Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272

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Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Estimation of unknown parameters and adaptive synchronizationof hyperchaotic systems

Francis Austin a,*, Wen Sun b, Xiaoqing Lu b

a Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, PR Chinab School of Mathematics and Statistics, Wuhan University, Wuhan 430072, PR China

a r t i c l e i n f o

Article history:Received 15 December 2008Received in revised form 3 March 2009Accepted 3 March 2009Available online 13 March 2009

PACS:05.45.�a

Keywords:Adaptive synchronizationHyperchaotic systemsLasalle invariance principle

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.03.002

* Corresponding author.E-mail address: maaustin@inet.polyu.edu.hk (F.

a b s t r a c t

This paper investigates the chaos synchronization of two hyperchaotic systems. Based onLasalle invariance principle, adaptive schemes are derived to make two unidirectional cou-pling and mutual coupling hyperchaotic systems asymptotically synchronized whether theparameters are given or uncertain, and unknown parameters are identified simultaneouslyin the process of synchronization. Numerical simulations of hyperchaotic Chen systems arepresented to show the effectiveness of the proposed chaos synchronization schemes.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

A chaotic system is a nonlinear deterministic system that displays complex and unpredictable behaviors and possesssome special features such as the extremely sensitive dependence on the initial conditions and bounded trajectories inthe phase space and at least one positive Lyapunov exponent [2,7]. Because of those properties, chaotic systems were con-sidered to be difficult to be controlled or synchronized in the past. Since the earlier work by Pecora and Carrol [1], research-ers have realized that the control and synchronization of chaos are not only possible but also have great potentialapplications in secret communication, chemical reaction, biological systems and so on [8,10,11]. During the last two decades,chaos control and synchronization have attracted a great deal attention from various fields. Many methods and techniquesfor handing chaos control and synchronization have been developed, such as impulsive control method [3], adaptive designmethod [4,6,9,12–16], backstepping design technique [5], OGY method, etc. Especially, adaptive design method becomesmore and more popular in many field because of its effectiveness in the applications of control and synchronization. In[12], based on the Lyapunov stability theorem, the author stabilized the uncertain hyperchaotic system to unstable equilib-rium via adaptive control. In [13–16], an adaptive control law is derived to make the states of two identical hyperchaoticsystems asymptotically synchronized based on the Lyapunov stability theory. But the methods mentioned above are validfor the unidirectional coupling chaotic systems. In this paper, a class of novel adaptive control schemes are proposed to maketwo coupled hyperchaotic systems asymptotically synchronized. The methods are valid for both the unidirectional coupledhyperchaotic systems and mutual coupled hyperchaotic systems. Numerical simulations of hyperchaotic Chen systems arepresented to show the effectiveness of the proposed chaos synchronization schemes.

. All rights reserved.

Austin).

F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272 4265

2. Theoretical analyses

2.1. Given parameters

In this section, we consider the synchronization of two coupled hyperchaotic systems with given parameters. We assumethat the two coupled hyperchaotic systems are

_xi ¼ FiðxÞ þ kiyxðyi � xiÞ; i ¼ 1;2; . . . ;n; ð2:1Þ

_yi ¼ FiðyÞ þ kixyðxi � yiÞ; i ¼ 1;2; . . . ;n; ð2:2Þ

where x ¼ ðx1; x2; . . . ; xnÞT 2 Rn; FiðxÞ is a nonlinear function, kixy and ki

yx are coupling strengths, kixy varies adaptively accord-

ing to the update law

_kixy ¼ ciðyi � xiÞ2; ð2:3Þ

where ci is a positive constant. FiðxÞ satisfies the uniform Lipschitz condition with Lipschitz condition constant l, i.e., for anyx; y 2 Rn, there exists a constant l > 0, satisfying

jFiðxÞ � FiðyÞj 6 l minjjxj � yjj; i ¼ 1;2; . . . ;n: ð2:4Þ

It is obvious that the condition (2.4) holds if oFi=oxj ði; j ¼ 1;2; . . . ;nÞ are bounded. If the two hyperchaotic systems are cou-pled unidirectionally, namely master–slave systems, then ki

yx ¼ 0; if the two hyperchaotic systems are coupled mutually, weassume ki

yx ¼ kixy, which is reasonable. our goal is to design and implement a scheme of coupling strengths which vary adap-

tively according to the update law (2.3), such that the coupled hyperchaotic system can be synchronous. To guarantee syn-chronization of the coupled hyperchaotic systems we must prove that the synchronization manifoldM ¼ fðxT ; yTÞ; x ¼ y; x; y 2 Rng in the systems (2.1) and (2.2) is globally attractive. We have the following theorem.

