Synchronization of mutual coupled chaotic systems via partial stability theory

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Synchronization of mutual coupled chaotic systems via

partial stability theory

Zheng-Ming Ge

a,*

, Yen-Sheng Chen

b

a_{Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC} b_{Division of Mechanics, Research Center for Applied Sciences,Academia Sinica, Taipei 115, Taiwan, ROC}

Accepted 10 November 2005

Communicated by Prof. G. Iovane

Abstract

A scheme is proposed to achieve chaos synchronization for mutual coupled systems via partial stability theory. Under this scheme, three criteria are given to ensure chaos synchronization. The ﬁrst criterion applies to the case with-out system perturbation and the other two apply to systems possessing vanishing and nonvanishing perturbations, respectively. Finally, coupled Lorenz systems are simulated to illustrate the theoretical analysis.

1. Introduction

Chaotic systems are thought diﬃcult to be synchronized or controlled in the past since they exhibit sensitive depen-dence on initial conditions. From the work of Pecora and Carroll[1], the researchers have realized that the synchronism of chaotic motions is possible. Hence chaos synchronization is of great interest in these years. In particular, it is pointed out that chaos synchronization has the potential in secure communication. Many engineers and scientists are attracted by this discipline.

Synchronization means that the state variables of a response system approach eventually to that of a driving system. Zero crossing of a Lyapunov exponent is used as a criterion of chaos synchronization widely. There is a drawback that we can only calculate finite evolution time in computer simulation but infinite evolution time is needed by definition of the Lyapunov exponent. On the other hand, it may be difficult to use the traditional Lyapunov direct method since the equation of state errors is not a pure function of state errors in general. In the paper of Ge and Chen[9], a general scheme is proposed to achieve chaos synchronization of unidirectional coupled systems via the partial stability theory. Preceding two obstacles can be overcome by this scheme. Furthermore, it not only applies for unidirectional coupled systems but also works for mutual coupled systems. The objective of this paper is to accomplish the theoretical analysis

*_{Corresponding author. Tel.: +886 35712121; fax: +886 35720634.}

E-mail address:[email protected](Z.-M. Ge).

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of chaos synchronization for mutual coupled systems via the partial stability theory. Some other achievement about synchronization of mutual coupled systems can be found in[2–8].

In this paper, three criteria are given to ensure synchronization for mutual coupled systems. The ﬁrst criterion suits for systems without perturbation and the other two suit for systems under vanishing and nonvanishing perturbations, respectively. The only assumption is that system equations meet the Lipschitz condition. Since there is no further restriction on the type of systems, all criteria derived work for nonlinear nonautonomous systems. When these criteria are used, a matrix should be negative deﬁnite and an estimation of a Lipschitz constant is needed in advance.

Theoretical analyses are arranged in Section2and coupled Lorenz systems are simulated to demonstrate analytical results in Section3. Conclusions follow sequentially in Section4.

2. Theoretical analyses

Consider the following mutual coupled system _x¼ fðt; xÞ þ G1ðt; x; yÞ;

_y¼ fðt; yÞ þ G2ðt; x; yÞ;

ð1Þ where x; y2 Rn_{and f : X}_{R R}n_{! R}n_{satisfy the Lipschitz condition}_{kf(t, x}

1) f(t, x2)k 6 Lkx1 x2k in x for all (t, x1)

and (t, x2) in X with a Lipschitz constant L. This constant L is not unique since any number larger than L is also a Lipschitz

constant. G1and G2are coupling functions which satisfy G1(t, x, y) = 0 and G2(t, x, y) = 0 for x(t) = y(t),"t P t0.

Deﬁne e = y x to be the state error. Then the error dynamic equation can be written as

_e¼ fðt; e þ xÞ fðt; xÞ þ G2ðt; x; e þ xÞ G1ðt; x; e þ xÞ. ð2Þ

In general the right hand side of Eq.(2)is not a function of the state error e only. As a result the traditional Lyapunov method might hardly be used. Herein, we take the ﬁrst equation of Eqs.(1) and (2)together with y = e + x to form an extended system of states x and e as follows

_x¼ fðt; xÞ þ G1ðt; x; e þ xÞ;

_e¼ fðt; e þ xÞ fðt; xÞ þ G2ðt; x; e þ xÞ G1ðt; x; e þ xÞ.

