Finite-Time Modified Projective Synchronization of Unknown R Ssler and Coullet Systems

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

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Finite-Time Modified Projective Synchronization of Unknown R o Ssler and Coullet Systems R. Z. Luo 1, F. Zhang 2 Department of Mathematics, Nanchang University, 330031, P. R. China luo_rz@163.com, 2Snowwhite0789@sohu.com

1

Abstract This paper discusses the finite-time modified projective

 ssler and Coullet systems. synchronization of unknown R o The unknown parameters are considered both in the drive system and the response system. By using the adaptive control technique, some sufficient conditions for the modified projective synchronization are presented. The corresponding numerical simulations are provided to illuminate the effectiveness of the proposed adaptive controllers.

Keywords Finite-Time Synchronization; Unknown Parameter; Chaotic System

Introduction Chaos synchronization is an important topic in the nonlinear science. Since Pecora and Carroll [Pecora, 1990] introduced a method to synchronization two identical chaotic systems with different initial conditions, chaos synchronization hasattracted a great deal of attention from various fields and has been developed and thoroughly studied over the past three decades because of its potential applications in secure communications, nano-oscillator, biological systems [Mohanty, 2005], etc. In the literature, there has several types of synchronization such as complete synchronization [Pecora, 1990], anti-synchronization [Zhao, 2011], phase synchronization [Taghvafard, 2011], generalized synchronization [Cai, 2011], lag synchronization [Taherionl, 1999], projective synchronization [Elabbasy, 2010], modified projective synchronization [Li, 2007], Modified function projective lag synchronization [Gao, 2013], combination synchronization [Luo, 2011], etc.

Amongst all kinds of chaos synchronization, the modified projective synchronization, which was proposed in [Li, 2007] where the chaotic systems can synchronize up to a constant scaling matrix, is the most noticeable one and has been extensively investigated [Li, 2007; Park, 2007; Park, 2008; Cai, 2010;

Bai, 2012] in recent years because it can obtain faster communication with its proportional feature.

However, most of researches mentioned above have concentrated on studying the infinite synchronization, which means that the trajectories of the response system can reach the trajectories of the drive system over the infinite time. From a practical point of view, however, it is more valuable that the synchronization objective is realized in finite time. To achieve faster convergence in control systems, finite-time control is a very useful technique. Moreover, the finite-time control techniques have demonstrated better robustness and disturbance rejection properties [Bhat, 1997].

So far as we know, less attention has been paid to the issue of finite-time modified projective synchronization of two different chaotic systems with unknown parameters. In this paper, adaptive control technique is adopted to synchronize two different chaotic  ssler and Coullet systems, with systems: the R o unknown parameters in finite time. A general scheme and parameters update laws are developed to deal with the modified synchronization of two different chaotic systems. Numerical simulations are presented to show the effectiveness of the proposed schemes.

This paper is organized as follows. In Section 2, the drive-response synchronization scheme is presented  ssler and Coullet systems with unknown for the R o parameters. In Section 3, some numerical simulation examples are given to verify the effectiveness of our method. A conclusion is given at the end.

Throughout this paper, if f ( x) : R → R is a vector n

n

function and V ( x) : R → R is a scalar function, then n

L f V ( x) is utilized to denote the Lie derivative of V ( x) along f ( x) , i.e., Lf V = ( x)

∂V ∂V ∂V ∂V = f ( x) f1 ( x) + f 2 ( x) +  + f n ( x). ∂x ∂x1 ∂x2 ∂xn

51

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Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

It is obvious that L f V ( x) is still a function from R n to R . If G ( x) is a n × m matrix, i.e.,

G ( x=) [ g1 ( x), g 2 ( x),, g m ( x)] , where gi ( x)(i = 1, 2, , m) is a function from R n to R n , we use the notation LG ( x )V ( x) for the Lie derivative, where [ Lg1 ( x )V ( x), Lg2 ( x )V ( x), , Lgm ( x )V ( x)].

LG (= x )V ( x )

Synchronization Criteria The Coullet chaotic system, proposed by Coullet and Arneodo [Coullet, 1979], is one of paradigms of chaos since it captures many features of chaotic system, including a simple cube part and three simple ordinary differential equations that depend on three positive real parameters.

