Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme
|
|
- Christal Lucas
- 6 years ago
- Views:
Transcription
1 Commun. Theor. Phys. (Beijing, China) 45 (2006) pp 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 , China Key Laboratory of Mathematics Mechanization, the Chinese Academy of Sciences, Beijing , 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: 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 wangqi dlut@yahoo.com.cn
2 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)
3 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) = b ss + δ ss b 11 + δ 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.
4 1052 WANG Qi Vol. 45 The well-known Lorenz system was given as ẏ 1 = 10y y 2, ẏ 2 = y 1 y y 1 y 2, ẏ 3 = y 1 y 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 = , 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) = 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) = 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)
5 No. 6 Bidirectional Partial Generalized Synchronization in Chaotic and Hyperchaotic Systems via a New Scheme 1053 A = ( ) ( ) y 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 = , 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) = and e 2 (0) = y 2 (0) x 2 3( 1) = 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) = and
6 1054 WANG Qi Vol. 45 e 2 (0) = y 2 (0) x 3 (1) = 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 y 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 B = , f 12 (t) = 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, (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 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.
7 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 = , 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)) = , e 2 (0) = y 2 (0) sinh (x 2 ( 1)) = and e 3 (0) = y 3 (0) tanh (x 3 ( 1)) = 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)) = , 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)) = , e 2 (0) = y 2 (0) sinh (x 2 (1)) = and e 3 (0) = y 3 (0) tanh (x 3 (1)) = 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.
8 1056 WANG Qi Vol 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) [4] N.F. Rulkov, et al., Phys. Rev. E 51 (1995) 980. [5] H.D.I. Abarbanel, et al., Phys. Rev. E 53 (1996) [6] S. Boccaletti, et al., Phys. Rep. 366 (2002) 1. [7] M.G. Rosenblum, et al., Phys. Rev. Lett. 78 (1997) [8] C. Li, et al., Phys. Lett. A 329 (2004) 301. [9] H.U. Voss, Phys. Rev. E 61 (2000) [10] S. Sivaprakasam, et al., Phys. Rev. Lett. 87 (2001) [11] M. Ho, et al., Phys. Lett. A 296 (2002) 43. [12] Z.Y. Yan, Chaos 15 (2005) ; Z.Y. Yan, Commun. Theor. Phys. (Beijing, China) 44 (2005) 72. [13] J.F. Heagy, et al., Phys. Rev. E 50 (1994) [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) [23] G. Chen and X. Dong, From Chaos to Order, World Scientific, Singapore (1998). [24] M. Hasler, et al., Phys. Rev. Lett. 58 (1998) [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) [29] E.N. Lorenz, J. Atmos. Sci. 20 (1963) 130. [30] A. Tamasevicius, et al., Electron. Lett. 32 (1996) 957.
Generalized-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal
Commun. Theor. Phys. (Beijing, China) 44 (25) pp. 72 78 c International Acaemic Publishers Vol. 44, No. 1, July 15, 25 Generalize-Type Synchronization of Hyperchaotic Oscillators Using a Vector Signal
More informationFunction Projective Synchronization of Discrete-Time Chaotic and Hyperchaotic Systems Using Backstepping Method
Commun. Theor. Phys. (Beijing, China) 50 (2008) pp. 111 116 c Chinese Physical Society Vol. 50, No. 1, July 15, 2008 Function Projective Synchronization of Discrete-Time Chaotic and Hyperchaotic Systems
More informationStudy on Proportional Synchronization of Hyperchaotic Circuit System
Commun. Theor. Phys. (Beijing, China) 43 (25) pp. 671 676 c International Academic Publishers Vol. 43, No. 4, April 15, 25 Study on Proportional Synchronization of Hyperchaotic Circuit System JIANG De-Ping,
More informationGeneralized projective synchronization of a class of chaotic (hyperchaotic) systems with uncertain parameters
Vol 16 No 5, May 2007 c 2007 Chin. Phys. Soc. 1009-1963/2007/16(05)/1246-06 Chinese Physics and IOP Publishing Ltd Generalized projective synchronization of a class of chaotic (hyperchaotic) systems with
