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Synchronization of Rossler and Chen chaotic dynamical systems using active control
1 January 2001
Physics Letters A 278 (2001) 191–197 www.elsevier.nl/locate/pla
Synchronization of Rossler and Chen chaotic dynamical systems using active control H.N. Agiza ∗ , M.T. Yassen Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 1 September 2000; accepted 20 November 2000 Communicated by A.R. Bishop
Abstract This Letter presents chaos synchronization of two identical Rossler and Chen systems by using active control. The proposed technique is applied to achieve chaos synchronization for the Rossler and Chen dynamical systems. We demonstrate that a coupled Rossler and Chen systems can be synchronized. Numerical simulations are used to show the effectiveness of the proposed control method. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction In the last few years, synchronization in chaotic dynamical systems has received a great deal of interest among scientists from various fields [1–14]. The idea of synchronizing two identical chaotic systems that from different initial conditions was introduced by Pecora and Carrols [1,2]. Synchronization in chaotic systems has been extensively investigated in the last few years [6,8,9] and many possible applications have been discussed by computer simulation and realized in laboratory condition [9,10]. More recently, synchronization of hyperchaotic systems was investigated [10,11] and the generalized synchronization was proposed [12–14], which makes secret communication more practicable. Recently, a coupled of Lorentz system is synchronized by using active control [15,16]. The aim of this * Corresponding author.
E-mail address: [email protected]. (H.N. Agiza).
Letter is to apply active control to synchronize both Rossler and Chen dynamical systems. This Letter is organized as follows: In Section 2 we apply active control to Rossler dynamical system and numerical simulation are used to show this process. In Section 3 active control is applied to Chen chaotic attractor and numerical experiments are performed to show such synchronization. Consider the system of differential equations x˙ = f (x),
(1)
y˙ = g(y, x),
(2)
where x ∈ R n , y ∈ R n , f, g : R n → R n are assumed to be analytic functions. Let x(t, x0 ) and y(t, y0 ) be solutions to (1) and (2), respectively. The solutions x(t, x0 ) and y(t, y0) are said to be synchronized if
lim x(t, x0 ) − y(t, y0) = 0.
t →∞
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 7 7 - 5
(3)
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H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
2. Synchronization of two Rossler systems using active control
We define the active control functions u1 (t), u2 (t) and u3 (t) as follows: u1 (t) = V1 (t),
The Rossler system [17] is described by the set of ordinary differential equations
u2 (t) = V2 (t),
x˙ = −y − z,
u3 (t) = x1 z1 − x2 z2 + V3 (t).
y˙ = x + ay,
(4)
This implies
z˙ = b + z(x − c).
x˙ 3 = −y3 − z3 + V1 (t),
The parameters a, b and c are taken in a range to insure the chaotic behaviour of (4). We assume that we have two Rossler systems and that the drive system with the subscript 1 is to control the response system with subscript 2. The drive and response systems are defined as follows:
y˙3 = x3 + ay3 + V2 (t),
x˙1 = −y1 − z1 , y˙1 = x1 + ay1,
(5)
z˙ 1 = b + z1 (x1 − c) and x˙2 = −y2 − z2 + u1 (t), y˙2 = x2 + ay2 + u2 (t),
(6)
z˙ 2 = b + z2 (x2 − c) + u3 (t). We have introduced three control functions u1 (t), u2 (t) and u3 (t) in (6). Our goal is to determine these functions u1 (t), u2 (t) and u3 (t). In order to estimate the control functions, we subtract (5) from (6). We define the error system as the differences between the Rossler systems (5) and (6) that is to be controlled and the controlling system using x3 = x2 − x1 ,
(9)
(10)
z˙ 3 = −cz3 + V3 (t). The error system (10) to be controlled is a linear system with a control input V1 (t), V2 (t) and V3 (t) as function of x3 , y3 and z3 . As long as these feedbacks stabilize the system, x3 , y3 and z3 converge to zero as time t tends to infinity. This implies that two Rossler systems are synchronized with feedback control. There are many possible choices for the control V1 (t), V2 (t) and V3 (t). We choose ! ! x3 V1 (t) V2 (t) = A y3 , z3 V3 (t) where A is a 3 × 3 constant matrix. In order to make the closed loop system will be stable, the proper choice of the elements of the matrix A is such that the feedback system must have all of the eigenvalues with negative real parts. Let the matrix A is chosen in the form ! −1 1 1 A = −1 −(1 + a) 0 . 0 0 c−1
Using this notation, we obtain
In this particular choice, the closed loop system (10) has the eigenvalues −1, −1 and −1. This choice will lead to the error states x3 , y3 and z3 converge to zero as time t tends to infinity and hence the synchronization of two Rossler systems is achieved.