In our drive-response type synchronization scheme, it is supposed that the Coullet system is the drive system, which is given by

 x= x2 , 1   x 2= x3 ,  3  x 3= a1 x3 + b1 x2 − c1 x1 − x1 ,

(1)

where a1 , b1 , c1 are unknown parameters. When

a1 =−0.45, b1 =−1.1, c1 =0.8 , system (1) is chaotic.

 ssler system [R o ssler, The response system is R o 1976], which is described by − y2 − y3 + u1 ,  y 1 =   y 2 = y1 + a2 y2 + u2 ,  y = b + y y − c y + u , 2 1 3 2 3 3  3

a1 , b1 , c1 , a2 , b2 , c2 , respectively. u1 Let = becomes

α1 +α 2 α2

y2 + y3 + sgn(e1 ), then system (3) e = f + g − BU ,

(4)

where e = (e1 , e2 , e3 ) , U = (0, u2 , u3 ) and T

T

α1    − sgn(e1 ) + α e2  2    α2  α2 = f  e3 + y3 − y1 − y2 a 2 ， α α 3 3    −α 3 x13 − y1 y3 + α 3 x3 a 1 + α 3 x2 b 1 − α 3 x1 c 1 − b 2 + y3 c 2      0    g=  − y2 (a2 − a 2) ，  α x (a − a ) + α x (b − ) − α x (c − c ) − (b − ) + y (c − c )   3 3 1 1 3 2 1 b1 3 1 1 1 2 b 2 3 2 2    0 0 0  0      B =  0 1 0  , U =  u2  .   0 0 1 u    3   Prior to moving toward designing the synchronizing controller, we then make the following assumptions:

A1. The unknown parameters a1, b1, c1, a2 , b2 , c2 are all norm bounded, i.e. | a1 |≤ aˆ1,| b1 |≤ bˆ1,| c1 |≤ cˆ1,| a2 |≤ aˆ 2,| b2 |≤ bˆ 2,| c2 |≤ cˆ 2,

where aˆ 1, bˆ1, cˆ1, aˆ 2, bˆ 2, cˆ 2 are known positive constants. The derivatives of unknown parameters (2)

where u1 , u2 , u3 are controllers, and a2 , b2 , c2 are unknown parameters. System (2) is chaotic if

u= u= u= 0 , and a2 = b2 = 0.2, c2 = 5.7 .

1 2 3

By defining the error variables as

e1= α1 x1 − y1 , e2= α 2 x2 − y2 , e3= α 3 x3 − y3 , where α1 , α 2 , α 3 are nonzero scaling factors, we get the following error system:

α1 α +α  e2 + 1 2 y2 + y3 − u1 e=1  α2 α2  α α  e=2 2 e3 + 2 y3 − y1− y2 (a2 −a 2)− y2 a 2−u2  α3 α3   e3= −α3 x13 − y1 y3 +α3 x3 a 1+α3 x2 b 1−α3 x1c 1−b 2+ y3 c 2+α3 x3 (a1−a 1)   + α 3 x2 (b1 − b 1) − α 3 x1 (c1 − c 1) − (b2 − b 2) + y3 (c2 − c 2) − u3 , (3) 52

where a 1, b 1, c 1, a 2, b 2, c 2 are the estimated values of

a1 , b1 , c1 , a2 , b2 , c2 are also all bounded, i.e. | a1 |≤ a1,| b1 |≤ b1,| c1 |≤ c1,| a 2 |≤ a 2,| b 2 |≤ b 2,| c 2 |≤ c 2, where a 1, b1, c1, a 2, b 2, c 2 are known positive constants.

ε, such that | ai − a i |≥ ε ,| bi − b i |≥ ε ,| ci − c i |≥ ε , i = 1, 2.

A2. There is a sufficient small positive constant

Note that | ai − a i |≥ 0,| bi − b i |≥ 0,| ci − c i |≥ 0 and ε is a positive design constant that can be chosen arbitrarily small, thus A2 is reasonable. The purpose of introducing ε is to avoid the unknown parameters from appearing in controllers and parameters update laws.