More information698 Zou Yan-Li et al Vol. 14 and L 2, respectively, V 0 is the forward voltage drop across the diode, and H(u) is the Heaviside function 8 < 0 u < 0;
Vol 14 No 4, April 2005 cfl 2005 Chin. Phys. Soc. 1009-1963/2005/14(04)/0697-06 Chinese Physics and IOP Publishing Ltd Chaotic coupling synchronization of hyperchaotic oscillators * Zou Yan-Li( ΠΛ) a)y,
More informationAdaptive feedback synchronization of a unified chaotic system
Physics Letters A 39 (4) 37 333 www.elsevier.com/locate/pla Adaptive feedback synchronization of a unified chaotic system Junan Lu a, Xiaoqun Wu a, Xiuping Han a, Jinhu Lü b, a School of Mathematics and
More informationBackstepping synchronization of uncertain chaotic systems by a single driving variable
Vol 17 No 2, February 2008 c 2008 Chin. Phys. Soc. 1674-1056/2008/17(02)/0498-05 Chinese Physics B and IOP Publishing Ltd Backstepping synchronization of uncertain chaotic systems by a single driving variable
More informationProjective synchronization of a complex network with different fractional order chaos nodes
Projective synchronization of a complex network with different fractional order chaos nodes Wang Ming-Jun( ) a)b), Wang Xing-Yuan( ) a), and Niu Yu-Jun( ) a) a) School of Electronic and Information Engineering,
More informationA SYSTEMATIC PROCEDURE FOR SYNCHRONIZING HYPERCHAOS VIA OBSERVER DESIGN
Journal of Circuits, Systems, and Computers, Vol. 11, No. 1 (22) 1 16 c World Scientific Publishing Company A SYSTEMATIC PROCEDURE FOR SYNCHRONIZING HYPERCHAOS VIA OBSERVER DESIGN GIUSEPPE GRASSI Dipartimento
More informationSynchronizing Chaotic Systems Based on Tridiagonal Structure
Proceedings of the 7th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-, 008 Synchronizing Chaotic Systems Based on Tridiagonal Structure Bin Liu, Min Jiang Zengke
More informationAnti-synchronization of a new hyperchaotic system via small-gain theorem
Anti-synchronization of a new hyperchaotic system via small-gain theorem Xiao Jian( ) College of Mathematics and Statistics, Chongqing University, Chongqing 400044, China (Received 8 February 2010; revised
More informationTime-delay feedback control in a delayed dynamical chaos system and its applications
Time-delay feedback control in a delayed dynamical chaos system and its applications Ye Zhi-Yong( ), Yang Guang( ), and Deng Cun-Bing( ) School of Mathematics and Physics, Chongqing University of Technology,
More informationChaos synchronization of complex Rössler system
Appl. Math. Inf. Sci. 7, No. 4, 1415-1420 (2013) 1415 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/070420 Chaos synchronization of complex Rössler
More informationChaos Synchronization of Nonlinear Bloch Equations Based on Input-to-State Stable Control
Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 308 312 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 2, February 15, 2010 Chaos Synchronization of Nonlinear Bloch Equations Based
More informationADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS
ADAPTIVE CHAOS SYNCHRONIZATION OF UNCERTAIN HYPERCHAOTIC LORENZ AND HYPERCHAOTIC LÜ SYSTEMS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationFunction Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping
Commun. Theor. Phys. 55 (2011) 617 621 Vol. 55, No. 4, April 15, 2011 Function Projective Synchronization of Fractional-Order Hyperchaotic System Based on Open-Plus-Closed-Looping WANG Xing-Yuan ( ), LIU
More informationImpulsive synchronization of chaotic systems
CHAOS 15, 023104 2005 Impulsive synchronization of chaotic systems Chuandong Li a and Xiaofeng Liao College of Computer Science and Engineering, Chongqing University, 400030 China Xingyou Zhang College
More informationComplete synchronization and generalized synchronization of one-way coupled time-delay systems
Complete synchronization and generalized synchronization of one-way coupled time-delay systems Meng Zhan, 1 Xingang Wang, 1 Xiaofeng Gong, 1 G. W. Wei,,3 and C.-H. Lai 4 1 Temasek Laboratories, National
More informationExperimental and numerical realization of higher order autonomous Van der Pol-Duffing oscillator