Theorem 2.1. Suppose that Fi ði ¼ 1;2; . . . ;nÞ satisfies the uniform Lipschitz condition with positive Lipschitz constant l and theupdate laws of the coupling strengths is (2.3), then the states of the systems (2.1) and (2.2) starting from any initial valuesconverge to the synchronous manifold M.

Proof. We prove the synchronous manifold M is attractive in the case that the two hyperchaotic systems are coupled mutu-ally. If the two hyperchaotic systems are coupled unidirectionally, the proof is similar. To this end, we select the scalar func-tion as

L1ðtÞ ¼ eT eþ 2Xn

i¼1

1ci

kixy � L

� �2; ð2:5Þ

where e ¼ ðe1; e2; . . . ; enÞT ¼ ðy1 � x1; y2 � x2; . . . ; yn � xnÞT ; L is a constant with 4L > nlþ 1. Differentiating the function L1ðtÞalong the trajectories of system (2.1) and (2.2), we obtain

_L1ðtÞ ¼ 2Xn

i¼1

eið _yi � _xiÞ þ 4Xn

i¼1

1ci

kixy � L

� �_ki

xy

¼ 2Xn

i¼1

ei FiðyÞ � kiyx þ ki

xy

� �ðyi � xiÞ � FiðxÞ

� �þ 4

Xn

i¼1

kixy � L

� �ðxi � yiÞ

2

¼ 2Xn

i¼1

eiðFiðyÞ � FiðxÞÞ � 4LXn

i¼1

e2i ð2:6Þ

6

Xn

i¼1

e2i þ l min

je2

j

� �� 4L

Xn

i¼1

e2i

6 ð1þ nl� 4LÞXn

i¼1

e2i 6 0:

Let the set W ¼ ðe; kÞ 2 R2n : e ¼ 0; k ¼ k1xy; k

2xy; . . . ; kn

xy

� �n o, then E ¼ fðe; kÞ 2 R2n : e ¼ 0; k ¼ k0 2 Rng is the largest invariant

set in W, according to the invariant principle, the trajectories of system (2.1) and (2.2) starting from any initial values con-verge asymptotically to the largest invariant set E, i.e., y! x as t !1, which implies the theorem. h

4266 F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272

2.2. Unknown parameters

For coupled hyperchaotic systems with unknown parameters, we assume the systems in the form as follows

_xi ¼ Fiðx;aiÞ þ kiyxðyi � xiÞ; ð2:7Þ

_yi ¼ Fiðy;biÞ þ kixyðxi � yiÞ; ð2:8Þ

where

Fiðx;piÞ ¼ fiðxÞ þXn

j¼1

pijfijðxÞ;

the update laws of coupling strengths and parameters are

_kixy ¼ ciðxi � yiÞ

2; ð2:9Þ

_bij ¼ �hijeifijðyÞ; i; j ¼ 1;2; . . . ;n; ð2:10Þ

respectively, where bij is the estimate value of the unknown parameter aij; ci is a positive constant, hij > 0; i; j ¼ 1;2; . . . ;n,are arbitrary constants, Fi ði ¼ 1;2; . . . ;nÞ satisfies the uniform Lipschitz condition with positive Lipschitz constant l, i.e.,

jFiðx;piÞ � Fiðy;piÞj 6 l minjjxj � yjj; i ¼ 1;2; . . . ;n ð2:11Þ

for any constant vector pi. If the two hyperchaotic systems are coupled unidirectionally, namely master–slave systems, thenki

yx ¼ 0; if the two hyperchaotic systems are coupled mutually, we assume kiyx ¼ ki

xy, which is reasonable. To guarantee syn-chronization of mutually coupled hyperchaotic systems, we must prove that the synchronization manifoldM ¼ fðxT ; yTÞ; x ¼ y; x; y 2 Rng in the systems (2.7) and (2.8) is globally attractive. We have the following theorem.

