Proceedings of the 3" World Congress on Intelligent Control and Automation June 2&JulY 2,2000, Hefei, P.R China

Generalized synchronization in coupled chaotic systems' Bang Yongdong Liu Yongqing Zhang Liqing

Department of Automatic Control Engineering, South China University of Technology, 5 10640, Guangzhou China

E-mail: yonder@mail.scut.edu.cn

Absrract - This paper proposes a simple analytical method for

constructing chaotically synchronizing systems in the generalized sense. By using the synchronization scheme

proposed herein, we have realized perfect generalized chaos

synchronization in two dimensional coupled maps. Numerical

simulations demonstrate the effectiveness of the proposed

scheme.

1. INTRODUCTION

Due to the theoretical significance and potential practical applications, synchronization of chaos between coupled chaotic systems has been an active topic of research recently [ I-8,101. Synchronization of chaos was initially understood to be the perfect coincidence of the chaotic dynamics of two coupled chaotic systems. This kind of synchronized chaos has been actually been observed experimentally between coupled identical chaotic systems; However, it is now recognized that this form of synchronization is only a particular case and, in fact, more other complicated forms of synchronization exist. Base on a drive-response system, H D I Abarbanel [8] er a1 have proposed a generalized concept of synchronized chaos. According to them, the dynamics of two coupled chaotic systems are synchronized if a functional relationship between their dynamical variables exists. This synchronization of chaos is called Generalized synchronization (GS) [ 8- 141. GS was taken to occur if there exists a mapping 0 from the

trajectories X(f) of the attractor in the driving space D

to the trajectories Y ( f ) in the response space R ; -

' Research supported by NNSF of China, grant no. 69934030

Y(t) = @(X(f)). Kocarev and Parlitz [ 101 have studied

the conditions for the occurrence of generalized synchronization in terms of asymptotic stability. They have showed that in the generalized synchronization the response is predictable, and the chaotic dynamics of the response system is conditionally equivalent to that of the driving systems.

Many methods for constructing synchronized chaotic systems, such as the Pecora-Carroll method [ 11, are based upon the decomposition of the states of chaotic systems, and it is proved by using conditional Lyapunov exponents (numerical investigations) whether the constructed systems are synchronized. As a GS method, T. Ushio [9] has proposed in-phase and anti-phase synchronization based on contraction mappings. On the basis of [9], we proposed a general approach for constructing a response system to implement generalized chaos synchronization with driving systems. Numerical results showing GS of dynamics between the driving and the response systems are presented.

2. GENERALIZED SYNCHRONIZATION OF CHAOS

We consider one-way coupled chaotic discrete-time systems of the form:

x(k + 1) = f ( x ( k ) ) , (1)

Here x ( k ) w(k) E R" denote the states of the driving

--

0- 7803-5999-X/00/$ IO. 00 02000 IEEE 325 1

and response systems at time k , respectively, f a

mapping from R" to itself, and g a coupling term.

Given dynamic(driving) system (l), We hope to construct

synchronization with driving systems (1).

0.4

0.2 a response system (2) to implement generalized chaos

T. Ushio has proposed in-phase and anti-phase synchronization based on contraction mappings (see [9]

d k ) 0

-0.2

-0.4 for more details).The phase synchronization of two

Fig. 1. A chaotic attractor of Eq.(3) with U = 1.4 and b = 0.3 equation holds, lim(lx(k)+ w(k)(I = C (constant), where k - m

\> : - -

/* .a+ .

-

x(k) and w(k).are the states of the systems S, and

S , , respectively. The use of contraction mappings is

straightforward but there is no systematic method for

whose eigenvalues are A* = -ax' f dmi. we see that for O<a;3(1-b)Z/4, IA:l<1,

finding contraction parts of systems. IAiI > 1 , and thus Eq.(3) has the stable fixed point A To overcome the difficult of finding contraction parts,

we will develop another method for constructing andunstablesaddle point B; chaotically synchronizing systems in the generalized sense. In order to demonstrate how this method works, we consider the ffenon mapping

for 3(1-b)2/4<a<(1-b)Z+(1+b)2,'b, the points

A 8 B are both unstable, and Eq.(3 ) has a stable period- 2 attractor. [;;l+;;)-" . (3)

We fix b and allow a to vary over a range. Eq.(3) with

We construct a response system described by

w(R+1)= f(w(k)M+g(x(k),w(k)). 1 x(k + 1) = f (x(k) , a) =

a = 1.4 and b = 0.3 has a chaotic attractor shown in In the generalized synchronization of chaos, the dynamics of coupled chaotic systems is functionally related, and it is possible that this functional relationship

Fig.]. When a > -(1- b)2/4, Eq.(3) have two fixed

is very complicated [9-141. However, we can choose this h c t i o n in a relatively simple way in our method.

In the following, we wish find a new function F such that

points A : (x: ,y: ) , B :(xf,yl)

- (1 - b)_+ J- xi = 2a

. yi = bx; ar(k + 1)+ ak.(k + 1) = F(ar(R)+ Nk)).

stable fixed point. For simplicity, we only choose We calculate its Jacobian matrix evaluated at The mapping Fcan be chosen freely as one which has a

F(.) = f ( . ,a , ) , where 0 < a, e 3(1 -b)2/4, Thus we

choose

1 (x*.y*) as

Since F(.) = f(., U , ) , where 0 < aI < 3(1- b)* /4 ,

we have

1 , [ - (1 - b) + d 0 ] / 2 a =i b [ - ( l - b ) + d b ) ' ] / 2 a

or limllm(k) + h ( k ) l l = C (constant), therefore

generalized chaos synchronization of x and w is achieved.

The results of the computer simulations at

a = 1 . 4 . b = 0 . 3 . a , = 0 . 3 . a = l . p = 2 are

k-bm

presented in Figs.(2)-(4). From Figs (2) and (4), we see

that w(k)+ f i ( k ) converge to the fixed point of the

mapping F as k increase. We calculate that the stable

fixed point of the mapping F is (1 0.3)T, thus states

x and w of Eqs. (3) and (4) are on the following lines after a transient,

x + 2w = (10.3j Fig. 4 only shows plots of x , ( k ) versus w,(k) after a

transient.

I I I 0 25 50 75 100

k Fig. 2. Plots of m, (A)+ fi, (k) versus number of iterations k

for a=l and p = 2

0.8 I

0.2 - 0 25 50 75 100

k Fig. 3.

for a = l a n d p = 2

Plots of a~,(k)+m,(&) versus number of iterations k

1.5 I I

Fig. 4. Plots of X, (k) versus w, (k ) of Eqs. (3) and (4)

3. CONCLUSION

In this paper, we have realized generalized chaos synchronization in two dimensional coupled maps. We have shown that the proposed scheme can realize perfect generalized chaos synchronization by only choosing the

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appropriate form of the function F . The proposed method can be M e r generalized to high dimensional chaotic maps and continuous chaotic systems.

chaos in electronic circuit experiments”, Physica D. vol. 112.

no.3-4. 1998, pp.459-471

[14]B R Hunt, E On, J A Yorke, “Differentiable generalized synchronization of chaos”, Phys Rev E, vo1.55, no.4, 1997,

pp.4029-4034

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