x˙3 = −y3 − z3 + u1 (t),
2.1. Numerical results
y3 = y2 − y1 ,
(7)
z3 = z2 − z1 .
y˙3 = x3 + ay3 + u2 (t),
(8)
z˙ 3 = x2 z2 − x1 z1 − cz3 + u3 (t). System (8) is called the error system and x3 , y3 and z3 are called the error states. The synchronization problem for Rossler system is to achieve the asymptotic stability of the zero solution of the error system (8).
Fourth-order Runge–Kutta method is used to solve the systems of differential equations (5) and (6) with time step size equal 0.001 in all numerical simulations. We select numerical values for the parameters in (4) as a = 0.2, b = 0.2, c = 5.7 to insure the chaotic behaviour of Rossler system. The initial values of the
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
193
(b)
(c) Fig. 1. Solution of the coupled Rossler system of equations with the active control deactivated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
drive system are x1 (0) = 0.5, y1 (0) = 1 and z1 (0) = 1.5 and the initial values of the response system are x2 (0) = 2.5, y2 (0) = 2 and z2 (0) = 2.5. Then the initial values for the error states are x3 (0) = 2, y3 (0) = 1 and z3 (0) = 1. The results of the simulation of the two identical Rossler systems without active control are shown
in Fig. 1: (a) displays the x1 and x2 , (b) displays the y1 and y2 , and (c) displays the z1 and z2 . Figs. 2(a)–(c) display the same sequence of signals with the active control. We observe in numerical simulations that the synchronization of derive and response of two identical Rossler systems occurred after t = 10.
194
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
(b)
(c) Fig. 2. Solution of the coupled Rossler system of equations with the active control activated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
z˙ = xy − bz.
3. Synchronization of two Chen systems using active control Recently, Chen [18] introduced a new chaotic attractor (Chen attractor). The nonlinear differential equations that describe Chen attractor are x˙ = a(y − x), y˙ = (c − a)x − xz + cy,
(11)
System (11) has chaotic behaviour [18] at the parameters values a = 35, b = 3 and c = 28. Our aim is to make synchronization of system (11) by using active control. For Chen chaotic attractor, the drive system is defined as follows: x˙ 1 = a(y1 − x1 ),
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
195
(b)
(c) Fig. 3. Solution of the coupled Chen system of equations with the active control deactivated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
y˙1 = (c − a)x1 − x1 z1 + cy1 ,
(12)
z˙ 1 = x1 y1 − bz1 , and the response system is given by
x3 = x2 − x1 ,
x˙2 = a(y2 − x2 ) + µ1 (t), y˙2 = (c − a)x2 − x2 z2 + cy2 + µ2 (t), z˙ 2 = x2 y2 − bz2 + µ3 (t).
We have introduced three control functions µ1 (t), µ2 (t) and µ3 (t) in Eq. (13). These functions are to be determined. Let the error states are
(13)
y3 = y2 − y1 , z3 = z2 − z1 .
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H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
(b)
(c) Fig. 4. Solution of the coupled Chen system of equations with the active control activated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
We define the active control functions µ1 (t), µ2 (t) and µ3 (t) as
Using this notation, we obtain the error system x˙3 = a(y3 − x3 ) + µ1 (t), y˙3 = (c − a)x3 + cy3 + x1 z1 − x2 z2 + µ2 (t), z˙ 3 = −bz3 + x2 y2 − x1 y1 + µ3 (t).