Theorem 2.2. Suppose Fi ði ¼ 1;2; . . . ;nÞ satisfies the uniform Lipschitz condition with positive Lipschitz constant l, the updatelaws of the coupling strengths and parameters are (2.9) and (2.10), respectively, then the states of the systems (2.7) and (2.8)starting from any initial values converge to the synchronous manifold M and b! a as t ! þ1.

Proof. We prove the synchronous manifold M is attractive in the case that the two hyperchaotic systems are coupled mutu-ally. If the two hyperchaotic systems are coupled unidirectionally, the proof is similar. To this end, we select the scalarfunction

L2ðtÞ ¼ eT eþXn

i¼1

Xn

j¼1

1hijðbij � aijÞ2 þ 2

Xn

i¼1

1ci

kixy � L

� �2; ð2:12Þ

where e ¼ ðe1; e2; . . . ; enÞT ¼ ðy1 � x1; y2 � x2; . . . ; yn � xnÞT ; L is a constant with 4L > nlþ 1. Differentiating the function L2ðtÞalong the trajectories of system (2.7) and (2.8), we obtain

_L2ðtÞ ¼ 2Xn

i¼1

eið _yi � _xiÞ þ 2Xn

i¼1

Xn

j¼1

1hijðbij � aijÞ _bij þ 4

Xn

i¼1

1ci

kixy � L

� �1 _ki

xy

¼ 2Xn

i¼1

eiðFiðy;biÞ � Fiðy;aiÞ þ Fiðy;aiÞ � Fiðx;aiÞÞ

þ 2Xn

i¼1

Xn

j¼1

1hijðbij � aijÞ _bij � 4L

Xn

i¼1

e2i ð2:13Þ

¼ 2Xn

i¼1

Xn

j¼1

eiðbij � aijÞfijðyÞ þ1hijðbij � aijÞ _bij

� �

þ 2Xn

i¼1

eiðFiðy;aÞ � Fiðx;aÞÞ � 4LXn

i¼1

e2i :

From the conditions of the theorem that _bij ¼ �eihijfijðyÞ; i ¼ 1;2; . . . ;n, we know that

eiðbij � aijÞfijðyÞ þ1hijðbij � aijÞ _bij ¼ 0; i ¼ 1;2; . . . ; n;

F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272 4267

which implies that

_L2ðtÞ ¼ 2Xn

i¼1

eiðFiðy;aÞ � Fiðx;aÞÞ � 2LXn

i¼1

e2i

6

Xn

i¼1

e2i þ l min

je2

j

� �� 4L

Xn

i¼1

e2i ð2:14Þ

6 ð1þ nl� 4LÞXn

i¼1

e2i 6 0:

Let the set W ¼ fðe; k; bÞ 2 R3n : e ¼ 0; k ¼ ðk1xy; k

2xy; . . . ; kn

xyÞg; bi 2 Rng, then E ¼ fðe; k; bÞ 2 R3n : e ¼ 0; k ¼ k0 2 Rn; b ¼ ag isthe largest invariant set in W, according to the invariant principle, the trajectories of system (2.7) and (2.8) starting fromany initial values converge asymptotically to the largest invariant set E, i.e., y! x and b! a as t ! þ1. The proof of thetheorem is completed. h

3. Numerical results

In this section, numerical simulations of hyperchaotic Chen systems are presented to demonstrate the effectiveness of theproposed adaptive methods. The hyperchaotic Chen system is described by

_x1 ¼ aðx2 � x1Þ þ x4;

_x2 ¼ dx1 � x1x3 þ cx2;

_x3 ¼ x1x2 � bx3;

_x4 ¼ x2x3 þ rx4;

8>>>><>>>>:

ð3:1Þ

where x1; x2; x3 and x4 are state variables, a; b; c; d and r are the parameters. When the parameters are given, two coupledhyperchaotic Chen systems is

_x1 ¼ aðx2 � x1Þ þ x4 þ k1yxðy1 � x1Þ;

_x2 ¼ dx1 � x1x3 þ cx2 þ k2yxðy2 � x2Þ;

_x3 ¼ x1x2 � bx3 þ k3yxðy3 � x3Þ;

_x4 ¼ x2x3 þ rx4 þ k4yxðy4 � x4Þ;

8>>>>>><>>>>>>:

ð3:2Þ

_y1 ¼ aðy2 � y1Þ þ y4 þ k1xyðx1 � y1Þ;

_y2 ¼ dy1 � y1y3 þ cy2 þ k2xyðx2 � y2Þ;