Remark 1. Many authors [Yassen, 2006; Park, 2007; Zhou, 2008; Yu, 2011] assumed that the unknown parameters of drive (or response) system are constants, however, in our paper the unknown parameters

ai , bi , ci , i =1, 2 may be time varying.

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2(  +1)( ˆ 2 + ˆ | | ) 2( a 1+1)( aˆ12 + aˆ1|a 1| ) 2( +1)( 2 + | | ) Now we give the definition of modified projective = Ω + b1 εb1 b1 a 1 + c1 εcˆ1 cˆ1 c 1 ε synchronization in finite time between the drive 2(  +1)( ˆ 2 + ˆ | | ) 2( +1)( 2 + | | ) 2( +1)( 2 + | | ) system (1) and the response system (2) and a lemma + a 2 aεˆ 2 aˆ 2 a 2 + b 2 bε 2 b 2 b 2 + c 2 εcˆ 2 cˆ 2 c 2 . which will be used in our proof.

Proof. Case 1. LBV1 = 0.

Definition 1. The drive system (1) and response system (2) are said to be modified projective Since LBV1 = 0, we conclude e2 = e3 = 0. Choose the synchronized in finite time if, for a suitable designed following Lyapunov function: V= V1 . Obviously, the feedback controller, there exists a constant T > 0 , time derivative of V along the trajectory of error lim ei = 0, system (3) is and ei = 0 for t > T , i = 1, 2, 3. such that t →T

Lemma 1 [Wang, 2009] Assume that a continuous, positive definite function V (t ) satisfies the following

 inequality, V (t ) ≤ −cV (t ), for η

differential

any

t > 0,V (t0 ) ≥ 0, where c > 0, 0 < η < 1 are all constants. Then, for any given t0 , V (t ) satisfies the following inequality, V 1−η (t ) ≤ V 1−η (t0 ) − c(1 − η )(t − t0 ), t0 ≤ t ≤ T , and V (t ) = 0 t0 + for t > T with T =

V 1−η ( t0 ) c (1−η )

.

Then we obtain the following results.

α i (i = 1, 2, 3) , the modified projective synchronization

between drive system (1) and response system (2) will occur in finite time by the following control laws (5) and (6) and the update rule (7):

α1 + α 2 y2 + y3 + sgn(e1 ), α2

time.

Case 2. LBV1 ≠ 0.

In this case, we choose the the Lyapunov candidate function as: 1 1 1 ( a1 − a 1) 2 + (b1 − b 1) 2 + (c1 − c 1) 2 2 2 2 1 1 1 2 2 + ( a2 − a 2) + (b2 − b 2) + (c2 − c 2) 2 .

2 2 2

V = V1 +

Theorem 1. For the given nonzero scaling factors

u1 =

1 dV 1 = − | e1 |= −(| e1 | + | e2 | + | e3 |) ≤ − 2 (e12 + e22 + e32 ) = − 2V12 . dt 2

By Lemma 1, one gets e1 = α1 x1 − y1 → 0 in finite

By differentiating the function of system (4), we get dV = dt

V along the trajectories

∂V1 ∂V1 ∂V1 f − BU + g + ( a1 − a 1)( a 1 − a 1) ∂e ∂e ∂e

+ (b1 − b 1)(b1 − b 1) + (c1 − c 1)(c1 − c 1) (5)

+(a2 − a 2)(a 2 − a 2) + (b2 − b 2)(b 2 − b 2) + (c2 − c 2)(c 2 − c 2)

L V + (| e1 | + | e2 | + | e3 |) + Ω = −(| e1 | + | e2 | + | e3 |) − Ω − y2 e2 (a2 − a 2) + α 3 x3e3 (a1 − a 1)  T f 1 , LBV1 ≠ 0, ( LBV1 ) 2 U = (||LBV1||) +α 3 x2e3 (b1 − b 1) − α 3 x1e3 (c1 − c 1) − e3 (b2 − b 2) 0, 0, LBV1 =  aˆ − a (6) + y3e3 (c2 − c 2) + (a1 − a 1)(a1 − (α 3 x3e3 + (a1 + 1) 1 1 )) ε and  aˆ 1 − a 1  a=1 α 3 x3e3 + (a 1 + 1) ε ,   = α x e + (  + 1) bˆ1 − b 1 , b1 3 2 3 b1 ε  cˆ − c c 1 = −α 3 x1e3 + (c1 + 1) 1 1 ,  ε  aˆ − a 2  a 2 = − y2 e2 + (a 2 + 1) 2 ,  ε  ˆ −  b 2 =−e3 + (b 2 + 1) b 2 b 2 , ε   cˆ − c 2 ,  c 2 = y3e3 + (c 2 + 1) 2 ε  Where V1=

