A Novel Adaptive Synchronization of Lu’s Chaotic System

8/3/2019 A Novel Adaptive Synchronization of Lus Chaotic System

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European Journal of Scientific Research

ISSN 1450-216X Vol.45 No.4 (2010), pp.630-636

EuroJournals Publishing, Inc. 2010

http://www.eurojournals.com/ejsr.htm

A Novel Adaptive Synchronization of Lus Chaotic System

K. Ratchagit

Department of Mathematics, Faculty of Science

Maejo University, Chiang Mai 50290, Thailand

E-mail: [email protected]: +66-053-873000; Fax: +66-053-873000

Abstract

In this paper, we study Lus system, and we study the stability of equilibrium point

of Lus system. Then, we study chaos synchronization of Lus system by using adaptive

control methods.

Keywords: Lus system; Synchronization; Adaptive control.

1. IntroductionChaos in control systems and controlling chaos in dynamical systems have both attracted increasing

attention in recent years. A chaotic system has complex dynamical behaviors that possess some special

features, such as being extremely sensitive to tiny variations of initial conditions, having boundedtrajectories in the phase space. Controlling chaos has focused on the nonlinear systems such as a Lus

system.

Lus system was first introduced in [2] which is described by.

.

.

( )x a y x

y xz cy

z xy bz

=

= +

=

(1.1)

where , , x y z are state variables, , ,a b c are positive constants.

The objective of this paper is as follow. We investigate adaptive synchronization for Lus

system when the parameters of the drive system are fully unknown and different with those of theresponse system.

2. Synchronization of the Lus SystemConsider two nonlinear systems:

( , ) (*)

( , ) ( , , ) (**)

x f t x

y g t y u t x y

=

= +

&

&

Where , , , [ , ],n r n nx y f g C + [ , ], 1,r n n nu C r+ + is the set of non-negative

real numbers. Assume that (*) is the drive system, (**) is the response system, and ( , , )u t x y is the

control vector.

Definition 2.1. Response system and drive system are said to be synchronic if for any initial

conditions 0 0( ), ( ) ,n

x t y t lim ( ) ( ) 0.t

x t y t +

=


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A Novel Adaptive Synchronization of Lus Chaotic System 631

In this section, we consider adaptive synchronization Lus systems. This approach cansynchronize the chaotic systems when the parameters of the drive system are fully unknown and

different with those of the response system. Assume that there are two Lus systems such that the drive

system (with the subscript 1) is to control the response system (with the subscript 2). The drive andresponse system are given, respectively, by

.

1 1 1

.

1 1 11

.

1 1 1 1

( ) x a y x

y x z cy

z x y bz

=

= +

=

(2.1)

where the parameters , ,a b c are unknown or uncertain, and..

2 1 2 2 1

.

2 2 2 1 2 2

.

2 2 2 1 2 3

( ) x a y x u

y x z c y u

z x y b z u

=

= +

=

(2.2)

where 1 1 1, ,a b c are parameters of the response system which need to be estimated, and [ ]T

1 2 3, ,u u u u= is

the controller we introduced in (2.2). We choose

1 1 2 1

2 2 2 1

3 3 2 1

( )( )

( )

x

y

z

u k e a x xu k e c y y

u k e b z z

= = +

=

(2.3)

where , , x y ze e e are the error states which are defined as follows:

2 1

2 1

2 1

x

y

z

e x x

e y y

e z z

=

=

=

and

1 2 2

1 2

1 2

( )a x x

b z

c z

a f x e y e

b f y e

c f x e

= =

= =

= =

&

&

&

(2.4)

where 1 2 3, , 0k k k and , , are positive real constants.

Theorem 2.1. Let 1 2 3, , 0k k k be properly chosen so that the following matrix inequality holds,

1

2

3

1( 1) 0

2

1 1( 1) 1 (1 ) 0

2 2

10 (1 )

2

k a a

P a k b

b k

+ = + + >

(2.5)

or 1 2 3, ,k k k can be chosen so that the following inequalities holds,

(ii) 1 0 A k a= > (iii) 22 1( 1) ( 1) 0

4 B A k a= + + >

(iv) 2 32 3 1( 1) (1 ) ( 1) 0.4 4

kC A k k A b a= + + >

Then the two Lus system (2.1) and (2.2) can be synchronized under the adaptive controls (2.3)

and (2.4).

Proof. It is easy to see from (2.1) and (2.2) that the error system is


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632 K. Ratchagit

1 2 2 1 1 1

2 2 1 2 1 1 1 2

2 2 1 2 1 1 1 3

( ) ( )x

y

z

e a y x a y x u

e x z c y x z cy u

e x y b z x y bz u

=

= + +

= +

&

&

&

(2.6)

Let 1 1 1, , .a b ce a a e b b e c c= = = Choose the Lyapunov function as follows:

2 2 2 2 2 21 1 1 1( ) .

