Generalized Synchronization of Different Dimensional Chaotic

Chaos, Solitons and Fractals 32 (2007) 773–779 www.elsevier.com/locate/chaos

Generalized synchronization of diﬀerent dimensional chaotic dynamical systems

Gang Zhang a,b,*, Zengrong Liu a, Zhongjun Ma a

a Department of Mathematics, Shanghai University, Shanghai 200444, China b Department of Mathematics, Shijiazhuang College, Shijiazhuang 050035, China

Accepted 9 November 2005

Abstract

Generalized synchronization between two different dimensional chaotic dynamical systems possesses significance in the research of synchronization. In order to study it, we propose an approach based on the Lyapunov stability theory. This method can be used widely. Numerical examples are given to demonstrate the effectiveness of this approach. 2006 Elsevier Ltd. All rights reserved.

1. Introduction

Synchronization is a fundamental phenomenon that enables coherent behavior in coupled systems. In 1990, Pecora and Carroll introduced a method [1] to synchronize two identical chaotic systems with different initial conditions. Since then, due to many potential applications in secure communication, biological science, chemical reaction, social science, and many other fields, the synchronization of coupled chaotic dynamical systems has been one of the most interesting topics in nonlinear science and many theoretical and experimental results have been obtained. A significant result is the discovery of a variety of different synchronization phenomena, such as complete synchronization (CS) [1–3], phase synchronization (PS) [4,5], lag synchronization (LS) or anticipated synchronization (AS) [6,7], and generalized synchronization (GS) [8–11], etc. At present, most of theoretical results about synchronization phenomena focus on structurally equivalent systems (identical systems or nonidentical systems whose nonidentity is resulted from a rather small parameter mismatch) [12]. However, we often need to discuss coherent behavior of strictly different dynamical systems (including different dimensional systems), especially the systems in biological science and social science [13,14]. For example, we can observe that both circulatory and respiratory systems behave in synchronous way, but their models are essentially different and they have different dimensions. So, the theoretical research of synchronization for strictly different dynamical systems is very important. Whereas this kind of research is just at the beginning stage. To the best of our knowledge, there are few theoretical results about generalized synchronization of different dimensional chaotic dynamical systems [15]. In this paper, we propose an approach to realize the globally generalized synchronization between two different dimensional dynamical systems.

* Corresponding author. Address: Department of Mathematics, Shanghai University, Shanghai 200444, China. E-mail address: [email protected] (G. Zhang).

The rest of the paper is organized as follows. In Section 2, we define a type of generalized synchronization with respect to a function relationship between two dynamical systems. Based on the Lyapunov stability theory, we propose a systematic and powerful approach to realize the generalized synchronization between two different dimensional dynamical systems. In Section 3, numerical simulations show that the method can satisfactorily ensure the occurrence of generalized synchronization between two different dimensional chaotic dynamical systems. Finally, brief conclusion remarks are given in Section 4.

2. Theoretical results

Consider the following coupled chaotic systems: x_ ¼ f ðxÞ; ð1Þ y_ ¼ gðyÞþuðy; xÞ; T n n n where x =(x1,x2,...,xn) 2 R is the n-dimensional state vector, f : R ! R is a continuous vector function, T m m m y =(y1,y2,...,ym) 2 R is the m-dimensional state vector, g : R ! R is a continuous vector function, and u(y,x) 2 Rm is a controller. Let x(t,x0) and y(t,y0,x0) be the solutions of drive system (the ﬁrst subsystem of systems (1)) and response system (the second subsystem of systems (1)), where x0 and y0 are initial values. For simplicity, we denote x(t,x0) and y(t,y0,x0) by x(t) and y(t), respectively. k•k denotes a vector norm. Deﬁnition. We say systems (1) are generalized synchronous with respect to the vector map /, if there exist a controller u(y,x) 2 Rm and a given map / : Rn ! Rm such that the solutions of systems (1) satisfy the following property: lim kyðtÞ/ðxðtÞk ¼ 0. t!1