_y3 ¼ y1y2 � by3 þ k3xyðx3 � y3Þ;

_y4 ¼ y2y3 þ ry4 þ k4xyðx4 � y4Þ:

8>>>>>><>>>>>>:

ð3:3Þ

The error dynamics for systems (3.2) and (3.3) are as follows

_e1 ¼ aðe2 � e1Þ þ e4 � k1xy þ k1

yx

� �e1;

_e2 ¼ de1 þ ce2 � y1e3 � y3e1 þ e1e3 � k1xy þ k1

yx

� �e2;

_e3 ¼ �be3 þ y1e2 þ y2e1 � e1e3 � k1xy þ k1

yx

� �e3;

_e4 ¼ re4 þ y2e3 þ y3e2 � e2e3 � k1xy þ k1

yx

� �e4:

8>>>>>>>>><>>>>>>>>>:

ð3:4Þ

If the two hyperchaotic Chen systems are coupled unidirectionally, then kiyx ¼ 0; if the two hyperchaotic Chen systems are

coupled mutually, we assume kiyx ¼ ki

xy. In both cases, kixy varies adaptively according to the update law

_kixy ¼ ciðyi � xiÞ2; i ¼ 1;2;3;4; ð3:5Þ

where ci is a positive constant. In the numerical simulations, we assume that a ¼ 35; b ¼ 3; c ¼ 12; d ¼ 7; r ¼ 0:3, the ini-tial condition, ðx1ð0Þ; x2ð0Þ; x3ð0Þ; x4ð0ÞÞ ¼ ð1;2;3;4Þ; ðy1ð0Þ; y2ð0Þ; y3ð0Þ; y4ð0ÞÞ ¼ ð8;7;6;5Þ and the coupling strengths,ðk1ð0Þ; k2ð0Þ; k3ð0Þ; k4ð0ÞÞ ¼ ð1;3;5;2Þ; c ¼ ðc1; c2; c3; c4Þ

T ¼ ð0:1;0:1;0:1;0:1ÞT . Figs. 1 and 2 display the synchronization er-rors e1; e2; e3; e4 and the update laws of the coupling strengths k1; k2; k3; k4 of unidirectional coupling hyperchaotic Chensystems with given parameters with time t. Figs. 3 and 4 show the synchronization errors e1; e2; e3; e4 and the update lawsof the coupling strengths k1; k2; k3; k4 of mutual coupling hyperchaotic Chen systems with given parameters with time t.Obviously, Figs. 1 and 3 show the effectiveness of the proposed chaos synchronization schemes.

0 5 10 15−10

−5

0

5

10

t

e 10 5 10 15

−20

−10

0

10

t

e 2

0 5 10 15−10

0

10

20

t

e 3

0 5 10 15−10

0

10

20

30

te 4

Fig. 1. Synchronization errors e1; e2; e3; e4 of unidirectional coupling hyperchaotic Chen systems with time t.

0 5 10 151

2

3

4

5

6

7

k1

k2

k3 k4

t

Fig. 2. The update laws of the coupling strengths k1; k2; k3; k4 with time t.

4268 F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272

In the case of unknown parameters, two coupled hyperchaotic Chen systems is

_x1 ¼ aðx2 � x1Þ þ x4 þ k1yxðy1 � x1Þ;

_x2 ¼ dx1 � x1x3 þ cx2 þ k2yxðy2 � x2Þ;

_x3 ¼ x1x2 � bx3 þ k3yxðy3 � x3Þ;

_x4 ¼ x2x3 þ rx4 þ k4yxðy4 � x4Þ;

8>>>>>>><>>>>>>>:

ð3:6Þ

0 5 10 15−5

0

5

10

t

e 10 5 10 15

−10

−5

0

5

10

t

e 2

0 5 10 15−5

0

5

10

t

e 3

0 5 10 15−5

0

5

10

15

te 4

Fig. 3. Synchronization errors e1; e2; e3; e4 of mutual coupling hyperchaotic Chen systems with time t.

0 5 10 151

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

k1

k2

k3

k4

t

Fig. 4. The update laws of the coupling strengths k1; k2; k3; k4 with time t.

