Synchronization of unidirectional coupled chaotic systems via partial stability

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Synchronization of unidirectional coupled chaotic systems

via partial stability

Zheng-Ming Ge

*

_{, Yen-Sheng Chen}

Department of Mechanical Engineering, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 30050, Taiwan, ROC Accepted 22 September 2003

Abstract

Chaos synchronization can be achieved by several methods but there is no easy uniﬁed criterion in general. In this paper, a general scheme is proposed to achieve chaos synchronization via stability with respect to partial variables. Three theorems for synchronization of unidirectional coupled non-autonomous (also autonomous) systems by linear feedback are developed for systems with and without system structure perturbations. The system, ﬂy-ball governor, is demonstrated as an example.

1. Introduction

Chaotic systems exhibit sensitive dependence on initial conditions. Because of this property, chaotic systems are diﬃcult to be synchronized or controlled. From the earlier works [1–3] (especially [3]), the researchers have realized that synchronization of chaotic motions are possible, synchronization of chaos was of great interest in these years [4–16]. In particular, it was pointed out that chaos synchronization has the potential in secure communication. Many engineers and scientists were attracted to this discipline [17–25].

Two kinds of chaos synchronization are discussed the most often. (1) Duplication: the ﬁrst method introduced by Pecora & Carroll [1] consists of a driving system and a response system. The former one evolves chaotic orbits and the latter is identical to the driving system except some partial states replaced by that of the driving one. (2) Coupling: the second kind consists of two identical chaotic systems except coupling term. Coupled systems can be unidirectional or mutual. Under certain conditions (appropriate coupling parameters and/or system parameters with enough evolution time) the response system will behave the same orbit with the driving system.

A more general case called generalized synchronization (GS) was studied in [48–53], this means that there is a functional relation between state variables of driving and response systems. This function need not be deﬁned on the whole phase space but on the attractor only. Three methods were proposed to detect GS in [48–50] respectively while another method measuring the smooth degree of this function in [52].

Synchronization means that the state variables of response system approach eventually to the ones of driving system. There are many control methods to synchronize chaotic systems such as observer-based design methods [26–29], adaptive control [30–38] and other control methods [39–47]. Zero crossing of Lyapunov exponent was used as a cri-terion 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 Lyapunov exponent. On the other hand, it is difficult to use Lyapunov direct method since the state error equation is not a pure function of state error in general. In this paper, we propose a general scheme to achieve chaos synchronization via partial stability due to Rumjantsev [55]. The upper

*_{Corresponding author. Tel.: +866-35712121; fax: +886-35720634.} E-mail address:[email protected](Z.-M. Ge).

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obstacles will be overcome by our method and it serves as a criterion for chaos synchronization by control methods. Criterions of unidirectional coupled nonautonomous systems by linear feedback are developed for systems with and without system structure perturbation. The system, ﬂy-ball governor, is demonstrated as an example.

2. Analysis

Consider the following unidirectional coupled nonautonomous systems _x1¼ fðt; x1Þ

_x2¼ fðt; x2Þ þ gðt; x2; x1Þ

ð1Þ

where x1, x22 Rn and f : X1 R Rn! Rn, g : X2 R R2n! Rn satisfy Lipschitz condition. X1, X2 are domains

containing the origin. Assume that the solutions of Eq. (1) have a priori bound then they must exist for inﬁnite time. That is, for givenðt0; x10; x20Þ 2 X1\ X2the solutions x1ðt; t0; x10; x20Þ, x2ðt; t0; x10; x20Þ of Eq. (1) exist for t P t0. At the

ﬁrst, we recall the deﬁnition of identical synchronization (complete synchronization).

Deﬁnition. The system (1) is identical synchronized if there is an invariant manifold S R R2n _s.t.

lim_t!1kx1ðt; t0; x10; x20Þ x2ðt; t0; x10; x20Þk ¼ 0 with ðt0; x10; x20Þ 2 X1\ X2.