µ1 (t) = V1 (t), (14)
µ2 (t) = x2 z2 − x1 z1 + V2 (t), µ3 (t) = x1 y1 − x2 y2 + V3 (t).
(15)
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
Hence x˙3 = a(y3 − x3 ) + V1 (t), y˙3 = (c − a)x3 + cy3 + V2 (t),
(16)
z˙ 3 = −bz3 + V3 (t). The control inputs V1 (t), V2 (t) and V3 (t) are functions of x3 , y3 and z3 and are chosen as ! ! x3 V1 (t) (17) V2 (t) = A y3 , z3 V3 (t) where the matrix A is given by A=
a−1 a−c 0
−a −(1 + c) 0
! 0 . 0 b−1
With this particular choice, the closed loop system has eigenvalues that are found to be −1, −1 and −1. This choice will lead to the error states x3 , y3 and z3 converge to zero as time t tends to infinity and this implies that the synchronization of two Chen systems is achieved. 3.1. Numerical results Fourth-order Runge–Kutta method is used to solve the systems of differential equations (12) and (13) with time step size equal 0.001 in all numerical simulations. The parameters are selected in (12) as follows: a = 35, b = 3, c = 28 to insure the chaotic behaviour of Chen system. The initial values of the drive system are x1 (0) = 0.5, y1 (0) = 1 and z1 (0) = 1 and the initial values of the response system are x2 (0) = 10.5, y2 (0) = 1 and z2 (0) = 38. Then the initial values of the error states are x3 (0) = 10, y3 (0) = 0 and z3 (0) = 37.
197
The results of the simulation of the two identical Chen systems without active control are shown in Fig. 3: (a) displays the x1 and x2 , (b) displays the y1 and y2 , and (c) displays the z1 and z2 . Figs. 4(a)–(c) show the synchronization is occurred after applying active control at t = 5.
4. Conclusion This Letter demonstrates that chaos in both Rossler and Chen systems can be easily controlled using active control techniques.
References [1] T.L. Carroll, L.M. Perora, Phys. Rev. Lett. 64 (1990) 821. [2] T.L. Carroll, L.M. Perora, IEEE Trans. Circuits Syst. 38 (1991) 453. [3] G. Chen, X. Dong, IEEE Trans. Circuits Syst. 40 (1993) 591. [4] G. Chen, Q. Xie, Int. J. Bifurcation Chaos 6 (11) (1996) 2153. [5] G. Chen, Control and Synchronization of Chaos, a Bibliography, Dept. of Elect. Eng., Univ. Houston, TX, 1997. [6] T. Kapitaniak, Chaos Solitons Fractals 6 (3) (1995) 237. [7] K.S. Halle, C.W. Wu, M. Itoh, L.O. Chua, Int. J. Bifurcation Chaos 3 (1993) 469. [8] L. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, U. Parlitz, Int. J. Bifurcation Chaos 3 (1993) 479. [9] K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65. [10] L. Kocarev, U. Parlitz, Phys. Rev. Lett. 74 (1995) 5028. [11] J.H. Peng, E.J. Ding, M. Ding, W. Yang, Phys. Rev. Lett. 76 (1996) 904. [12] N.F. Rulkov et al., Phys. Rev. E 51 (1995) 980. [13] L. Kocarev, U. Parlitz, Phys. Rev. Lett. 76 (1996) 1816. [14] K. Murali, M. Lakshmanan, Phys. Lett. A 241 (1998) 303. [15] E. Bai, K.E. Lonngrn, Chaos Solitons Fractals 10 (9) (1998) 1571. [16] E. Bai, K.E. Lonngrn, Chaos Solitons Fractals 11 (2000) 1041. [17] X. Yu, Chaos Solitons Fractals 8 (9) (1997) 1577. [18] G. Chen, Int. J. Bifurcation Chaos 9 (7) (1999) 1465.
