (t τ) + C(y 1 (t), y 2 (t), y 3 (t), t)). (22) Then the error dynamical system (21) becomes ė(t) = (ė 1 (t), ė 2 (t)) T = ( + B)(e 1 (t), e 2 (t)) T. (23) It can be shown that there exist many types of solutions for δ ij such that all eigenvalues of system (21) all have negative real parts, that is to say, lim t e(t) = lim t y s (t) H(x s (t τ)) = 0. For example, there exist one family of solutions for δ ij : δ 11 = 7, δ 12 = 4, δ 21 = 0 and δ 22 = 2 such that system (21) is globally asymptotically stable. In what follows we would like to use the numerical simulations to verify the effectiveness of the above-designed controllers. Let the initial values of system (12) and (16) be H = (x 2 2(t τ), x 2 3(t τ)), α = 10.814, β = 14, a = 1.3, b = 0.11, c = 4, d = π, [x 1 (0) = 1, x 2 (0) = 0, x 3 (0) = 0] and [y 1 (0) = 2, y 2 (0) = 2, y 3 (0) = 2]. Case ii(a) Partial generalized lag synchronization. In the case τ > 0, without loss of generality, we set τ = 1. Thus by calculation, the initial values of the error dynamical system (21) is e 1 (0) = y 1 (0) x 2 2( 1) = 0.816 511 871 and e 2 (0) = y 2 (0) x 2 3( 1) = 40.815 439 88. The dynamics of partial generalized lag synchronization errors for the drive system (12) and the response system (16) is shown in Figs. 3(a) and 3(b). Fig. 3 (a) and (b) Partial generalized lag synchronization errors; (c) and (d) Partial generalized (complete) synchronization errors; (e) and (f) Partial generalized anticipated synchronization errors. Case ii(b) Partial generalized (complete) synchronization. In this case τ = 0. Thus the initial values of the error dynamical system (21) are e 1 (0) = y 1 (0) x 2 2(0) = 2 and e 2 (0) = y 2 (0) x 3 (0) = 2. Similarly, we also display the dynamics of generalized synchronization errors for the drive system (12) and the response system (16) as shown in Figs. 3(c) and 3(d). Case ii(c) Partial generalized anticipated synchronization. In this case τ < 0, and without loss of generality, we set τ = 1. Thus the initial values of the error dynamical system (21) are e 1 (0) = y 1 (0) x 2 2(1) = 1.972 326 479 and

1054 WANG Qi Vol. 45 e 2 (0) = y 2 (0) x 3 (1) = 1.667 540 254. The dynamics of generalized anticipated synchronization errors for the drive system (12) and the response system (16) are shown in Figs. 3(e) and 3(f). 3.2 Bidirectional Partial Generalized Synchronization Between Modified Chua System and Hyperchaotic Tamasevicius Namajunas Cenys System Here we will consider multivariable bidirectional partial generalized synchronization between the modified Chua system (12) and the hyperchaotic Tamasevicius Namajunas Cenys (TNC) system, [30] ẏ 1 = y 2, ẏ 2 = y 3 + 0.7y 2 y 1, ẏ 3 = y 4 + 3y 2, ẏ 4 = 3y 3 30(y 4 1)H(y 4 1), (24) where H(z) is the Heaviside function, i.e., H(z < 0) = 0 and H(z 0) = 1. System (24) contains an opamp, two LC circuits, and a diode. The projection of hyperchaotic attractor in (y 1, y 2, y 3 ) space is displayed in Fig. 4. Fig. 4 Hyperchaotic attractor of TNC system in (y 1, y 2, y 3) space. Now let us consider the case when we wish to synchronize x 1, x 2, x 3 and y 1, y 2, y 3, i.e., x s (t) = [x 1 (t), x 2 (t), x 3 (t)] T, y s = [y 1 (t), y 2 (t), y 3 (t)] T, x m (t) = 0, y n (t) = [y 4 (t)]. Then 0 0 0 B = 0 0.7 0, f 12 (t) = 0 0 0 y 2 (t) f 21 (t) = y 3 (t) y 1 (t) y 4 (t) + 3y 2 (t) βx 2 (t) x 3 (t) + x 1 (t) α(x 2 (t) F (x 3 (t))) 0 0 0, A = 0 1 0, 0 0 0. (25) Let the error