- Email: [email protected]
Chaos synchronization between two different chaotic dynamical systems
Chaos, Solitons and Fractals 27 (2006) 549–554 www.elsevier.com/locate/chaos
Chaos synchronization between two different chaotic dynamical systems Ju H. Park
*
Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea Accepted 31 March 2005
Abstract This work presents chaos synchronization between two different chaotic systems by nonlinear control laws. First, synchronization problem between Genesio system and Rossler system has been investigated, and then the similar approach is applied to the synchronization problem between Genesio system and a new chaotic system developed recently in the literature. The control performances are verified by two numerical examples. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction During the last two decades, synchronization in chaotic dynamic systems has received a great deal of interest among scientists from various research fields [2–24] since Pecora and Carroll [1] introduced a method to synchronization two identical chaotic systems with different initial conditions. The idea of synchronization is to use the output of the master system to control the slave system so that the output of the response system follows the output of the master system asymptotically. A wide variety of approaches have been proposed for the synchronization of various chaotic systems which include PC method [1], OGY method [3,20], active control approach [12], adaptive control method [15,17–19], time-delay feedback approach [4], and backstepping design technique [9], etc. However, most of the methods mentioned above synchronize two identical chaotic systems. In fact, in many practical world such as laser array and biological systems, it is hardly the case that every component can be assumed to be identical. As a result, more and more applications of chaos synchronization in secure communications make it much more important to synchronize two different chaotic systems in recent years [23]. In this regard, some works on synchronization of two different chaotic systems have been performed [22,23]. This article considers the synchronization problems of two different chaotic systems. The Genesio system [6,21] is taken as drive system. The Rossler system and a new chaotic system devised by Chen and Lee [24] are selected as response system. In general, it is known that nonlinear control is an effective method for making two different chaotic systems be synchronized. Thus, a class of nonlinear control scheme is applied to solve the synchronization problems
*
Tel.: +82 53 8102491; fax: +82 53 8104629. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.049
550
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
of this work. Then, the stability of error signals between drive and response system is easily proved by linear control theory. This article is organized as follows: In Section 2, the synchronization problem between Genesio system and Rossler system is considered. The result in Section 2 is extended to the problem between Genesio system and a new chaotic system in Section 3. In Section 4, we provide two numerical examples to demonstrate the effectiveness of the proposed method. Finally concluding remark is given in Section 5.
2. Synchronization between Genesio and Rossler systems In order to observe the chaos synchronization behavior in Genesio and Rossler systems, it is assumed that Genesio system drives the Rossler system. Thus, the drive and response systems are as follows: 8 x_ ¼ y; > > < y_ ¼ z; ð1Þ > > : 2 z_ ¼ ax by cz þ x ; and
8 x_ 1 ¼ y 1 z1 þ u1 ; > > < y_ 1 ¼ x1 þ b1 y 1 þ u2 ; > > : z_ 1 ¼ a1 þ z1 ðx1 c1 Þ þ u3 ;
ð2Þ
where a, b, c, a1, b1 and c1 are positive scalars, and u1, u2 and u3 are the control inputs to be designed. The aim of this section is to determine the control laws ui for the global synchronization of two different chaotic systems. The Genesio system is chaotic for the parameters, a = 6, b = 2.92, c = 1.2, and the Rossler system is chaotic for the parameters, a1 = 0.2, b1 = 0.2, c1 = 5.7. For chaotic synchronization of above two different systems, the error dynamical system is described by 8 e_ 1 ¼ y 1 z1 y þ u1 ; > > < e_ 2 ¼ x1 þ b1 y 1 z þ u2 ; ð3Þ > > : 2 e_ 3 ¼ a1 þ z1 ðx1 c1 Þ þ ax þ by þ cz x þ u3 ; where
8 e1 ðtÞ ¼ x1 ðtÞ xðtÞ; > > < e2 ðtÞ ¼ y 1 ðtÞ yðtÞ; > > : e3 ðtÞ ¼ z1 ðtÞ zðtÞ.