Indian Journal of Pure & Applied Physics Vol. 47, November 2009, pp. 823-827 Experimental and numerical realization of higher order autonomous Van der Pol-Duffing oscillator V Balachandran, * & G Kandiban
More informationComputers and Mathematics with Applications. Adaptive anti-synchronization of chaotic systems with fully unknown parameters
Computers and Mathematics with Applications 59 (21) 3234 3244 Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa Adaptive
More informationCONTROLLING CHAOTIC DYNAMICS USING BACKSTEPPING DESIGN WITH APPLICATION TO THE LORENZ SYSTEM AND CHUA S CIRCUIT
Letters International Journal of Bifurcation and Chaos, Vol. 9, No. 7 (1999) 1425 1434 c World Scientific Publishing Company CONTROLLING CHAOTIC DYNAMICS USING BACKSTEPPING DESIGN WITH APPLICATION TO THE
More informationChaos synchronization of nonlinear Bloch equations
Chaos, Solitons and Fractal7 (26) 357 361 www.elsevier.com/locate/chaos Chaos synchronization of nonlinear Bloch equations Ju H. Park * Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical
More informationNew communication schemes based on adaptive synchronization
CHAOS 17, 0114 2007 New communication schemes based on adaptive synchronization Wenwu Yu a Department of Mathematics, Southeast University, Nanjing 210096, China, Department of Electrical Engineering,
More information[2] B.Van der Pol and J. Van der Mark, Nature 120,363(1927)
Bibliography [1] J. H. Poincaré, Acta Mathematica 13, 1 (1890). [2] B.Van der Pol and J. Van der Mark, Nature 120,363(1927) [3] M. L. Cartwright and J. E. Littlewood, Journal of the London Mathematical
More informationComplete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different 4D Nonlinear Dynamical Systems
Mathematics Letters 2016; 2(5): 36-41 http://www.sciencepublishinggroup.com/j/ml doi: 10.11648/j.ml.20160205.12 Complete Synchronization, Anti-synchronization and Hybrid Synchronization Between Two Different
More informationDynamical Behavior And Synchronization Of Chaotic Chemical Reactors Model
Iranian Journal of Mathematical Chemistry, Vol. 6, No. 1, March 2015, pp. 81 92 IJMC Dynamical Behavior And Synchronization Of Chaotic Chemical Reactors Model HOSSEIN KHEIRI 1 AND BASHIR NADERI 2 1 Faculty
More informationK. Pyragas* Semiconductor Physics Institute, LT-2600 Vilnius, Lithuania Received 19 March 1998
PHYSICAL REVIEW E VOLUME 58, NUMBER 3 SEPTEMBER 998 Synchronization of coupled time-delay systems: Analytical estimations K. Pyragas* Semiconductor Physics Institute, LT-26 Vilnius, Lithuania Received
More informationTHE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS
THE ACTIVE CONTROLLER DESIGN FOR ACHIEVING GENERALIZED PROJECTIVE SYNCHRONIZATION OF HYPERCHAOTIC LÜ AND HYPERCHAOTIC CAI SYSTEMS Sarasu Pakiriswamy 1 and Sundarapandian Vaidyanathan 1 1 Department of
More informationGLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC QI AND HYPERCHAOTIC JHA SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationHYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL
HYBRID CHAOS SYNCHRONIZATION OF HYPERCHAOTIC LIU AND HYPERCHAOTIC CHEN SYSTEMS BY ACTIVE NONLINEAR CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationLag anti-synchronization of delay coupled chaotic systems via a scalar signal
Lag anti-synchronization of delay coupled chaotic systems via a scalar signal Mohammad Ali Khan Abstract. In this letter, a chaotic anti-synchronization (AS scheme is proposed based on combining a nonlinear
More informationGeneralized projective synchronization between two chaotic gyros with nonlinear damping
Generalized projective synchronization between two chaotic gyros with nonlinear damping Min Fu-Hong( ) Department of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042, China
More informationCommun Nonlinear Sci Numer Simulat
Commun Nonlinear Sci Numer Simulat 14 29) 1494 151 Contents lists available at ScienceDirect Commun Nonlinear Sci Numer Simulat journal homepage: www.elsevier.com/locate/cnsns The function cascade synchronization
More informationGLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN SPROTT J AND K SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi,
More informationRobust Gain Scheduling Synchronization Method for Quadratic Chaotic Systems With Channel Time Delay Yu Liang and Horacio J.