ð3Þ

If the partial state variable e in Eq.(3)is asymptotically stable about e = 0, then x and y in Eq.(1)are synchronized. The stability of partial state variables can be veriﬁed via the partial stability theory. A brief review of the partial stability theory can be found in the appendix of paper[9]or in[10]. Although the acquirement of the extended system Eq.(3)

doubles the order of the original error dynamic equation Eq.(2), only partial variables e are handled. This scheme does not increase any diﬃculty due to the increase of the order. Furthermore, the usage of the partial stability theory is sim-ilar to the traditional Lyapunov method.

The proposed scheme not only applies to mutual coupled systems but also applies to unidirectional cases. Actually, it reduces to unidirectional cases if G1= 0 is satisﬁed[9]. The rest mission is to choose appropriate controllers G1and G2

to guarantee the occurrence of synchronization. There are many forms of G1 and G2 for choice. We choose

G1= C1(y x) and G2= C2(x y) and Eq.(1)can be rewritten as

_x¼ fðt; xÞ þ C1ðy xÞ;

_y¼ fðt; yÞ þ C2ðx yÞ;

ð4Þ

where C1, C22 Mn·nare two constant matrices whose entries represent the coupling strength. An extended system can

be obtained as

_x¼ fðt; xÞ þ C1e;

_e¼ fðt; e þ xÞ fðt; xÞ ðC1þ C2Þe.

ð5Þ

A synchronization criterion of Eq.(5)is derived as follows.

Theorem 1. The partial state e in Eq.(5)uniformly asymptotically approaches 0 if LIn (C1+ C2) is negative deﬁnite.

This means that two subsystems in Eq.(4)are synchronized if LIn (C1+ C2) is negative deﬁnite.

Proof. Choose a function Vðx; eÞ ¼1

2e

T_{e which is positive deﬁnite with respect to e and possesses an inﬁnitesimal upper}

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_

Vðx; eÞ ¼ eT_{_e 6 Lkek}2

eT_ðC

1þ C2Þe ¼ eT½LIn ðC1þ C2Þe.

The state error e approaches 0 uniformly asymptotically if LIn (C1+ C2) is negative deﬁnite by Theorem A2 in the

appendix of[9]. h

A special case C1= C2= diag(c1, c2, . . . , cn) with ci> 0, i = 1, . . . , n is commonly used. Since the time derivative of

V(x, e) along x(t) and e(t) satisﬁes _Vðx; eÞ 6 ðL 2cminÞkek 2

, the synchronization criterion reduces to cmin> L/2,

cmin6ci, i = 1, . . . , n. When c = c1, . . . , cn, synchronization occurs if c > L/2. This means that the synchronization of

mutual coupled chaotic systems is guaranteed by the large coupling strength c. If the system is autonomous, the thresh-old value of c for the occurrence of synchronization is one half of the largest Lyapunov exponent of the chaotic system

[2].

If perturbations exist in the system, similar criterion can also be obtained. Consider a mutual coupled nonautono-mous system with the perturbations in the form of

_x¼ fðt; xÞ þ Df1ðt; x; yÞ þ C1ðy xÞ;

_y¼ fðt; yÞ þ Df2ðt; x; yÞ þ C2ðx yÞ;

ð6Þ

where Df1(t, x, y) and Df2(t, x, y) are the vanishing perturbation. Vanishing perturbation means that Dfj(t, x, y) = 0

when-ever x(t) = y(t),"t for j = 1,2. Dfj(t, x, y) can be rephrased as Dfj(t, x, e) for j = 1,2. Then an extended system can be

obtained as

_x¼ fðt; xÞ þ Df1ðt; x; eÞ þ C1e;

_e¼ fðt; e þ xÞ fðxÞ þ Df2ðt; x; eÞ Df1ðt; x; eÞ ðC1þ C2Þe.

ð7Þ

Theorem 2. Assume that $Kj> 0) kDfjk < Kjkek, j = 1, 2. Then null solution of the partial state e of Eq. (7) is

uniformly asymptotically stable if (L + K1+ K2)In (C1+ C2) is negative deﬁnite, i.e., the two subsystems in Eq.(6)

are synchronized if (L + K1+ K2)In (C1+ C2) is negative deﬁnite.