ð4Þ
Now we define the control functions u1,u2 and u3 as follows: 8 > < u1 ¼ 2y 1 þ z1 e1 ; u2 ¼ z x1 2b1 y 1 þ b1 y; > : u3 ¼ a1 þ x2 z1 x1 ax by cz1 þ c1 z1 . Hence the error system (3) becomes 8 > < e_ 1 ¼ e1 þ e2 ; e_ 2 ¼ b1 e2 ; > : e_ 3 ¼ ce3 .
ð5Þ
The error system (5) is a linear system of the form, w_ ¼ Aw. Thus by linear control theory, if the system matrix A is Hurwitz, the system is asymptotically stable. Hence the error system (5) with 2 3 1 1 0 6 7 A ¼ 4 0 b1 0 5 ð6Þ 0 0 c
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
551
has all eigenvalues with negative real parts. This guarantees the asymptotic stability of the system (5), which implies that the Rossler system (2) synchronize the Genesio system (1). 3. Synchronization between Genesio and a new chaotic system Recently, on studying anti-control of chaos, Chen and Lee [24] introduced a new chaotic system, which is described by the following nonlinear differential equation, 8 < x_ 1 ¼ y 1 z1 þ a1 x1 ; y_ ¼ x1 z1 b1 y 1 ; ð7Þ : 1 z_ 1 ¼ ð1=3Þx1 y 1 c1 z1 ; where a1, b1 and c1 are positive scalars. The system (7) is chaotic for the parameters, a1 = 5.0, b1 = 10, and c1 = 3.8. In order to observe the chaos synchronization behavior in Genesio and the new systems (7), it is assumed that Genesio system drives the new chaotic system. Thus, the drive and response systems are as follows: 8 < x_ ¼ y; y_ ¼ z; ð8Þ : z_ ¼ ax by cz þ x2 ; and 8 < x_ 1 ¼ y 1 z1 þ a1 x1 þ u1 ; y_ ¼ x1 z1 b1 y 1 þ u2 ; : 1 z_ 1 ¼ ð1=3Þx1 y 1 c1 z1 þ u3 ;
ð9Þ
where u1, u2 and u3 are the control laws to be designed. The aim of this section is to determine the control laws ui for the global synchronization of two different chaotic systems. For chaotic synchronization of two different systems, the error dynamical system is expressed by 8 > < e_ 1 ¼ y 1 z1 þ a1 x1 y þ u1 ; e_ 2 ¼ x1 z1 b1 y 1 z þ u2 ; ð10Þ > : e_ 3 ¼ ð1=3Þx1 y 1 c1 z1 þ ax þ by þ cz x2 þ u3 ; where
8 > < e1 ðtÞ ¼ x1 ðtÞ xðtÞ; e2 ðtÞ ¼ y 1 ðtÞ yðtÞ; > : e3 ðtÞ ¼ z1 ðtÞ zðtÞ.
ð11Þ
Now we define the control functions u1, u2 and u3 as follows: 8 > < u1 ¼ y 1 þ y 1 z1 2a1 x1 þ a1 x; u2 ¼ z x1 z1 þ b1 y; > : u3 ¼ ð1=3Þx1 y 1 þ c1 z þ x2 ax by cz. Hence the error system (10) becomes 8 > < e_ 1 ¼ a1 e1 þ e2 ; e_ 2 ¼ b1 e2 ; > : e_ 3 ¼ c1 e3 . Hence, the error system (12) with 3 2 1 0 a1 7 6 b1 0 5 A¼4 0 0 0 c1
ð12Þ
ð13Þ
has all eigenvalues with negative real parts. Thus, one can conclude that the new chaotic system (9) synchronize the Genesio system (8) by linear control theory mentioned in Section 2.
552
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
4. Numerical examples In this section, to verify and demonstrate the effectiveness of the proposed method, we consider two numerical examples. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. Example 1. Consider the systems given in (1) and (2). The system parameters of two systems in simulation are a = 6, b = 2.92, c = 1.2, a1 = 0.2, b1 = 0.2 and c1 = 5.7 so that the two systems exhibit the chaotic behavior. The initial values of the drive system and response system are taken as (x(0), y(0), z(0)) = (3, 4, 2) and (x1(0), y1(0), z1(0)) = (3, 3, 3),
10 x x1
5 0 -5
0
5
10
15
20 25 Time (sec)
30
35
10
40
y y1
5 0 -5 -10 0
5
10
15
20 25 Time (sec)
30
35
40
20 z z1
10 0 -10 -20
0
5
10
15
20 25 Time (sec)
30
35
40
Fig. 1. State trajectories.
e1 (t)
5 0 -5 -10 0
5
10
15
20 25 Time (sec)
30
35
40
5
10
15
20 25 Time (sec)
30
35
40
5
10
15
20 25 Time (sec)
30
35
40
8 e2 (t)
6 4 2 0
0
e3 (t)
1 0.5 0 -0.5
0
Fig. 2. Synchronization errors.