In Eq. (1) g is the coupling function. Assume that gðt; x1; x1Þ ¼ 0, i.e. the synchronized sub-manifold of Eq. (1)

agrees with the original uncoupled one while synchronization occurs. In order to discuss the transversal stability of synchronization manifold, deﬁne e¼ x2 x1 to be the state error. Then the error equations can be written as

_e¼ fðt; e þ x1Þ fðt; x1Þ þ gðt; e þ x1; x1Þ ð2Þ

Notice that the right hand side of Eq. (2) is not a pure function of t and error e, as a result that the Lyapunov direct method might hardly be used. On the other hand, the variational equation or Lyapunov exponents can be used to clarify transversal stability. Josic [54] analyzed that synchronization manifolds will persist under perturbation if such manifolds possess a property of k-hyperbolicity. Herein, we add the upper half (lower half also works) of Eq. (1) with x2

replaced by x2¼ e þ x1 to Eq. (2), then extended equations are obtained as following

_x1¼ fðt; x1Þ

_e¼ fðt; e þ x1Þ fðt; x1Þ þ gðt; e þ x1; x1Þ

ð3Þ

If the partial variables e in Eq. (3) are asymptotically stable about e¼ 0, the synchronization manifold is stable in transversal directions. This can be done via stability with respect to partial variables. The theory of partial stability can be found in Appendix A.

In the following, three theorems will be derived for a special form of Eq. (1). The ﬁrst theorem is suitable for the case without system structure perturbation and the other two are the cases for systems under structure perturbations. These theorems will be applied to an example, the ﬂy-ball governor, in the next section. Consider unidirectional coupled nonautonomous systems as

_x1¼ fðt; x1Þ

_x2¼ fðt; x2Þ þ Cðx1 x2Þ

ð4Þ

where f satisﬁes Lipschitz condition with Lipschitz constant L and C2 Mnnis a constant matrix whose entries represent

the coupling strength of the linear feedback termðx1 x2Þ. Deﬁne e ¼ x2 x1, an extended equation can be obtained as

_x1¼ fðt; x1Þ

_e¼ fðt; e þ x1Þ fðt; x1Þ Ce:

ð5Þ

Theorem 1. The partial state e asymptotically approaches to 0 in Eq. (5) if LpffiffiffinIn C is negative definite, i.e. the systems

in the form of Eq. (4) are synchronized if LpffiffiffinIn C is negative definite.

Proof. Choose a function Vðx1; eÞ ¼12e

T_{e positive deﬁnite with respect to e and with inﬁnitesimal upper bound, then}

_ V ¼ eT_{_e}_¼X n i¼1 ei½fiðt; x1þ eÞ fiðt; x1Þ eTCe 6 Lkek Xn i¼1 jeij eTCe 6 L ffiffiffi n p kek2 eT_Ce_{¼ e}T_ðLpffiffiffi_n_I n CÞe

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The state error e approaches 0 asymptotically if LpffiffiffinIn C is negative definite by Theorem A.2in Appendix A. In upper

deviations, property of norm equivalent on finite dimensional vector space and Lipschitz condition were used. h Remark 1. From the matrix theory, LpffiffiffinIn C is negative definite if and only if all its eigenvalues are negative. For the

case C¼ diag ðc1;c2; . . . ;cnÞ with ci>0 for i¼ 1; . . . ; n, synchronization occurs if cmin> L

ffiffiffi n p

, cmin6ci, i¼ 1; . . . ; n.

This is because the time derivative of Vðx; eÞ can be written as _V ðx; eÞ 6 ðLpffiffiffin cminÞnkek 2

. Moreover, the result is global by Theorem A.4 if f is globally Lipschitz.