states e(t) = (y 1 (t), y 2 (t), y 3 (t)) T H(x 1 (t τ), x 2 (t τ), x 3 (t τ)), where τ R and H(x 1 (t τ), x 2 (t τ), x 3 (t τ)) is a smooth column vector function. Then we have the error dynamical system Let ė(t) = Be (t) + BH (x 1 (t τ), x 2 (t τ), x 3 (t τ)) + f 21 (t) + U (x (t τ), y (t), t) DH (x 1 (t τ), x 2 (t τ), x 3 (t τ)) (Ax s (t τ) + f 12 (t τ) + C (y s (t), y n (t), t)). (26) U = (u 1, u 2, u 3 ) = e (t) BH (x 1 (t τ), x 2 (t τ), x 3 (t τ)) f 21 (t) + DH (x 1 (t τ), x 2 (t τ), x 3 (t τ)) (Ax s (t τ) + f 12 (t τ) + C (y s (t), y n (t), t)). (27) Then the error dynamical system (26) becomes ė 1 (t) e 1 (t) δ 11 δ 12 δ 13 e 1 (t) ė(t) = ė 2 (t) = ( + B) e 2 (t) = δ 21 δ 22 + 0.7 δ 23 e 2 (t). (28) ė 3 (t) e 3 (t) δ 31 δ 32 δ 33 e 3 (t) It can be shown that there exist many types of solutions for δ ij such that all eigenvalues of system (26) all have negative real parts, that is to say, lim t e(t) = lim t y s (t) H (x s (t τ)) = 0. For example, there exists one family of solutions for δ ij : δ 21 = δ 31 = δ 32 = 0, δ 11 = δ 33 = 1, δ 22 = 3.7 and δ 12 = δ 13 = δ 23 = 5, such that system (26) is globally asymptotically stable.

No. 6 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme 1055 In what follows we use the numerical simulations to verify the effectiveness of the above-designed controllers. Let the initial values of systems (12) and (24) be H = (sin (x 1 (t τ)), sinh (x 2 (t τ)), tanh (x 3 (t τ))), α = 10.814, β = 14, a = 1.3, b = 0.11, c = 4, d = π, [x 1 (0) = 1, x 2 (0) = 0, x 3 (0) = 0] and [y 1 (0) = 2, y 2 (0) = 3, y 3 (0) = 2, y 4 (0) = 3]. Case iii(a) Partial generalized lag synchronization. In this case τ > 0, without loss of generality, we set τ = 1. Thus by calculation, the initial values of the error dynamical system (26) is e 1 (0) = y 1 (0) sin (x 1 ( 1)) = 1.126 493 917, e 2 (0) = y 2 (0) sinh (x 2 ( 1)) = 4.315 526 607 and e 3 (0) = y 3 (0) tanh (x 3 ( 1)) = 2.999 995 855. The dynamics of partial generalized lag synchronization errors for the drive system (12) and the response system (24) is shown in Figs. 5(a) 5(c). Fig. 5 (a) (c) Partial generalized lag synchronization errors; (d) (f) Partial generalized (complete) synchronization errors; (g) (i) Partial generalized anticipated synchronization errors. Case iii(b) Partial generalized (complete) synchronization. In this case τ = 0. Thus the initial values of the error dynamical system (26) are e 1 (0) = y 1 (0) sin(x 1 (0)) = 1.158 529 015, e 2 (0) = y 2 (0) sinh(x 2 (0)) = 3 and e 3 (0) = y 3 (0) tanh(x 3 (0)) = 2. Similarly, we also display the dynamics of generalized synchronization errors for the drive system (12) and the response system (24) as shown in Figs. 5(d) 5(f). Case iii(c) Partial generalized anticipated synchronization. In this case τ < 0. Without loss of generality, we set τ = 1. Thus the initial values of the error dynamical system (26) are e 1 (0) = y 1 (0) sin (x 1 (1)) = 1.155 671 128, e 2 (0) = y 2 (0) sinh (x 2 (1)) = 2.832 878 070 and e 3 (0) = y 3 (0) tanh (x 3 (1)) = 2.957 510 201. The dynamics of generalized anticipated synchronization errors for the drive system (12) and the response system (24) is shown in Figs. 5(g) 5(i). Remark 4 Here we can conclude that partial chaos synchronization not only can take place between two chaotic systems, but also can take place between chaotic and hyperchaotic system, which to our knowledge, has not been studied before.