1 2

(e + e + e ) and 2 1

2 2

2 3

ˆ − + (b1 − b 1)(b1 − (α 3 x2 e3 + (b1 + 1) b1 b 1 ))

ε

+ (c1 − c 1)(c1 − ( −α 3 x1e3 + (c1 + 1) + ( a2 − a 2)( a 2 − ( − y2 e2 + ( a 2 + 1)

cˆ1 − c 1

ε

aˆ 2 − a 2

ε ˆ − + (b2 − b 2)(b 2 − (−e3 + (b 2 + 1) b 2 b 2 )) ε + (c2 − c 2)(c 2 − ( y3e3 + (c 2 + 1)

))

cˆ 2 − c 2

ε

))

))

= −(| e1 | + | e2 | + | e3 |) − Ω + (a1 − a 1)(a 1 − (a 1 + 1)

aˆ 1 − a 1

ε

)

ˆ− cˆ − c +(b1 − b 1)(b1 − (b1 + 1) b1 b 1 ) + (c1 − c 1)(c1 − (c1 + 1) 1 1 ) ε ε (7)

+(a2 − a 2)(a 2 − (a 2 + 1)

ˆ− aˆ 2 − a 2 ) + (b2 − b 2)(b 2 − (b 2 + 1) b 2 b 2 ) ε ε 53

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+(c2 − c 2)(c 2 − (c 2 + 1)

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

cˆ 2 − c 2

ε

In view of the A1 and A2, we have

)

= −(| e1 | + | e2 | + | e3 |) − Ω + ( a1 − a 1) a 1 + (b1 − b 1)b1

+(c1 − c 1)c1 + (a2 − a 2)a 2 + (b2 − b 2)b 2 + (c2 − c 2)c 2 (a − a )(a − a ) (a − a )(a − aˆ ) −(a1 + 1) 1 1 1 1 + (a1 + 1) 1 1 1 1 ε ε (b − )(b − ˆ ) (b − )(b − ) −(b1 + 1) 1 b 1 1 b 1 + (b1 + 1) 1 b 1 1 b1

ε

−(c1 + 1)

−(a 2 + 1) −(b 2 + 1) −(c 2 + 1)

ε

(c1 − a 1)(c1 − c 1)

+ (c1 + 1)

ε (a2 − a 2)(a2 − a 2)

+ (a 2 + 1)

ε

(b2 − b 2)(b2 − b 2)

ε

(c2 − c 2)(c2 − c 2)

ε

ε (a2 − a 2)(a2 − aˆ 2)

ε

(c2 − c 2)(c2 − cˆ 2)

Note that

1 2), in finite time. and a i → ai , b i → ai , c i → ci (i =, It is worth mentioning that the adaptive feedback scheme proposed in this paper can also be extended to investigate the special cases in which some of the example, if the unknown parameters a1 , b1 , c1 , a2 , b2 , c2 are all constants, and we have the following Corollary 1.

(b2 − b 2)(b2 − bˆ 2)

ε

By Lemma 1, it can be concluded that ei → 0(i = 1, 2, 3)

unknown parameters a1 , b1 , c1 , a2 , b2 , c2 are constants. For

ε

+ (b 2 + 1)

+ (c 2 + 1)

(c1 − c 1)(c1 − cˆ1)

dV ≤ −(| e1 | + | e2 | + | e3 |)− | a1 − a 1 | − | b1 − b 1 | − | c1 − c 1 | dt 1 − | a2 − a 2 | − | b2 − b 2 | − | c2 − c 2 |≤ − 2V 2 .

(8)

(a1 − a 1)(a1 − aˆ1)= a12 − a1 aˆ1 − a 1a1 + aˆ1a 1 ≤ 2(aˆ12 + aˆ1 | a 1 |).