Proof. Choose a function Vðx; eÞ ¼1

2e

T_{e which is positive deﬁnite with respect to e and possesses an inﬁnitesimal upper}

bound. By the Cauchy-Schwarz inequality and the Lipschitz condition, _Vðx; eÞ satisﬁes _

Vðx; eÞ 6 eT_{½ðL þ K}

1þ K2ÞIn ðC1þ C2Þe.

Hence the null solution of Eq. (7) is uniformly asymptotically e-stable if (L + K1+ K2)In (C1+ C2) is negative

deﬁnite. h

When C1= C2= diag (c1, c2, . . . , cn) with ci> 0 for i = 1, . . . , n, synchronization occurs if cmin> (L + K1+ K2)/2,

where cminis the minimum of ci. If c = c1, . . . , cn, the synchronization criterion reduces to c > (L + K1+ K2)/2.

More-over, by Theorem A4[11], the synchronization inTheorem 1 and 2are global if f is globally Lipschitzian.

If perturbations Df1(t, x1, x2) and Df2(t, x1, x2) are not vanishing, it is diﬃcult to design a controller to guarantee

the occurrence of asymptotically partial stability as that inTheorem 2. The reason is that the origin is not an equi-librium point anymore. The stability under constantly acting perturbation small on the average[11]must be studied instead.

Theorem 3. Assume that the functions f and Df(x) are continuous and bounded. The null solution of Eq. (6) is uniformly e-stable under constantly acting perturbation small on the average if LIn 2C is negative deﬁnite.

Proof. FromTheorem 1, the partial state e uniformly asymptotically approaches 0 in Eq.(5)if LIn 2C is negative

deﬁnite. By corollary in[11], the null solution of Eq.(7)is uniformly e-stable under constantly acting perturbation small on the average if LIn 2C is negative deﬁnite with the assumption that f and Df(x) are continuous and bounded. This

completes the proof. h

If C = diag(c1, c2, . . . , cn) with ci> 0 for i = 1, . . . , n, practical synchronization occurs if cmin> L, where cmin6ci,

i = 1, . . . , n. Moreover, the larger cminis, the smaller bounds of the state errors are. This criterion is global if f is globally

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 e₁ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.05 0 e₂ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 t e 3

Fig. 1. State errors versus time for mutual coupled Lorenz systems without perturbation while c = 0.6.

0 0.1 0.2 0.3 0.4 0.5 0.6 -16 -14 -12 -10 -8 -6 -4 -2 0 2 coupling Lyapunov spectra

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0 50 100 150 200 250 300 -20 -10 0 10 20 e₁ 0 50 100 150 200 250 300 -20 0 20 40 e₂ 0 50 100 150 200 250 300 -40 -20 0 20 t e 3

Fig. 3. State errors versus time for mutual coupled Lorenz systems without perturbation while c = 0.6.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1 -0.5 0 e₁ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.1 -0.05 0 e₂ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 t e 3

Fig. 4. State errors versus time for mutual coupled Lorenz systems with vanishing perturbations Df1= cost Æ (y1 y2) and

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3. Numerical illustrations Consider a Lorenz system

_x¼ rðx yÞ; _y¼ rx y xz; _z¼ xy bz;

where r = 10, r = 28 and b = 8/3 ensure that there exists chaotic behavior. WhenTheorem 1is applied, an estimation of a Lipschitz constant is needed. By Cauchy–Schwarz inequality, we have

jf1ðx2Þ f1ðx1Þj ¼ j re1þ re2j 6 k½ r r 0kkx2 x1k; jf2ðx2Þ f2ðx1Þj ¼ jre1 e2 x2e3 z1e1jkx2 x1k 6 k½ r þ B3 1 B1kkx2 x1k; jf3ðx2Þ f3ðx1Þj ¼ jx2y2 x1y1 be3j 6 k½ B2 B1 b kkx2 x1k; for any x2¼ ½ x2 y2 z2 T ; x1¼ ½ x1 y1 z1 T

, wherejx(t)j 6 B1,jy(t)j 6 B2,jz(t)j 6 B3"t > t0. Hence a Lipschitz

con-stant is obtained as L¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k½ r r 0k2þ k½ r þ B3 1 B1k 2 þ k½ B2 B1 b k 2 q .