Remark. There are many exact classes of GS based on the mathematical property of the map / [16,17]. Here, the map / is an arbitrary continuously differentiable function. So, with this definition, our study about GS has more extensive meaning than before. It is well known that many systems, such as Lorenz system, Chen system, Chua’s circuit, Ro¨ssler system, hyperchaotic Ro¨ssler system, and Lu¨ system, can be written as the form y_ ¼ By þ GðyÞ; ð2Þ where B =(bij)m·m is a constant matrix, G(y) is the nonlinear part. So, without loss of generality, we can describe systems (1) as x_ ¼ f ðxÞ; ð3Þ y_ ¼ By þ GðyÞþuðy; xÞ. In order to study the generalized synchronization of systems (3), we define the generalized synchronization error e = y /(x), where / : Rn ! Rm is a continuously differentiable map. The error system e is given by e_ ¼ y_ /_ ðxÞ¼By þ GðyÞþuðy; xÞD/ðxÞf ðxÞ; ð4Þ where D/(x) is the Jacobian matrix of the map /(x): 0 1 o o o /1ðxÞ /1ðxÞ /1ðxÞ o o ... o B x1 x2 xn C o o o B /2ðxÞ /2ðxÞ /2ðxÞ C B o o ... o C B x1 x2 xn C D/ðxÞ¼B C. B . . . . C @ . . .. . A o o o /mðxÞ /mðxÞ ... /mðxÞ ox1 ox2 oxn From the definition of generalized synchronization, we know that the study of the generalized synchronization between two coupled systems can be translated into the analysis of the asymptotical stability of zero solution of error system. So, we discuss the asymptotical stability of zero solution of system (4) and draw the following conclusion. Theorem 1. In the systems (3), for any continuously differentiable map / :Rn ! Rm, let us design the controller uðx; yÞ¼eð/ðxÞyÞþD/ðxÞf ðxÞB/ðxÞGðyÞ; G. Zhang et al. / Chaos, Solitons and Fractals 32 (2007) 773–779 775

T where e =(ij)m·m is a constant matrix. If (B e) +(B e) is a negative deﬁnite matrix, then systems (3) are globally generalized synchronous with respect to the vector map /.

Proof. Substituting the controller uðx; yÞ¼eð/ðxÞyÞþD/ðxÞf ðxÞB/ðxÞGðyÞ into the error system (4), we get e_ ¼ y_ /_ ðxÞ¼By þ GðyÞD/ðxÞf ðxÞþeð/ðxÞyÞþD/ðxÞf ðxÞB/ðxÞGðyÞ¼ðB eÞe. ð5Þ Construct a Lyapunov function in the form of V(t)=eT(t)e(t). If (B e)T +(B e) is a negative definite matrix, then the derivative of V along the solution of (5) is V_ ¼ e_ Te þ eTe_ ¼ eT½ðB eÞT þðB eÞe < 0. From the Lyapunov stability theory, the zero solution of error system (4) is globally asymptotically stable, i.e., systems (3) are globally generalized synchronous with respect to the vector map /. h For a given map /, linear controller usually can not be found to achieve generalized synchronization between two different dimensional chaotic dynamical systems. The controller should contain the information of f, G and /. From the view of mechanical control, our controller seems a little bit complicated, but it is appropriate for biological and social systems, and we also note that similar controller has been adopted in [18]. Following examples will show that our approach is effective.

3. Numerical results

In order to show the eﬀectiveness of our approach, two examples are used to discuss two kinds of cases: n < m and n > m, respectively. (1) n < m We choose Lorenz system as drive system and hyperchaotic Ro¨ssler system as response system. The Lorenz system can be described by following nonlinear ODE: 0 1 0 1 x_ 1 aðx2 x1Þ B C B C @ x_ 2 A ¼ @ cx1 x1x3 x2 A; ð6Þ

x_ 3 x1x2 bx3

8 which has a chaotic attractor when a = 10, b ¼ 3, c = 28. The Lorenz chaotic attractor is shown in Fig. 1. The hyperchaotic Ro¨ssler system can be described by following nonlinear ODE: 0 1 0 1 y_ 1 y2 y3 B C B C B y_ C B y þ ay þ y C B 2 C B 1 2 4 C @ A ¼ @ A; ð7Þ y_ 3 y1y3 þ b