F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272 4269

_y1 ¼ b11ðy2 � y1Þ þ y4 þ k1xyðx1 � y1Þ;

_y2 ¼ b21y1 � y1y3 þ b22y2 þ k2xyðx2 � y2Þ;

_y3 ¼ y1y2 � b33y3 þ k3xyðx3 � y3Þ;

_y4 ¼ y2y3 þ b44y4 þ k4xyðx4 � y4Þ:

8>>>>><>>>>>:

ð3:7Þ

The update laws of coupling strengths and parameters are

_kixy ¼ ciðxi � yiÞ

2; i ¼ 1;2;3;4 ð3:8Þ

_b11 ¼ �h11e1ðy2 � y1Þ; _b21 ¼ �h21e2y1;

_b22 ¼ �h22e2y2;_b33 ¼ �h33e3y3;

_b44 ¼ �h44e4y4; ð3:9Þ

4270 F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272

respectively, where bij is an estimated value of the unknown parameter aij; ci; hij are positive constants. In the same way asabove, the error dynamics for systems (3.6) and (3.7) can be determined. The synchronization errors e1; e2; e3; e4 and theupdate laws of the coupling strengths k1; k2; k3; k4 of unidirectional coupling hyperchaotic Chen systems with unknownparameters with time t are displayed in Figs. 5 and 6(a). Synchronization errors e1; e2; e3; e4 and the update laws of the cou-pling strengths k1; k2; k3; k4 of the mutually coupled hyperchaotic Chen systems with unknown parameters with time t areshown in Figs. 7 and 8(a). Figs. 6(b) and 8(b) show that the estimated parameters b ¼ ðb11; b21; b22; b33; b44Þ converge to theparameters a ¼ ða11;a21;a22;a33;a44Þ ¼ ða; d; c; b; rÞ ¼ ð35;7;12;3;0:3Þ as t !1 with the initial conditionsbð0Þ ¼ ðb11ð0Þ; b21ð0Þ; b22ð0Þ; b33ð0Þ; b44ð0ÞÞ ¼ ð1;2;4;6;8Þ.

0 100 200 300−10

0

10

20

t

e 1

0 100 200 300−20

−10

0

10

te 2

0 100 200 300−20

−10

0

10

t

e 3

0 100 200 300−20

−10

0

10

20

t

e 4

Fig. 5. Synchronization errors e1; e2; e3; e4 of unidirectional coupling hyperchaotic Chen systems with time t.

0 100 200 3000

2

4

6

8

10

12

14

16

18

20

k1

k2 k3

k4

t0 100 200 300

−5

0

5

10

15

20

25

30

35

40

β11

β33

β22

β21

β44

t

a b

Fig. 6. (a) The update laws of the coupling strengths k1; k2; k3; k4 with time t; (b) changing parameters b ¼ ðb11 ;b21; b22;b33; b44Þ of unidirectional couplinghyperchaotic Chen systems with time t.

0 100 200 300−10

0

10

20

t

e 10 100 200 300

−20

−10

0

10

t

e 2

0 100 200 300−20

−10

0

10

t

e 3

0 100 200 300−20

−10

0

10

20

te 4

Fig. 7. Synchronization errors e1; e2; e3; e4 of mutual coupling hyperchaotic Chen systems with time t.

0 100 200 3000

5

10

15

k1

k2

k3k4

t0 100 200 300

−10

−5

0

5

10

15

20

25

30

35

40

β11

β33

β22β21

β44

t

a b

Fig. 8. (a) The update laws of the coupling strengths k1; k2; k3; k4 with time t; (b) changing parameters b ¼ ðb11 ;b21; b22;b33; b44Þ of mutual couplinghyperchaotic Chen systems with time t.

F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272 4271

4. Conclusion

This article investigates adaptive synchronization of two hyperchaotic systems and identification of unknown parame-ters. Based on Lasalle invariance principle, adaptive schemes are derived to make two unidirectional coupling and mutualcoupling hyperchaotic systems asymptotically synchronized whether the parameters are given or uncertain, and unknownparameters are identified simultaneously in the process of synchronization. Numerical simulations of hyperchaotic Chensystems are presented to show the effectiveness of the proposed chaos synchronization schemes.

4272 F. Austin et al. / Commun Nonlinear Sci Numer Simulat 14 (2009) 4264–4272

Acknowledgements

The authors thank the referees and the editor for their valuable comments and suggestions on improvement of this paper.The work is supported by the National Nature Science Foundation of China (No. 70571059).

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