Physics Letters A 278 (2001) 191–197 www.elsevier.nl/locate/pla
Synchronization of Rossler and Chen chaotic dynamical systems using active control H.N. Agiza ∗ , M.T. Yassen Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Received 1 September 2000; accepted 20 November 2000 Communicated by A.R. Bishop
Abstract This Letter presents chaos synchronization of two identical Rossler and Chen systems by using active control. The proposed technique is applied to achieve chaos synchronization for the Rossler and Chen dynamical systems. We demonstrate that a coupled Rossler and Chen systems can be synchronized. Numerical simulations are used to show the effectiveness of the proposed control method. 2001 Elsevier Science B.V. All rights reserved.
1. Introduction In the last few years, synchronization in chaotic dynamical systems has received a great deal of interest among scientists from various fields [1–14]. The idea of synchronizing two identical chaotic systems that from different initial conditions was introduced by Pecora and Carrols [1,2]. Synchronization in chaotic systems has been extensively investigated in the last few years [6,8,9] and many possible applications have been discussed by computer simulation and realized in laboratory condition [9,10]. More recently, synchronization of hyperchaotic systems was investigated [10,11] and the generalized synchronization was proposed [12–14], which makes secret communication more practicable. Recently, a coupled of Lorentz system is synchronized by using active control [15,16]. The aim of this * Corresponding author.
E-mail address: [email protected]. (H.N. Agiza).
Letter is to apply active control to synchronize both Rossler and Chen dynamical systems. This Letter is organized as follows: In Section 2 we apply active control to Rossler dynamical system and numerical simulation are used to show this process. In Section 3 active control is applied to Chen chaotic attractor and numerical experiments are performed to show such synchronization. Consider the system of differential equations x˙ = f (x),
(1)
y˙ = g(y, x),
(2)
where x ∈ R n , y ∈ R n , f, g : R n → R n are assumed to be analytic functions. Let x(t, x0 ) and y(t, y0 ) be solutions to (1) and (2), respectively. The solutions x(t, x0 ) and y(t, y0) are said to be synchronized if
lim x(t, x0 ) − y(t, y0) = 0.
t →∞
0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 7 7 7 - 5
(3)
192
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
2. Synchronization of two Rossler systems using active control
We define the active control functions u1 (t), u2 (t) and u3 (t) as follows: u1 (t) = V1 (t),
The Rossler system [17] is described by the set of ordinary differential equations
u2 (t) = V2 (t),
x˙ = −y − z,
u3 (t) = x1 z1 − x2 z2 + V3 (t).
y˙ = x + ay,
(4)
This implies
z˙ = b + z(x − c).
x˙ 3 = −y3 − z3 + V1 (t),
The parameters a, b and c are taken in a range to insure the chaotic behaviour of (4). We assume that we have two Rossler systems and that the drive system with the subscript 1 is to control the response system with subscript 2. The drive and response systems are defined as follows:
y˙3 = x3 + ay3 + V2 (t),
x˙1 = −y1 − z1 , y˙1 = x1 + ay1,
(5)
z˙ 1 = b + z1 (x1 − c) and x˙2 = −y2 − z2 + u1 (t), y˙2 = x2 + ay2 + u2 (t),
(6)
z˙ 2 = b + z2 (x2 − c) + u3 (t). We have introduced three control functions u1 (t), u2 (t) and u3 (t) in (6). Our goal is to determine these functions u1 (t), u2 (t) and u3 (t). In order to estimate the control functions, we subtract (5) from (6). We define the error system as the differences between the Rossler systems (5) and (6) that is to be controlled and the controlling system using x3 = x2 − x1 ,
(9)
(10)
z˙ 3 = −cz3 + V3 (t). The error system (10) to be controlled is a linear system with a control input V1 (t), V2 (t) and V3 (t) as function of x3 , y3 and z3 . As long as these feedbacks stabilize the system, x3 , y3 and z3 converge to zero as time t tends to infinity. This implies that two Rossler systems are synchronized with feedback control. There are many possible choices for the control V1 (t), V2 (t) and V3 (t). We choose ! ! x3 V1 (t) V2 (t) = A y3 , z3 V3 (t) where A is a 3 × 3 constant matrix. In order to make the closed loop system will be stable, the proper choice of the elements of the matrix A is such that the feedback system must have all of the eigenvalues with negative real parts. Let the matrix A is chosen in the form ! −1 1 1 A = −1 −(1 + a) 0 . 0 0 c−1
Using this notation, we obtain
In this particular choice, the closed loop system (10) has the eigenvalues −1, −1 and −1. This choice will lead to the error states x3 , y3 and z3 converge to zero as time t tends to infinity and hence the synchronization of two Rossler systems is achieved.