y_ 4 cy3 þ dy4 which has a chaotic attractor when a = 0.25, b =3,c = 0.5, d = 0.05. The projections of chaotic attractor in (y1,y2,y3) and (y4,y2,y3) are shown in Figs. 2 and 3. System (7) can be rewritten as the form y_ ¼ By þ GðyÞ, where 0 1 0 1 0 1 10 0 B C B C B 1 a 01C B 0 C B C B C B ¼ @ A; GðyÞ¼@ A. 00 00 y1y3 þ b 00cd 0 With the parameters above, let u(x,y)=e(/(x) y) B/(x) G(y)+D/(x)f(x), where /ðx ; x ; x Þ¼ðx ; x ; x ; x þ x þ x ÞT; 1 2 03 1 2 13 1 2 3 100 B C B 010C D/ðxÞ¼B C; @ 001A 111 776 G. Zhang et al. / Chaos, Solitons and Fractals 32 (2007) 773–779

Fig. 1. The chaotic attractor of Lorenz system.

Fig. 2. Chaotic behavior in (y1,y2,y3) of hyperchaotic Ro¨ssler system.

0 1 1 1 10 B C B 10:50 1C e ¼ B C. @ 00 1 0A 000:50:25 So, 0 1 10 0 0 B C B 0 0:25 0 0 C B e ¼ B C. @ 0010A 0000:2 G. Zhang et al. / Chaos, Solitons and Fractals 32 (2007) 773–779 777

Fig. 3. Chaotic behavior in (y4,y2,y3) of hyperchaotic Ro¨ssler system.

0 e1 −10

−20 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

−

e2 5

−10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

−

e3 5

−10 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

−10 e4 −20

−30 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 − t (10 2)

Fig. 4. State errors versus time e1=y1 x1, e2=y2 x2, e3=y3 x3, e4=y4 (x1 + x2 + x3).

It is easy to know that (B e)T +(B e) is a negative deﬁnite matrix. Then the conditions of Theorem 1 are satisﬁed. With the help of Matlab, we get the numeric results that are showed in Fig. 4 where x(0) = (10,10,10), y(0) = (1,1,1,1), and ei = yi /i(x(t)), i = 1,2,3,4. 778 G. Zhang et al. / Chaos, Solitons and Fractals 32 (2007) 773–779

(2) n > m

We choose hyperchaotic Ro¨ssler system as drive system and Lorenz system as response system. 0 1 x x B 2 3 C B x þ ax þ x C B 1 2 4 C x_ ¼ f ðxÞ¼@ A; ð8Þ x1x3 þ b cx þ dx 0 3 4 1 aðy2 y1Þ @ A y_ ¼ gðyÞ¼ cy1 y1y3 y2 . ð9Þ

y1x2 by3 System (9) can be rewritten as the form y_ ¼ By þ GðyÞ, where 0 1 0 1 aa 0 0 B C B C B ¼ @ c 10A; GðyÞ¼@ y1y3 A.

00b y1y2 With the parameters above, let u(x,y)=e(y /(x)) B/ (x) G(y)+D/(x)f(x), where / x ; x ; x ; x x2; x ; x x T; ð 1 2 03 4Þ¼ð 1 2 31þ 4Þ 2x 000 B 1 C D/ðxÞ¼@ 0100A; 0011 0 1 0100 B C e ¼ @ 28 0 0 A; 000

−1000 e1 −2000

−3000 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

5 e2 0

−5 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

−

e3 20

−40

−60 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 − t (10 2)

2 Fig. 5. State errors versus time e1 ¼ y1 x1, e2=y2 x2, e3=y3 x3 x4. G. Zhang et al. / Chaos, Solitons and Fractals 32 (2007) 773–779 779

So, 0 1 10 0 0 B C B e ¼ @ 0 10A. 8 00 3 We can see (B e)T +(B e) is a negative deﬁnite matrix, and the conditions of Theorem 1 are satisﬁed. With the help of Matlab, we get the numeric results that are showed in Fig. 5 where x(0) = (50,1,30,30), y(0) = (10,10,10), and ei = yi /i(x(t)), i =1,2,3.

4. Conclusion

We have studied the generalized synchronization of chaotic dynamical systems with different dimensions. Based on the Lyapunov stability theory, a method to realize generalized synchronization for two different dimensional chaotic systems is proposed. The validity of this approach is verified theoretically and numerically.

Acknowledgements

This work is supported by NNSF: Nos. 10372054 and 70431002.

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