In the same way, we have 2 (b1 − b 1)(b1 − bˆ1) ≤ 2(bˆ1 + bˆ1 | b 1 |),

(c1 − c 1)(c1 − cˆ1) ≤ 2(cˆ12 + cˆ1 | c 1 |), ( a2 − a 2)( a2 − aˆ 2) ≤ 2( aˆ 22 + aˆ 2 | a 2 |),

Corollary 1. For the given nonzero scaling factors

α i (i = 1, 2, 3) , if the unknown parameters a1, b1, c1, a2 , b2 , c2

are all constants, then the modified projective synchronization between drive system (1) and response system (2) will occur in finite time by the following control laws (9) and (10) and the update rule (11):

u1 =

2 (b2 − b 2)(b2 − bˆ 2) ≤ 2(bˆ 2 + bˆ 2 | b 2 |), (c2 − c 2)(c2 − cˆ 2) ≤ 2(cˆ 22 + cˆ 2 | c 2 |).

54

(9)

L V + (| e1 | + | e2 | + | e3 |) + Ω  T f 1 , LBV1 ≠ 0, ( LBV1 ) U = (||LBV1||)2  0, 0, LBV1 = 

By substituting the above inequations into (8), one gets

dV ≤ −(| e1 | + | e2 | + | e3 |) + (a1 − a 1)a1 + (b1 − b 1)b1 + (c1 − c 1)c1 dt (a − a )2 +(a2 − a 2)a 2 + (b2 − b 2)b2 + (c2 − c 2)c 2 − (a1 + 1) 1 1 ε 2 2 (b1 − b 1) (c1 − c 1) (a2 − a 2)2  −(b1 + 1) − (c1 + 1) − (a 2 + 1) ε ε ε 2 2 (b − ) (c − c ) −(b 2 + 1) 2 b 2 − (c 2 + 1) 2 2 ε ε |a −a | ≤ −(| e1 | + | e2 | + | e3 |)+ | (a1 − a 1) | (| a1 | −a1 1 1 ) ε | b1 − b 1 | |c −c | )+ | (c1 − c 1) | (| c1 | −c1 1 1 ) + | (b1 − b 1) | (| b1 | −b1 ε ε | a2 − a 2 | |a − | )+ | (b2 − b 2) | (| b2 | −b 2 2 b 2 ) + | (a2 − a 2) | (| a 2 | −a 2 ε ε 2 2 |c −c | |a −a | |b − | + | (c2 − c 2) | (| c 2 | −c 2 2 2 ) − 1 1 − 1 b 1 ε ε ε 2 2 2 2 |c −c | |a −a | |b −a | |c −c | − 1 1 − 2 2 − 2 2 − 2 2 . ε ε ε ε

α1 + α 2 y2 + y3 + sgn(e1 ), α2

(10)

and

 aˆ 1 − a 1  a 1 = α 3 x3e3 + ε ,    = α x e + bˆ1 − b 1 , 3 2 3  b1 ε  cˆ − c c 1 = −α 3 x1e3 + 1 1 ,  ε  aˆ − a 2  a 2 = − y2 e2 + 2 ,  ε  ˆ −  b 2 =−e3 + b 2 b 2 , ε   cˆ − c y3e3 + 2 2 ,  c 2 = ε  where V1=

+

2( bˆ12 +bˆ1|a 1|) ε

1 2

+

(e12 + e22 + e32 ) and Ω =

2( cˆ12 + cˆ1|c 1|) ε

+

2( aˆ 22 + aˆ 2|a 2|) ε

+

(11)

2( a ˆ 12 + a ˆ 1|a 1| )

2( bˆ 22 +bˆ 2|b 2|) ε

ε

+

2( cˆ 22 + cˆ 2|c 2|) ε

.

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

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Proof. The proof is straightforward since in this case

a= c 2 , are shown in Figs. 2 and 3, respectively.

2 b= 2

order to avoid tedious repetitions.