604 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 56, NO. 3, MARCH 2009 Robust Gain Scheduling Synchronization Method for Quadratic Chaotic Systems With Channel Time Delay Yu Liang
More informationADAPTIVE SYNCHRONIZATION FOR RÖSSLER AND CHUA S CIRCUIT SYSTEMS
Letters International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1579 1597 c World Scientific Publishing Company ADAPTIVE SYNCHRONIZATION FOR RÖSSLER AND CHUA S CIRCUIT SYSTEMS A. S. HEGAZI,H.N.AGIZA
More information3. Controlling the time delay hyper chaotic Lorenz system via back stepping control
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol 10, No 2, 2015, pp 148-153 Chaos control of hyper chaotic delay Lorenz system via back stepping method Hanping Chen 1 Xuerong
More informationGLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationStability and Projective Synchronization in Multiple Delay Rössler System
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.7(29) No.2,pp.27-214 Stability and Projective Synchronization in Multiple Delay Rössler System Dibakar Ghosh Department
More informationAdaptive synchronization of uncertain chaotic systems via switching mechanism
Chin Phys B Vol 19, No 12 (2010) 120504 Adaptive synchronization of uncertain chaotic systems via switching mechanism Feng Yi-Fu( ) a) and Zhang Qing-Ling( ) b) a) School of Mathematics, Jilin Normal University,
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC NEWTON-LEIPNIK SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
More informationPhase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
Commun. Theor. Phys. (Beijing, China) 47 (2007) pp. 265 269 c International Academic Publishers Vol. 47, No. 2, February 15, 2007 Phase Synchronization of Coupled Rossler Oscillators: Amplitude Effect
More informationMULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED FROM CHEN SYSTEM
International Journal of Bifurcation and Chaos, Vol. 22, No. 2 (212) 133 ( pages) c World Scientific Publishing Compan DOI: 1.1142/S21812741332 MULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED
More informationADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM
ADAPTIVE DESIGN OF CONTROLLER AND SYNCHRONIZER FOR LU-XIAO CHAOTIC SYSTEM WITH UNKNOWN PARAMETERS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University
More informationGlobal Chaos Synchronization of Hyperchaotic Lorenz and Hyperchaotic Chen Systems by Adaptive Control
Global Chaos Synchronization of Hyperchaotic Lorenz and Hyperchaotic Chen Systems by Adaptive Control Dr. V. Sundarapandian Professor, Research and Development Centre Vel Tech Dr. RR & Dr. SR Technical
More informationGLOBAL CHAOS SYNCHRONIZATION OF PAN AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROL
GLOBAL CHAOS SYNCHRONIZATION OF PAN AND LÜ CHAOTIC SYSTEMS VIA ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 and Karthikeyan Rajagopal 2 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationBifurcation control and chaos in a linear impulsive system
Vol 8 No 2, December 2009 c 2009 Chin. Phys. Soc. 674-056/2009/82)/5235-07 Chinese Physics B and IOP Publishing Ltd Bifurcation control and chaos in a linear impulsive system Jiang Gui-Rong 蒋贵荣 ) a)b),
More informationADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM
ADAPTIVE CONTROL AND SYNCHRONIZATION OF A GENERALIZED LOTKA-VOLTERRA SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical University Avadi, Chennai-600
More informationTracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single Input
ISSN 1746-7659, England, UK Journal of Information and Computing Science Vol. 11, No., 016, pp.083-09 Tracking the State of the Hindmarsh-Rose Neuron by Using the Coullet Chaotic System Based on a Single
More informationGeneralized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems
Generalized Function Projective Lag Synchronization in Fractional-Order Chaotic Systems Yancheng Ma Guoan Wu and Lan Jiang denotes fractional order of drive system Abstract In this paper a new synchronization
More informationPhase Synchronization of Van der Pol-Duffing Oscillators Using Decomposition Method
Adv. Studies Theor. Phys., Vol. 3, 29, no. 11, 429-437 Phase Synchronization of Van der Pol-Duffing Oscillators Using Decomposition Method Gh. Asadi Cordshooli Department of Physics, Shahr-e-Rey Branch,
More informationSYNCHRONIZATION IN SMALL-WORLD DYNAMICAL NETWORKS
International Journal of Bifurcation and Chaos, Vol. 12, No. 1 (2002) 187 192 c World Scientific Publishing Company SYNCHRONIZATION IN SMALL-WORLD DYNAMICAL NETWORKS XIAO FAN WANG Department of Automation,
More informationThe Application of Contraction Theory in Synchronization of Coupled Chen Systems
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.1,pp.72-77 The Application of Contraction Theory in Synchronization of Coupled Chen Systems Hongxing
More informationADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS
ADAPTIVE CONTROLLER DESIGN FOR THE ANTI-SYNCHRONIZATION OF HYPERCHAOTIC YANG AND HYPERCHAOTIC PANG SYSTEMS Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationPhase Desynchronization as a Mechanism for Transitions to High-Dimensional Chaos
Commun. Theor. Phys. (Beijing, China) 35 (2001) pp. 682 688 c International Academic Publishers Vol. 35, No. 6, June 15, 2001 Phase Desynchronization as a Mechanism for Transitions to High-Dimensional
More informationResearch Article Mathematical Model and Cluster Synchronization for a Complex Dynamical Network with Two Types of Chaotic Oscillators
Applied Mathematics Volume 212, Article ID 936, 12 pages doi:1.11/212/936 Research Article Mathematical Model and Cluster Synchronization for a Complex Dynamical Network with Two Types of Chaotic Oscillators
More informationA Non-symmetric Digital Image Secure Communication Scheme Based on Generalized Chaos Synchronization System
Commun. Theor. Phys. (Beijing China) 44 (2005) pp. 1115 1124 c International Academic Publishers Vol. 44 No. 6 December 15 2005 A Non-symmetric Digital Image Secure Communication Scheme Based on Generalized
More informationA Novel Hyperchaotic System and Its Control
1371371371371378 Journal of Uncertain Systems Vol.3, No., pp.137-144, 009 Online at: www.jus.org.uk A Novel Hyperchaotic System and Its Control Jiang Xu, Gouliang Cai, Song Zheng School of Mathematics
More informationADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM
ADAPTIVE STABILIZATION AND SYNCHRONIZATION OF HYPERCHAOTIC QI SYSTEM Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR Dr. SR Technical University Avadi, Chennai-600 062,
More informationConstruction of a New Fractional Chaotic System and Generalized Synchronization
Commun. Theor. Phys. (Beijing, China) 5 (2010) pp. 1105 1110 c Chinese Physical Society and IOP Publishing Ltd Vol. 5, No. 6, June 15, 2010 Construction of a New Fractional Chaotic System and Generalized
More informationAverage Range and Network Synchronizability
Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 115 120 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 1, January 15, 2010 Average Range and Network Synchronizability LIU Chao ( ),
More informationChaos, Solitons and Fractals
Chaos, Solitons and Fractals 41 (2009) 962 969 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos A fractional-order hyperchaotic system
More informationLinear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays
Chin. Phys. B Vol. 21, No. 4 (212 4842 Linear matrix inequality approach for synchronization control of fuzzy cellular neural networks with mixed time delays P. Balasubramaniam a, M. Kalpana a, and R.