2x y z a b cV t e e e e e e

= + + + + +

Then the differentiation ofV along trajectories of (2.6) is

[ ]

[ ]

[ ]

[ ] [ ]

[ ]

1 2 2 1 1 1

2 2 1 2 1 1 1 2

2 2 1 2 1 1 1 3

1 2 1 2 2 1 2 1 2 2

1 1 2 2 2 1 2 1

1 1 1

( ) ( )

1 1 1

= + + + + +

=

+ + +

+ +

+ + +

= + +

+ + +

& & & & & & &x x y y z z a a b b c c

x

y

z

a a b b c c

x x

y

V e e e e e e e e e e e e

e a y x a y x u

e x z c y x z cy u

e x y b z x y bz u

e f e f e f

e a y ay ay ay e a x ax ax ax

e x z x z x z x z e [ ]

[ ] [ ]

1 2 1 2 2

2 2 1 1 1 2 1 2 1 1 2 1 1 1 1

1 2 3

1 1 1

+

+ + + +

+ + +

y

z z

x y z a a b b c c

c y cy cy cy

e x y x y x y x y e bz b z b z b z

e u e u e u e f e f e f

2 2 1 2

2 2

2

2

2 2 1 1 2 1

2 2

2 3 2 1

2 2 21 2 3

2

( )

( ) ( )

1( )

1 1

( ) ( 1) ( 1)

1(1 )

= + +

+ +

+ + +

+

+ +

= + + +

+ + +

a x x y a x x a x

x x y y y z b z y z

c z z x x

y z z a a

b b c

x y z x y

y z a a

y e e ae e e e x x e x e e

e e e e e e y e e be e

x e e c x x e k e x x e

k e k e c x x e e f

e f e f

k a e k e k e a e e

b e e e f y e 2

2 2

2 2 2

1 2 3

1 1

( ) ( 1) ( 1)

(1 )

+

+ + +

+ + +

+

=

x x x

b b z c c z

x y z x y

y z

T

x e e

e f y e e f x e

k a e k e k e a e e

b e e

e Pe

whereT

x y ze e e e = and P is as in (2.5). Since ( )V t is positive definite and ( )V t& is negative

semidefinite, it follows that 1 1 1, , , , , . x y ze e e a b c L From ( ) ,T

V t e Pe & we can easily show that the square

of , , x y ze e e are integrable with respect to t, namely, 2, , . x y ze e e L From (2.6), for any initial conditions,

we have ( ), ( ), ( ) . x y ze t e t e t L& & & By the well-known Barbalats Lemma, we conclude that ( , , ) (0,0,0) x y ze e e

as .t + Therefore, in the closed-loop system, 2 1 2 1 2 1( ) ( ), ( ) ( ), ( ) ( )x t x t y t y t z t z t as .t + This

implies that the two Lus systems have synchronized under the adaptive controls (2.3) and (2.4).


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Numerical Simulations

The numerical simulations are carried out using the Fourth-order Runge-Kutta method. The initial

states are 1 1 1(0) 0.65, (0) 0, (0) 0 x y z= = = for the drive system and 2 (0) 0.2, x = 2 2(0) 0.1, (0) 0.1y z= = forthe response system. The parameters of the drive system are 5, 10, 0.5.a b c= = = The control parameters

are chosen as follows 1 2 35, 50, 21k k k= = = which satisfy (2.5). Choose 1. = = = The initial values of

the parameters 1 1 1, ,a b c are all chosen to be 0. As shown in Fig. 1, the response system synchronizes

with the drive system. The changing parameters of 1 1 1, ,a b c are shown in Fig. 2-4.

Figure 1: Synchronization error ( , , ) x y z

e e e for system (2.1) and (2.2) with time t.

Figure 2: Changing parameter1a of the system (2.2) with time t.


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634 K. Ratchagit

Figure 3: Changing parameter1b of the system (2.2) with time t.

Figure 4: Changing parameter1c of the system (2.2) with time t.


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3. Summary and Concluding RemarksIn this paper, we give sufficient conditions for stability of equilibrium points of synchronization of two

Lus systems using adaptive control which control the chaotic behavior of Lus system to its

equilibrium points. Numerical Simulations are also given to verify results we obtained.

AckhowledgmentThis work was supported by the Thai Research Fund Grant, the Higher Education Commission and

Maejo University, Thailand. The author would like to thank the anonymous referee for his/her valuablecomments and remarks which greatly improved the final version of the paper.

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A Novel Adaptive Synchronization of Lu’s Chaotic System