x˙3 = −y3 − z3 + u1 (t),
2.1. Numerical results
y3 = y2 − y1 ,
(7)
z3 = z2 − z1 .
y˙3 = x3 + ay3 + u2 (t),
(8)
z˙ 3 = x2 z2 − x1 z1 − cz3 + u3 (t). System (8) is called the error system and x3 , y3 and z3 are called the error states. The synchronization problem for Rossler system is to achieve the asymptotic stability of the zero solution of the error system (8).
Fourth-order Runge–Kutta method is used to solve the systems of differential equations (5) and (6) with time step size equal 0.001 in all numerical simulations. We select numerical values for the parameters in (4) as a = 0.2, b = 0.2, c = 5.7 to insure the chaotic behaviour of Rossler system. The initial values of the
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
193
(b)
(c) Fig. 1. Solution of the coupled Rossler system of equations with the active control deactivated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
drive system are x1 (0) = 0.5, y1 (0) = 1 and z1 (0) = 1.5 and the initial values of the response system are x2 (0) = 2.5, y2 (0) = 2 and z2 (0) = 2.5. Then the initial values for the error states are x3 (0) = 2, y3 (0) = 1 and z3 (0) = 1. The results of the simulation of the two identical Rossler systems without active control are shown
in Fig. 1: (a) displays the x1 and x2 , (b) displays the y1 and y2 , and (c) displays the z1 and z2 . Figs. 2(a)–(c) display the same sequence of signals with the active control. We observe in numerical simulations that the synchronization of derive and response of two identical Rossler systems occurred after t = 10.
194
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
(b)
(c) Fig. 2. Solution of the coupled Rossler system of equations with the active control activated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
z˙ = xy − bz.
3. Synchronization of two Chen systems using active control Recently, Chen [18] introduced a new chaotic attractor (Chen attractor). The nonlinear differential equations that describe Chen attractor are x˙ = a(y − x), y˙ = (c − a)x − xz + cy,
(11)
System (11) has chaotic behaviour [18] at the parameters values a = 35, b = 3 and c = 28. Our aim is to make synchronization of system (11) by using active control. For Chen chaotic attractor, the drive system is defined as follows: x˙ 1 = a(y1 − x1 ),
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
195
(b)
(c) Fig. 3. Solution of the coupled Chen system of equations with the active control deactivated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
y˙1 = (c − a)x1 − x1 z1 + cy1 ,
(12)
z˙ 1 = x1 y1 − bz1 , and the response system is given by
x3 = x2 − x1 ,
x˙2 = a(y2 − x2 ) + µ1 (t), y˙2 = (c − a)x2 − x2 z2 + cy2 + µ2 (t), z˙ 2 = x2 y2 − bz2 + µ3 (t).
We have introduced three control functions µ1 (t), µ2 (t) and µ3 (t) in Eq. (13). These functions are to be determined. Let the error states are
(13)
y3 = y2 − y1 , z3 = z2 − z1 .
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H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
(a)
(b)
(c) Fig. 4. Solution of the coupled Chen system of equations with the active control activated. (a) Signals x1 and x2 , (b) signals y1 and y2 , (c) signals z1 and z2 . (x1 , y1 and z1 —; x2 , y2 and z2 – – –).
We define the active control functions µ1 (t), µ2 (t) and µ3 (t) as
Using this notation, we obtain the error system x˙3 = a(y3 − x3 ) + µ1 (t), y˙3 = (c − a)x3 + cy3 + x1 z1 − x2 z2 + µ2 (t), z˙ 3 = −bz3 + x2 y2 − x1 y1 + µ3 (t).