More informationA Generalization of Some Lag Synchronization of System with Parabolic Partial Differential Equation
American Journal of Theoretical and Applied Statistics 2017; 6(5-1): 8-12 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.s.2017060501.12 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online)
More informationSome explicit formulas of Lyapunov exponents for 3D quadratic mappings
Some explicit formulas of Lyapunov exponents for 3D quadratic mappings Zeraoulia Elhadj 1,J.C.Sprott 2 1 Department of Mathematics, University of Tébessa, (12002), Algeria. E-mail: zeraoulia@mail.univ-tebessa.dz
More informationResearch Article Adaptive Control of Chaos in Chua s Circuit
Mathematical Problems in Engineering Volume 2011, Article ID 620946, 14 pages doi:10.1155/2011/620946 Research Article Adaptive Control of Chaos in Chua s Circuit Weiping Guo and Diantong Liu Institute
More informationExperimental observation of direct current voltage-induced phase synchronization
PRAMANA c Indian Academy of Sciences Vol. 67, No. 3 journal of September 2006 physics pp. 441 447 Experimental observation of direct current voltage-induced phase synchronization HAIHONG LI 1, WEIQING
More informationFinite-time hybrid synchronization of time-delay hyperchaotic Lorenz system
ISSN 1746-7659 England UK Journal of Information and Computing Science Vol. 10 No. 4 2015 pp. 265-270 Finite-time hybrid synchronization of time-delay hyperchaotic Lorenz system Haijuan Chen 1 * Rui Chen
More informationSynchronization of different chaotic systems and electronic circuit analysis
Synchronization of different chaotic systems and electronic circuit analysis J.. Park, T.. Lee,.. Ji,.. Jung, S.M. Lee epartment of lectrical ngineering, eungnam University, Kyongsan, Republic of Korea.
More informationAn Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson Equation
Commun. Theor. Phys. (Beijing, China) 50 (008) pp. 309 314 c Chinese Physical Society Vol. 50, No., August 15, 008 An Improved F-Expansion Method and Its Application to Coupled Drinfel d Sokolov Wilson
More informationSynchronization of identical new chaotic flows via sliding mode controller and linear control
Synchronization of identical new chaotic flows via sliding mode controller and linear control Atefeh Saedian, Hassan Zarabadipour Department of Electrical Engineering IKI University Iran a.saedian@gmail.com,
More informationDynamical analysis and circuit simulation of a new three-dimensional chaotic system
Dynamical analysis and circuit simulation of a new three-dimensional chaotic system Wang Ai-Yuan( 王爱元 ) a)b) and Ling Zhi-Hao( 凌志浩 ) a) a) Department of Automation, East China University of Science and
More informationControlling chaos in Colpitts oscillator
Chaos, Solitons and Fractals 33 (2007) 582 587 www.elsevier.com/locate/chaos Controlling chaos in Colpitts oscillator Guo Hui Li a, *, Shi Ping Zhou b, Kui Yang b a Department of Communication Engineering,
More informationNetwork synchronizability analysis: The theory of subgraphs and complementary graphs
Physica D 237 (2008) 1006 1012 www.elsevier.com/locate/physd Network synchronizability analysis: The theory of subgraphs and complementary graphs Zhisheng Duan a,, Chao Liu a, Guanrong Chen a,b a State
More informationFinite Time Synchronization between Two Different Chaotic Systems with Uncertain Parameters
www.ccsenet.org/cis Coputer and Inforation Science Vol., No. ; August 00 Finite Tie Synchronization between Two Different Chaotic Systes with Uncertain Paraeters Abstract Wanli Yang, Xiaodong Xia, Yucai
More informationA new four-dimensional chaotic system