Consider unidirectional coupled nonautonomous systems under system perturbation as _x1¼ fðt; x1Þ

_x2¼ fðt; x2Þ þ Dfðt; x1; eÞ þ Cðx1 x2Þ

ð6Þ

where C2 Mnnis a constant matrix whose entries represent the coupling strength of the linear feedback termðx1 x2Þ

and Dfðt; x1; eÞ is the system perturbation with Dfðt; x1; 0Þ ¼ 0. Deﬁne e ¼ x2 x1, an extended equation can be

ob-tained as

_x1¼ fðt; x1Þ

_e¼ fðt; e þ x1Þ fðt; x1Þ þ Dfðt; x1; eÞ Ce

ð7Þ

Theorem 2. Assume that9Ki>0 such thatjDfij < Ki, i¼ 1; . . . ; n, i.e. 9K > 0 ) kDfk < K. C is a diagonal matrix such

that C¼ diag ðc1;c2; . . . ;cnÞ with ci>0 for i¼ 1; . . . ; n. Then the Eq. (7) is asymptotically e-stable if cmin>ðL þ KÞ ffiffiffin

p with c_min6c_i for i¼ 1; . . . ; n, i.e. the systems in the form of Eq. (6) are synchronized if cmin>ðL þ KÞ

ffiffiffi n p

. Proof. Choose a function Vðx1; eÞ ¼12e

T_{e positive deﬁnite with respect to e and with inﬁnitesimal upper bound, then}

_ V ¼ eT_{_e} ¼X n i¼1 ei½fiðt; x1þ eÞ fiðt; x1Þ þ Dfi eTCe 6 Lkek Xn i¼1 jeij þ K Xn i¼1 jeij cminkek 2 6_ðLpffiffiffi_n_c minÞkek 2 þ Kpffiffiffinkek

There are three cases to discuss. The first case: _Vðx1; eÞ ¼ 0 for kek ¼ 0; the second case: _V ðx1; eÞ < ½ðL þ KÞ ffiffiffin

p cminkek

2

for kek > 1; the third case: _V ðx1; eÞ < ðL þ KÞ ffiffiffin

p

cmin for kek 6 1. Hence, _V ðx1; eÞ < 0 if ðL þ KÞ ffiffiffin

p c_min<0. h

Remark 2. This result is global by Theorem A.4 if f is globally Lipschitz.

Theorem 3. Assume that 9K > 0 ) kDfk < Kkek. Then the Eq. (7) is asymptotically e-stable if ðL þ KÞpffiffiffinIn C is

negative definite, i.e. the systems in the form of Eq. (6) are synchronized ifðL þ KÞpffiffiffinIn C is negative definite.

Proof. Choose a function Vðx1; eÞ ¼12e

T_{e positive deﬁnite with respect to e, then}

_ V ¼ eT_{_e}_¼X n i¼1 ei½fiðt; x1þ eÞ fiðt; x1Þ þ Dfi eTCe 6ðL þ KÞkek Xn i¼1 jeij eTCe 6 eT½ðL þ KÞ ffiffiffi n p In Ce

Hence, the Eq. (7) is asymptotically e-stable ifðL þ KÞpffiffiffinIn C is negative definite. h

Remark 3. ðL þ KÞpffiffiffinIn C is negative definite if and only if all its eigenvalues are negative. When C ¼ diag ðc1;

c₂; . . . ;c_nÞ with ci>0 for i¼ 1; . . . ; n, synchronization occurs if cmin>ðL þ KÞ

ffiffiffi n p

, where c_minis the minimum one in c_i. Furthermore, this result is global by Theorem A.4 if f is globally Lipschitz.

3. Examples

A system, ﬂy-ball governor with and without system structure perturbation, is demonstrated as an example in this section. The system equation is as following

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_x¼ y

_y¼ rz2_{sin x cos x}_{sin x Cy}

_z¼ k cos x F

where r¼ 0:25, C ¼ 0:7, F ¼ 1:942and k ¼ 5:13 ensure that there exists chaotic behavior. The chaotic attractor is shown in Fig. 1.