From numerical simulation, B1= 20, B2= 28, B3= 49, then L = 87.87. The mutual coupled systems are in the form of

Eq.(4)with C1= C2= diag(c, . . . , c) and c = 44 > L/2. The initial value is x0= [1,0.01, 3, 0, 0, 5]T. The simulated

re-sults are shown inFigs. 1–6. InFig. 1, three state errors approach zero as time evolves. Lyapunov exponents versus coupling strength c are shown inFig. 2. There is a zero-crossing of one Lyapunov spectrum while c 0.41. This value of c is a threshold value where synchronization occurs. Fujisaka and Yamada[2]proved that synchronization of linear mutual coupled autonomous systems occurs if the coupling strength larger than one half of the largest Lyapunov expo-nent. The largest Lyapunov exponent of the Lorenz system is 0.82 and its half is 0.41. This coincides with the value of c

0 0.5 1 1.5 2 2.5 3 3.5 4 -1 -0.5 0 0.5 e₁ 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.1 -0.05 0 e₂ 0 0.5 1 1.5 2 2.5 3 3.5 4 -1 0 1 2 t e 3

Fig. 5. State errors versus time for mutual coupled Lorenz systems with nonvanishing perturbations Df2= 2 sin(20pt), Df4= r(t),

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at the zero-crossing of the Lyapunov spectrum. Choose c = 0.6, the simulated result inFig. 3shows that the state errors still converge to zero but the transient time of convergence is long. This fact agrees with our intuition.

If there exist vanishing perturbations in mutual coupled Lorenz systems as _x1¼ rðx1 y1Þ þ Df1þ cðx2 x1Þ; _y1¼ rx1 y1 x1z1þ cðy2 y1Þ; _z1¼ x1y1 bz1þ cðz2 z1Þ; _x2¼ rðx2 y2Þ þ cðx1 x2Þ; _y₂¼ rx2 y2 x2z2þ cðy1 y2Þ; _z2¼ x2y2 bz2þ Df6þ cðz1 z2Þ.

System perturbations are bounded sincejDf1j = jcost Æ (y1 y2)j 6 kek and jDf6j = jx1 x2j 6 kek. Choose c = 45 to

satisfy c > (L + K1+ K2)/2. InFig. 4, state errors approach zero as time goes to inﬁnity although there are persistent

acting perturbations.

If not all perturbations are vanishing as Df1= Df3= Df5= 0, Df2= 2 sin(20pt), Df4= r(t) and Df6= 5 cos(30pt),

where r(t) is the unit normal random variable. These perturbations are bounded on the average since RtþT

t supfjDf2jgds 6 2T ;

RtþT

t supfjDf4jgds 6 T and

RtþT

t supfjDf6jgds 6 5T ; 8t 2 ½0; 1Þ; T > 0. The initial condition

is the same and c = 44. State errors versus time are shown in Fig. 5 and they are bounded as time evolves. If c= 130, results are shown inFig. 6. As coupling strength c increases, the error bounds decrease.

When these criteria are used, a matrix should be negative deﬁnite and an estimation of a Lipschitz constant is needed in advance. Moreover, this estimation is often conservative. An adaptive method can improve these two shortcomings

[12]. 0 0.5 1 1.5 2 2.5 3 3.5 4 -1 -0.5 0 0.5 e₁ 0 0.5 1 1.5 2 2.5 3 3.5 4 -0.1 -0.05 0 e₂ 0 0.5 1 1.5 2 2.5 3 3.5 4 -1 0 1 2 t e 3

Fig. 6. State errors versus time for mutual coupled Lorenz systems with nonvanishing perturbations Df2= 2 sin(20pt), Df4= r(t),

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4. Conclusions

A general scheme to achieve the chaos synchronization of mutual coupled systems via the partial stability theory is proposed in this paper. By the procedure of the proposed scheme, three criteria are proven to ensure the chaos synchro-nization for a general kind of mutual coupled systems. The ﬁrst theorem applies for the system without perturbation. The other two theorems suit for systems possessing vanishing and nonvanishing perturbations, respectively. All these criteria work for nonlinear nonautonomous systems. Numerical simulations show that these criteria are eﬀective.

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