1056 WANG Qi Vol. 45 4 Summary and Conclusion In summary, by directly extending the definition by M. Hasler, [24] we have defined a bidirectional partial generalized (lag, complete, and anticipated) synchronization of chaotic systems. A systematic and powerful scheme has been developed to investigate the bidirectional partial generalized (lag, complete, and anticipated) synchronization between the drive system and response system based on the active control idea. The modified Chua system, Lorenz system, and the hyperchaotic Tamasevicius Namajunas Cenys system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme for different smooth vector function H. So we can get partial chaos synchronization between chaotic and hyperchaotic systems. The scheme can be also extended to research bidirectional partial generalized (lag, complete, and anticipated) synchronization between other dynamical systems. This will be studied further. Acknowledgments The author thanks the referees for their valuable suggestions and is very grateful to Dr. Yan Zhen-Ya for his enthusiastic guidance and help. References [1] L.M. Pecora and T.L. Carroll, Phys. Rev. Lett. 64 (1990) 821. [2] T.L. Caroll and L.M. Pecora, IEEE Trans. Circuits Syst. I: Fundam. Theory Appl. 38 (1991) 453. [3] L. Kocarev and U. Parlitz, Phys. Rev. Lett. 76 (1996) 1816. [4] N.F. Rulkov, et al., Phys. Rev. E 51 (1995) 980. [5] H.D.I. Abarbanel, et al., Phys. Rev. E 53 (1996) 4528. [6] S. Boccaletti, et al., Phys. Rep. 366 (2002) 1. [7] M.G. Rosenblum, et al., Phys. Rev. Lett. 78 (1997) 4193. [8] C. Li, et al., Phys. Lett. A 329 (2004) 301. [9] H.U. Voss, Phys. Rev. E 61 (2000) 5115. [10] S. Sivaprakasam, et al., Phys. Rev. Lett. 87 (2001) 154101. [11] M. Ho, et al., Phys. Lett. A 296 (2002) 43. [12] Z.Y. Yan, Chaos 15 (2005) 013101; Z.Y. Yan, Commun. Theor. Phys. (Beijing, China) 44 (2005) 72. [13] J.F. Heagy, et al., Phys. Rev. E 50 (1994) 1874. [14] J.Y. Hsieh, et al., Int. J. Control 72 (1999) 882. [15] K. Tanaka and H.O. Wang, IEEE World Congress on Fuzzy Systems Proceedings 1 (1998) 434. [16] A.W. Hubler, Helv. Phys. Acta 62 (1989) 434. [17] T. Yang, et al., Phys. D 110 (1997) 18. [18] Y. Wang, et al., Chaos 14 (2004) 199. [19] E. Bai and K.E. Lonngren, Chaos, Solitons & Fractals 8 (1997) 51. [20] H.N. Agiza and M.T. Yassen, Phys. Lett. A 278 (2001) 191. [21] M.C. Ho and Y.C. Huang, Phys. Lett. A 301 (2002) 424. [22] G. Grassi and S. Mascolo, Electron. Lett. 34 (1998) 1844. [23] G. Chen and X. Dong, From Chaos to Order, World Scientific, Singapore (1998). [24] M. Hasler, et al., Phys. Rev. Lett. 58 (1998) 6843. [25] S. Yanchuk, et al., Math. Comput. Simu. 54 (2001) 491. [26] D.J. Wagg, Int. J. Bifur. Chaos 12 (2002) 561. [27] J. Lu and Y. Xi, Chaos, Solitons & Fractals 17 (2003) 825. [28] K. S. Tang, et al., IEEE Trans. Circuits Syst. I 48(11) (2001) 1369. [29] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [30] A. Tamasevicius, et al., Electron. Lett. 32 (1996) 957.