3.1. Unidirectional coupled ﬂy-ball governors without perturbation

Consider the following unidirectional coupled systems without system perturbation as in the form of Eq. (5) _x1¼ y1

_y1¼ rz21sin x1cos x1 sin x1 Cy1

_z1¼ k cos x1 F

_x2¼ y2þ cðx1 x2Þ

_y2¼ rz22sin x2cos x2 sin x2 Cy2þ cðy1 y2Þ

_z2¼ k cos x2 F þ cðz1 z2Þ

where c¼ 1. The initial value x0¼ ð1; 1; 1; 3; 3; 3Þ T

is adopted in all simulated results. In Fig. 2, three state errors versus time are shown and the state errors approach zero as time evolves. Fig. 3 shows that synchronization sub-manifolds represent diagonal-like since x2! x1, y2! y1, z2! z1 as t! 1. The three Lyapunov exponents versus coupling

strength c are shown in Fig. 4. There is a zero crossing when c 0:155. This c is a threshold value which synchroni-zation occurs.

3.2. Unidirectional coupled ﬂy-ball governors with perturbationkDf k < K

Consider the following unidirectional coupled systems with system perturbation as in the form of Eq. (6)

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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -4 -2 0 2 4 x1 x2 -5 -4 -3 -2 -1 0 1 2 3 4 -5 0 5 y1 y2 0 2 4 6 8 10 12 0 5 10 15 z1 z2

Fig. 3. Synchronization sub-manifold of unidirectional coupled ﬂy-ball governor without system perturbation.

0 10 20 30 40 50 60 70 80 90 100 -2 -1 0 1 t x1-x2 0 10 20 30 40 50 60 70 80 90 100 -2 -1 0 1 t y1-y2 0 10 20 30 40 50 60 70 80 90 100 -2 0 2 4 6 t z1-z2

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_x1¼ y1

_z1¼ k cos x1 F

_x2¼ y2þ sinðtz2ðy1 y2ÞÞ þ cðx1 x2Þ

The ﬁrst error dynamics is _e1¼ e2 ce1þ sinðtz2ðy1 y2ÞÞ, then the system perturbation is jDf1j ¼

j sinðtz2ðy1 y2ÞÞj 6 1. For c ¼ 7:3, the state errors approach zero as time goes to inﬁnite as shown in Fig. 5.

Syn-chronization sub-manifolds are shown in Fig. 6. They represent diagonal-like since the state errors are asymptotically stable.

3.3. Unidirectional coupled ﬂy-ball governors with perturbationkDfk < Kkek

Consider the following unidirectional coupled systems with system perturbation as in the form of Eq. (7) _x1¼ y1

_z1¼ k cos x1 F

_x2¼ y2þ 100ðz2 z1Þ þ cðx1 x2Þ

where c¼ 5:1. The system perturbation is jDf1j ¼ j100ðz2 z1Þj 6 100kek. In Fig. 7, the state errors approach zero as

time goes to inﬁnite. Synchronization sub-manifolds are shown in Fig. 8. They represent diagonal-like since the state errors are asymptotically stable.

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 -1 0 0.2 -0.2 -0.4 -0.6 -0.8 -1.2 Lyapunov spectra coupling strength Lyapunov exponents

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0 5 10 15 20 25 30 -2 -1 0 1 t x1 -x 2 0 5 10 15 20 25 30 -2 -1 0 1 t y1 -y 2 0 5 10 15 20 25 30 -2 -1 0 1 t z1 -z 2

Fig. 5. State errors versus time of unidirectional coupled ﬂy-ball governor with system perturbationjDf1j 6 1 for c ¼ 7:3.