It is clear that the adaptive parameters converge to the expected values quickly. These simulation results have illustrated the effectiveness of the proposed method.

a 1 = b1 = c1 = a 2 = b 2 = c 2 = 0, which is omitted in

Numerical Simulation In this section, the Fourth-order Runge-Kutta method is used to obtain solutions for systems (1) and (2). In the simulation process, the ε is selected as 0.0001, the parameters are chosen to be a1 =−0.45, b1 =−1.1, c1 =0.8 and

a2 = b2 = 0.2, c2 = 5.7 in all simulations so that the drive and response systems exhibit chaotic behavior if no control is applied. Thus, we can take

aˆ 1 =−0.45, bˆ1 =−1.1, cˆ1 =0.8, aˆ 2 =bˆ 2 =0.2, cˆ 2 =5.7 and a 1 = b1 = c1 = a 2 = b 2 = c 2 = 0. Assume that the synchronization factors are α1 = 2, α 2 =−1, α 3 =1, and the initial states of the drive and response system are chosen as ( x1 (0), x2 (0), x3 (0)) = (1,1.2, 0.3) , ( y1 (0), y2 (0), y3 (0)) T

T

T

= (0.3, −4,1.2)T , respectively, then the error system has the initial values e1 =1.7, e2 =2.8, e3 =−0.9. Furthermore, the initial values of estimated parameters are chosen

1 . Synchronization c= as a= 1 2 2 b=1 c=1 a= b= 2 of systems (1) and (2) via control laws (9) and (10) and the update rule (11) are shown in Figs. 1-3.

FIG. 2. TIME RESPONSE OF THE ADAPTIVE VECTOR PARAMETER

a=1 b=1 c 1 .

FIG. 1. ERROR SIGNALS BETWEEN DRIVE SYSTEM (1) AND RESPONSE SYSTEM (2).

 ssler and The synchronization errors between the R o Coullet systems are revealed in Fig. 1. It is observed that the synchronization errors are regulated to zero 1.7 2 + 2.82 + 0.92 ≈ 3.3971 . The time responses of the adaptive parameter vectors a= 1 b=1 c 1 and within

FIG. 3. TIME RESPONSE OF THE ADAPTIVE VECTOR PARAMETER

a= c2.

2 b= 2

55

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Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

Conclusions

A., Vol. 72, 1979, pp. 268-270.

In this paper, the finite-time modified projective  ssler and Coullet synchronization of unknown R o systems has been investigated, then a novel nonlinear control scheme for finite-time chaos synchronization is proposed using the finite-time Lyapunov theory and the adaptive method. Finally, numerical simulations are provided to illustrate the effectiveness of the proposed scheme.

The scheme proposed in this paper is under the assumption that there is no time delays in transmission of information between the drive and response systems. However, it is well known that many communication processes contain time delays in transmission of information due to finite signal transmission times, switching speeds, memory effects, etc. Thus, the finite-time modified projective  ssler and Coullet synchronization between delayed R o systems with unknown parameters would be an interesting issues for further research.

Elabbasy, E. M. and El-Dessoky, M. M. “Adaptive feedback

 control for the projective synchronization of the L u dynamical

system

and

its

application

to

secure

communication,” Chin. J. Phys. (Taipei), Vol. 48(6), 2010, pp. 863-872.

Gao, Y. B. & Sun, B. H. & Lu, G. P. “Modified function projective lag synchronization of chaotic systems with disturbance

estimations",

Applied

Mathematical

Modelling, Vol. 37, 2013, pp.4993-5000.

Li, G. H. “Modified projective synchronization of chaotic system", Chaos, Solitons

&

Fractals, Vol. 32, 2007, pp.

1786-1790.

Luo, R. Z. & Wang, Y. L. & Deng, S.C. “Combination synchronization of three classic chaotic systems using active backstepping design", CHAOS, Vol. 21, 2011, 043114.

Mohanty

P,

“Nanotechnology:

nano-oscillators

get

it

together", Nature, Vol. 437, 2005, pp. 325-326.

ACKNOWLEDGMENTS

Park, Ju H. “Adaptive modified projective synchronization

We are grateful to anonymous reviewers for their valuable comments and suggestions, which has led to better presentation of this paper. This work was supported by the Natural Science Foundation of Jiangxi Province under grant No. 20122BAB201005 and the National Natural Science Foundation of China under grant No. 11361043.

of a unified chaotic system with an uncertain parameter", Chaos, Solitons and Fractals, Vol. 34, 2007, pp. 1552-1559.