Chin. Phys. B Vol. 19 No. 12 2010) 120510 A new four-imensional chaotic system Chen Yong ) a)b) an Yang Yun-Qing ) a) a) Shanghai Key Laboratory of Trustworthy Computing East China Normal University Shanghai
More informationAnti-synchronization Between Coupled Networks with Two Active Forms
Commun. Theor. Phys. 55 (211) 835 84 Vol. 55, No. 5, May 15, 211 Anti-synchronization Between Coupled Networks with Two Active Forms WU Yong-Qing ( ï), 1 SUN Wei-Gang (êå ), 2, and LI Shan-Shan (Ó ) 3
More informationControlling a Novel Chaotic Attractor using Linear Feedback
ISSN 746-7659, England, UK Journal of Information and Computing Science Vol 5, No,, pp 7-4 Controlling a Novel Chaotic Attractor using Linear Feedback Lin Pan,, Daoyun Xu 3, and Wuneng Zhou College of
More informationHYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL
HYBRID CHAOS SYNCHRONIZATION OF UNCERTAIN LORENZ-STENFLO AND QI 4-D CHAOTIC SYSTEMS BY ADAPTIVE CONTROL Sundarapandian Vaidyanathan 1 1 Research and Development Centre, Vel Tech Dr. RR & Dr. SR Technical
More informationRational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional Dispersive Long Wave Equation
Commun. Theor. Phys. (Beijing, China) 43 (005) pp. 975 98 c International Academic Publishers Vol. 43, No. 6, June 15, 005 Rational Form Solitary Wave Solutions and Doubly Periodic Wave Solutions to (1+1)-Dimensional
More informationSYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA S CANONICAL CIRCUIT. THE CASE OF UNCERTAINTY IN CHAOS SYNCHRONIZATION
International Journal of Bifurcation and Chaos, Vol. 16, No. 7 (2006) 1961 1976 c World Scientific Publishing Company SYNCHRONIZING CHAOTIC ATTRACTORS OF CHUA S CANONICAL CIRCUIT. THE CASE OF UNCERTAINTY
More informationImpulsive Stabilization for Control and Synchronization of Chaotic Systems: Theory and Application to Secure Communication
976 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: FUNDAMENTAL THEORY AND APPLICATIONS, VOL. 44, NO. 10, OCTOBER 1997 Impulsive Stabilization for Control and Synchronization of Chaotic Systems: Theory and
More informationarxiv: v1 [nlin.cd] 23 Jan 2019
Synchronization of Chaotic Oscillators With Partial Linear Feedback Control K. Mistry, 1 S. Dash, 1, a) 1, b) and S. Tallur Indian Institute of Technology (IIT) Bombay, Mumbai, India c) (Dated: 24 January
More informationBIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ HYPERCHAOTIC SYSTEM
Journal of Applied Analysis and Computation Volume 5, Number 2, May 215, 21 219 Website:http://jaac-online.com/ doi:1.11948/21519 BIFURCATIONS AND SYNCHRONIZATION OF THE FRACTIONAL-ORDER SIMPLIFIED LORENZ
More informationAdaptive synchronization of chaotic neural networks with time delays via delayed feedback control
2017 º 12 È 31 4 ½ Dec. 2017 Communication on Applied Mathematics and Computation Vol.31 No.4 DOI 10.3969/j.issn.1006-6330.2017.04.002 Adaptive synchronization of chaotic neural networks with time delays
More informationSynchronization of an uncertain unified chaotic system via adaptive control
Chaos, Solitons and Fractals 14 (22) 643 647 www.elsevier.com/locate/chaos Synchronization of an uncertain unified chaotic system via adaptive control Shihua Chen a, Jinhu L u b, * a School of Mathematical
More informationADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM
International Journal o Computer Science, Engineering and Inormation Technology (IJCSEIT), Vol.1, No., June 011 ADAPTIVE CHAOS CONTROL AND SYNCHRONIZATION OF HYPERCHAOTIC LIU SYSTEM Sundarapandian Vaidyanathan
More informationControlling the Period-Doubling Bifurcation of Logistic Model
ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.20(2015) No.3,pp.174-178 Controlling the Period-Doubling Bifurcation of Logistic Model Zhiqian Wang 1, Jiashi Tang
More informationADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT
International Journal of Bifurcation and Chaos, Vol. 12, No. 7 (2002) 1599 1604 c World Scientific Publishing Company ADAPTIVE FEEDBACK LINEARIZING CONTROL OF CHUA S CIRCUIT KEVIN BARONE and SAHJENDRA
More informationGeneralized function projective synchronization of chaotic systems for secure communication
RESEARCH Open Access Generalized function projective synchronization of chaotic systems for secure communication Xiaohui Xu Abstract By using the generalized function projective synchronization (GFPS)
More informationCharacteristics and synchronization of time-delay systems driven by a common noise
Eur. Phys. J. Special Topics 87, 87 93 (2) c EDP Sciences, Springer-Verlag 2 DOI:.4/epjst/e2-273-4 THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS Regular Article Characteristics and synchronization of time-delay
More informationSTUDY OF SYNCHRONIZED MOTIONS IN A ONE-DIMENSIONAL ARRAY OF COUPLED CHAOTIC CIRCUITS
International Journal of Bifurcation and Chaos, Vol 9, No 11 (1999) 19 4 c World Scientific Publishing Company STUDY OF SYNCHRONIZED MOTIONS IN A ONE-DIMENSIONAL ARRAY OF COUPLED CHAOTIC CIRCUITS ZBIGNIEW
More informationHyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system
Nonlinear Dyn (2012) 69:1383 1391 DOI 10.1007/s11071-012-0354-x ORIGINAL PAPER Hyperchaos and hyperchaos control of the sinusoidally forced simplified Lorenz system Keihui Sun Xuan Liu Congxu Zhu J.C.
More informationSynchronizing Hyperchaotic Systems by Observer Design
78 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 6, NO., APRIL 1999 M. Kunz, and K. Vogelsang, Eds. Leipzig, Germany: Verlag im Wissenschaftszentrum Leipzig,
More informationSynchronization and control in small networks of chaotic electronic circuits
Synchronization and control in small networks of chaotic electronic circuits A. Iglesias Dept. of Applied Mathematics and Computational Sciences, Universi~ of Cantabria, Spain Abstract In this paper, a
More informationA New Circuit for Generating Chaos and Complexity: Analysis of the Beats Phenomenon
A New Circuit for Generating Chaos and Complexity: Analysis of the Beats Phenomenon DONATO CAFAGNA, GIUSEPPE GRASSI Diparnto Ingegneria Innovazione Università di Lecce via Monteroni, 73 Lecce ITALY Abstract:
More informationUSING DYNAMIC NEURAL NETWORKS TO GENERATE CHAOS: AN INVERSE OPTIMAL CONTROL APPROACH
International Journal of Bifurcation and Chaos, Vol. 11, No. 3 (2001) 857 863 c World Scientific Publishing Company USING DYNAMIC NEURAL NETWORKS TO GENERATE CHAOS: AN INVERSE OPTIMAL CONTROL APPROACH
More informationGenerating a Complex Form of Chaotic Pan System and its Behavior
Appl. Math. Inf. Sci. 9, No. 5, 2553-2557 (2015) 2553 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090540 Generating a Complex Form of Chaotic Pan
More informationA New Dynamic Phenomenon in Nonlinear Circuits: State-Space Analysis of Chaotic Beats
A New Dynamic Phenomenon in Nonlinear Circuits: State-Space Analysis of Chaotic Beats DONATO CAFAGNA, GIUSEPPE GRASSI Diparnto Ingegneria Innovazione Università di Lecce via Monteroni, 73 Lecce ITALY giuseppe.grassi}@unile.it
More informationResearch Article Design of PDC Controllers by Matrix Reversibility for Synchronization of Yin and Yang Chaotic Takagi-Sugeno Fuzzy Henon Maps
Abstract and Applied Analysis Volume 212, Article ID 35821, 11 pages doi:1.1155/212/35821 Research Article Design of PDC Controllers by Matrix Reversibility for Synchronization of Yin and Yang Chaotic
More information