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 0 2 x1 x2 -3 -2 -1 0 1 2 3 4 -2 0 2 4 y1 y2 1 2 3 4 5 6 7 8 9 2 4 6 8 10 z1 z2

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-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -4 -2 0 2 4 x1 x2 -5 -4 -3 -2 -1 0 1 2 3 -5 0 5 y1 y2 0 2 4 6 8 10 12 0 5 10 15 z1 z2

Fig. 8. Synchronization sub-manifold of unidirectional coupled ﬂy-ball governor with system perturbationjDf1j 6 100kek for c ¼ 5:1.

0 5 10 15 20 25 30 35 40 45 50 -20 -10 0 10 20 t x1 -x 2 0 5 10 15 20 25 30 35 40 45 50 -2 -1 0 1 2 t y1 -y 2 0 5 10 15 20 25 30 35 40 45 50 -2 -1 0 1 t z1 -z 2

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4. Conclusions

There are many methods to ensure chaos synchronization such as zero crossing of Lyapunov spectra, Lyapunov direct method and control methods. The realization of Lyapunov exponent needs numerical calculation for infinite evolution time, therefore this method is not complete in practice. On the other hand, it is difficult to use Lyapunov direct method since the state error equation is not a pure function of time and state error in general. Control methods might be appropriate to some kinds of systems. In this paper, a general scheme to achieve chaos synchronization via partial stability was proposed. The upper drawbacks can be overcome by this method. Three theorems were proven to ensure chaos synchronization for a general kind of unidirectional coupled nonautonomous (also autonomous) sys-tems by linear feedback coupling term. The first theorem is for the case without system perturbation and the other two theorems are for the case under perturbations. The fly-ball governor was illustrated as an example to show these results.

Acknowledgement

This research was supported by the National Science Council, Republic of China, under grant number NSC 91-2212-E-009-025.

Appendix A

The content of this appendix follows [55–57]. Consider a diﬀerential system

_x¼ fðt; xÞ ðA:1Þ

where f :½t0;1Þ X ! Rn, fðt; 0Þ ¼ 0 8t 2 ½t0;1Þ and X Rnis a region containing the origin. Assume that f is smooth

enough to ensure that the solution of (A.1) exists uniquely. To shorten the notation, write x¼ ðy1; . . . ; ym; z1; . . . ; znmÞ T , kyk ¼ Pm i¼1y 2 i 1=2 ,kzk ¼ Pnm_i¼1 z2 i 1=2 andkxk ¼ Pn i¼1x 2 i 1=2 ¼ ðkyk2_{þ kzk}2_Þ1=2

with 0 < m 6 n. We assume that the solution of (A.1) is z-extendable, i.e. any solution of (A.1) exists for all t P t0 andkyðtÞk 6 H , H is a constant. Write

Q¼ fðt; xÞjt P t0;kykH ; 0 6 kzk < þ1g and ~Q¼ fðt; xÞjt P t0;kxk < 1g.

Deﬁnition A.1. The solution of (A.1) is stable with respect to y (y-stable) if 8e > 0, 8t02 ½0; 1Þ, 9dðt0;eÞ > 0,

8x02 Bd:¼ fxj kxk < dðt0;eÞg such that kyðt; t0; x0Þk < e 8t P t0. The solution of (A.1) is uniformly y-stable if dðt0;eÞ is

independent of t0in the deﬁnition of y-stable.

The solution of (A.1) is asymptotically stable with respect to y (asymptotically stable) if it is (1) stable and (2) y-attractive, i.e.8t02 ½0; 1Þ, 9d0ðt0Þ > 0, 8e0>0,8x02 Bd0:¼ fxj kxk < d0ðt0Þg, 9T ðt0; x0;e0Þ such that kyðt; t0; x0Þk < e0

8t P t0þ T . The solution of (A.1) is uniformly asymptotically y-stable if it is (1) uniformly y-stable and (2) uniformly

y-attractive, i.e. d0ðt0Þ is independent of t0 and Tðt0; x0;e0Þ is independent of t0; x0 in the deﬁnition of y-attractive.