µ1 (t) = V1 (t), (14)
µ2 (t) = x2 z2 − x1 z1 + V2 (t), µ3 (t) = x1 y1 − x2 y2 + V3 (t).
(15)
H.N. Agiza, M.T. Yassen / Physics Letters A 278 (2001) 191–197
Hence x˙3 = a(y3 − x3 ) + V1 (t), y˙3 = (c − a)x3 + cy3 + V2 (t),
(16)
z˙ 3 = −bz3 + V3 (t). The control inputs V1 (t), V2 (t) and V3 (t) are functions of x3 , y3 and z3 and are chosen as ! ! x3 V1 (t) (17) V2 (t) = A y3 , z3 V3 (t) where the matrix A is given by A=
a−1 a−c 0
−a −(1 + c) 0
! 0 . 0 b−1
With this particular choice, the closed loop system has eigenvalues that are found to be −1, −1 and −1. This choice will lead to the error states x3 , y3 and z3 converge to zero as time t tends to infinity and this implies that the synchronization of two Chen systems is achieved. 3.1. Numerical results Fourth-order Runge–Kutta method is used to solve the systems of differential equations (12) and (13) with time step size equal 0.001 in all numerical simulations. The parameters are selected in (12) as follows: a = 35, b = 3, c = 28 to insure the chaotic behaviour of Chen system. The initial values of the drive system are x1 (0) = 0.5, y1 (0) = 1 and z1 (0) = 1 and the initial values of the response system are x2 (0) = 10.5, y2 (0) = 1 and z2 (0) = 38. Then the initial values of the error states are x3 (0) = 10, y3 (0) = 0 and z3 (0) = 37.
197
The results of the simulation of the two identical Chen systems without active control are shown in Fig. 3: (a) displays the x1 and x2 , (b) displays the y1 and y2 , and (c) displays the z1 and z2 . Figs. 4(a)–(c) show the synchronization is occurred after applying active control at t = 5.
4. Conclusion This Letter demonstrates that chaos in both Rossler and Chen systems can be easily controlled using active control techniques.
References [1] T.L. Carroll, L.M. Perora, Phys. Rev. Lett. 64 (1990) 821. [2] T.L. Carroll, L.M. Perora, IEEE Trans. Circuits Syst. 38 (1991) 453. [3] G. Chen, X. Dong, IEEE Trans. Circuits Syst. 40 (1993) 591. [4] G. Chen, Q. Xie, Int. J. Bifurcation Chaos 6 (11) (1996) 2153. [5] G. Chen, Control and Synchronization of Chaos, a Bibliography, Dept. of Elect. Eng., Univ. Houston, TX, 1997. [6] T. Kapitaniak, Chaos Solitons Fractals 6 (3) (1995) 237. [7] K.S. Halle, C.W. Wu, M. Itoh, L.O. Chua, Int. J. Bifurcation Chaos 3 (1993) 469. [8] L. Kocarev, K.S. Halle, K. Eckert, L.O. Chua, U. Parlitz, Int. J. Bifurcation Chaos 3 (1993) 479. [9] K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71 (1993) 65. [10] L. Kocarev, U. Parlitz, Phys. Rev. Lett. 74 (1995) 5028. [11] J.H. Peng, E.J. Ding, M. Ding, W. Yang, Phys. Rev. Lett. 76 (1996) 904. [12] N.F. Rulkov et al., Phys. Rev. E 51 (1995) 980. [13] L. Kocarev, U. Parlitz, Phys. Rev. Lett. 76 (1996) 1816. [14] K. Murali, M. Lakshmanan, Phys. Lett. A 241 (1998) 303. [15] E. Bai, K.E. Lonngrn, Chaos Solitons Fractals 10 (9) (1998) 1571. [16] E. Bai, K.E. Lonngrn, Chaos Solitons Fractals 11 (2000) 1041. [17] X. Yu, Chaos Solitons Fractals 8 (9) (1997) 1577. [18] G. Chen, Int. J. Bifurcation Chaos 9 (7) (1999) 1465.