Park, Ju H. “Adaptive controller design for modified projective

synchronization

of

Genesio-Tesi

chaotic

system with uncertain parameters", Chaos, Solitons and Fractals, Vol. 34, 2007, 1154-1159.

Park, Ju H. “Adaptive control for modified projective

REFERENCES

synchronization of a four-dimensional chaotic system

Bai, J. & Yu, Y. G. & Wang, S. & Song, Y. “Modified

with uncertain parameters", Journal of Computational

projective synchronization of uncertain fractional order

and Applied Mathematics, Vol. 213, 2008, pp. 288- 293.

hyperchaotic systems", Commun Nonlinear Sci Numer

Pecora, L.M. and Carroll, T.L. “Synchronization in chaotic systems", Physical Review Letters, Vol. 64, 1990, pp. 821-

Simulat, Vol. 17, 2012, pp. 1921-1928.

Bhat,

S.

&

Bernstein,

homogeneous

D.

systems",

“Finite-time in:

stability

Proceedings

of

of

ACC,

Albuquerque, NM, 1997, pp. 2513-2514.

824.

 ssler, Ro

O. E. “An equation for continuous chaos", Phys

Lett A, Vol. 57, 1976, pp. 397-398.

Cai, N. & Jing, Y. W. & Zhang, S. Y. “Modified projective

Taghvafard, H. and Erjaee, G.H. “Phase and anti-phase

synchronization of chaotic systems with disturbances via

synchronization of fractional order chaotic systems via

active sliding mode control", Commun Nonlinear Sci

active control", Commun Nonlinear Sci Numer Simulat,

Numer Simulat, Vol. 15, 2010, pp. 1613-1620.

Vol. 16, 2011, pp. 4079-4088.

Cai, N. & Li, W. Q. & Jing, Y. W. “Finite-time generalized

Taherionl,

S.

&

Lai,

Y.

C.

“Observability

of

lag

synchronization of chaotic systems with different order",

synchronization of coupled chaotic oscillators", Phys Rev

Nonlinear Dyn., Vol. 64, 2011, pp. 385-393.

E, Vol. 59, 1999, pp. 6247-6250.

Coullet, & Tresser, P. C. & Arneodo, A. “Transition to

Yu, Y. G. & Li, H. X. “Adaptive hybrid projective

stochasticity for a class of forced oscillators", Phys. Lett.

synchronization of uncertain chaotic systems based on

56

Communications in Control Science and Engineering (CCSE) Volume 1 Issue 4, October 2013

www.as-se.org/ccse

backstepping design", Nonlinear Analysis: Real World

and synchronization of a novel hyperchaotic system with

Applications, Vol.12, 2011, pp. 388-393.

uncertain

Wang, H. &Han, Z. Z. & Xie, Q. Y. & Zhang, W. “Finite-time chaos synchronization of unified chaotic system with uncertain parameters", Communications in Nonlinear Science and Numerical Simulation, Vol. 14, 2009, pp.

2239-2247.

Yassen, M. T. “Adaptive chaos control and synchronization for uncertain new chaotic dynamical system", Physics Letters A, Vol. 350, 2006, pp.36-43.

Zhao, H. Y. and Zhang, Q. “Global impulsive exponential anti-synchronization of delayed chaotic neural networks", Neurocomputing, Vol. 74, 2011, pp. 563-567.

Zhou, X. B. & Wu, Y. & Li, Y. & Xue, H. Q. “Adaptive control

parameters",

Applied

Mathematics

and

Computation, Vol. 203, 2008, pp. 80-85.

R. Z. Luo received the M. Sc. in Operational Research and Cybernetics in 2001, and the Ph.D. in Operational Research nd Cybernetics in 2005 all from Shanghai University, China.

He has published more than 20 papers in chaos control and synchronization. He is currently an associate professor in the Department of Mathematics at Nanchang University, China.

His research areas include chaos control and synchronization, Network control and synchronization, and nonlinear control.

F．Zhang, Lecturer in Nanchang University, obtained her master degree of basic mathematics in 2003. Now she is majoring in Management Science and Engineering for her Doctor degree.

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