The solution of (A.1) is globally y-attractive if Bd¼ Rnin the deﬁnition of y-attractive. Furthermore, if Bd¼ Rnand

9d0_ðt

0Þ > 0 can be replaced by 8d0 the solution of (A.1) is globally uniformly y-attractive. The solution of (A.1) is

globally asymptotically y-stable if it is (1) y-stable and (2) globally y-attractive. The solution of (A.1) is globally uni-formly asymptotically y-stable if it is (1) uniuni-formly y-stable and (2) globally uniuni-formly y-attractive.

The next definition extends the notation of definite functions with respect to partial variables. Let Vðt; xÞ 2 Cð½t0;1Þ Rn; RÞ with V ðt; 0Þ ¼ 0 and V defined on Q.

Definition A.2. A t implicit positive (negative) semi-definite function VðxÞ is called positive (negative) definite with respect to y if VðxÞ can vanish only when y ¼ 0.

A positive (negative) semi-definite function Vðt; xÞ is called positive (negative) definite with respect to y if there is a positive (negative) definite function WðyÞ such that V ðt; xÞ P W ðyÞ ðV ðt; xÞ 6 W ðyÞÞ.

Deﬁnition A.3. A function Vðt; xÞ is called bounded if 9M > 0 such that jV ðt; xÞj 6 M. A bounded function V ðt; xÞ possesses an inﬁnitesimal upper bound if8~e > 0, 9 ~dð~eÞ > 0, for t P t0andkxk < ~dð~eÞ such that jV ðt; xÞj 6 ~e. A bounded

function Vðt; xÞ possesses an inﬁnitesimal upper bound with respect to x1; . . . ; xkðm 6 k 6 nÞ if 8~e > 0, 9 > ~dð~eÞ > 0, for

tP t0,

Pk i¼1x2i < ~d

2

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The following four theorems still hold when the undisturbed motion has nonzero z.

Theorem A.1. Suppose there exists a positive deﬁnite function Vðt; xÞ with respect to x1; . . . ; xkðk 6 nÞ such that _Vðt; xÞ

is negative semi-deﬁnite or vanishes, then the undisturbed motion is stable with respect to x1; . . . ; xkðk 6 nÞ.

Theorem A.2. Suppose there exists a positive deﬁnite function Vðt; xÞ with respect to x1; . . . ; xkðk 6 nÞ such that V ðt; xÞ

possesses an inﬁnitesimal upper bound and _Vðt; xÞ is negative deﬁnite with respect to x1; . . . ; xk, then the undisturbed motion

is asymptotically stable with respect to x1; . . . ; xk.

Theorem A.3. Suppose there exist a function V :½0; 1Þ X ! R such that for some functions a; b; c 2 K and every ðt; xÞ 2 Q: ðiÞ aðkykÞ 6 V ðt; xÞ; V ðt; 0Þ ¼ 0; ðiiÞ Vðt; xÞ 6 b X k i¼1 x2 i !1=2 0 @ 1 A; m 6 k 6 n; ðiiiÞ _V__{ðt; xÞ 6 c} X k i¼1 x2 i !1=2 0 @ 1 A;

then the origin is uniformly asymptotically y-stable.

Theorem A.4. Suppose there exist a function V :½0; 1Þ X ! R such that for some functions a, b, c 2 K, a : Rþ_{! R}þ

with r! þ1 ) aðrÞ ! þ1 and every ðt; xÞ 2 ~Q: ðiÞ aðkykÞ 6 V ðt; xÞ; V ðt; 0Þ ¼ 0; ðiiÞ Vðt; xÞ 6 b X k i¼1 x2_i !1=2 0 @ 1 A m 6 k 6 n; ðiiiÞ V_ðt; xÞ 6 c X k i¼1 x2 i !1=2 0 @ 1 A; ðivÞ X n i¼1 x2_i ! þ1 ) V ðt; xÞ ! þ1;

then the origin is globally asymptotically y-stable.

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