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
553
respectively. Thus, the initial errors are e1(0) = 6, e2(0) = 7, and e3(0) = 1. The simulation results are illustrated in Figs. 1 and 2. As we expect, one can observe that response system starts to trace drive system and finally becomes the same after t P 20. Example 2. Consider the systems given in (8) and (9). The parameters of two systems in simulation are a = 6, b = 2.92, c = 1.2, a1 = 5.0, b1 = 10 and c1 = 3.8 so that the two systems exhibit the chaotic behavior. The initial conditions of the drive and response system are taken as (x(0), y(0), z(0)) = (3, 5, 5) and (x1(0), y1(0), z1(0)) = (5, 5, 5), respectively. The simulation results are also illustrated in Figs. 3 and 4. From the figures, it can be seen that the synchronization error converge to zero and two different systems are indeed achieving chaos synchronization. 10
x x1
5 0 -5 0
2
4
6
8
10
12
14
16
18
Time (sec) 10
y y1
5 0 -5 -10 0
2
4
6
8 10 Time (sec)
12
14
16
18
20 z z1
10 0 -10 -20
0
2
4
6
8
10
12
14
16
18
Time (sec)
Fig. 3. State trajectories.
e1 (t)
5 0 -5 -10 0
2
4
6
8 10 12 Time (sec)
14
16
18
2
4
6
8 10 Time (sec)
12
14
16
18
2
4
6
8 10 Time (sec)
12
14
16
18
e2 (t)
5 0 -5 -10 0
10 e3 (t)
5 -0 -5 0
Fig. 4. Synchronization errors.
554
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
5. Concluding remark This article demonstrates that chaos in two different systems can be easily controlled using nonlinear control techniques. The stability of the resulting closed-loop error signals is easily proved by linear control theory. Two numerical simulations have been done to show the effectiveness of control scheme proposed.
Acknowledgement The author would like to thank E.K. Park, J.H. Park and J.Y. Park for their enthusiastic guidance and encouragement.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. Chen G, Dong X. From chaos to order. Singapore: World Scientific; 1998. Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Park JH, Kwon OM. A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons & Fractals 2005;23:495–501. Chua LO, Itah M, Kosarev L, Eckert K. Chaos synchronization in ChuaÕs circuits. J Circuit Syst Comput 1993;3:93–108. Chen M, Han Z. Controlling and synchronizing chaotic Genesio system via nonlinear feedback control. Chaos, Solitons & Fractals 2003;17:709–16. Lu¨ J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26. Lu J, Wu X, Lu¨ J. Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 2002;305:365–70. Wu X, Lu J. Parameter identification and backstepping control of uncertain Lu¨ system. Chaos, Solitons & Fractals 2003;18:721–9. Li D, Lu JA, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons & Fractals 2005;23:79–85. Lu¨ J, Zhou T, Zhang S. Chaos synchronization between linearly coupled chaotic systems. Chaos, Solitons & Fractals 2002;14:529–41. Agiza HN, Yassen MT. Synchronization of Rossler and Chen dynamical systems using active control. Phys Lett A 2001;278:191–7. Park JH. Stability criterion for synchronization of linearly coupled unified chaotic systems. Chaos, Solitons & Fractals 2005;23:1319–25. Chen HK. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;23:1245–51. Wang Y, Guan ZH, Wang HO. Feedback an adaptive control for the synchronization of Chen system via a single variable. Phys Lett A 2003;312:34–40. Park JH. On synchronization of unified chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;25:699–704. Lu J, Wu X, Han X, Lu¨ J. Adaptive feedback synchronization of a unified chaotic system. Phys Lett A 2004;329:327–33. Han X, Lu JA, Wu X. Adaptive feedback synchronization of Lu¨ systems. Chaos, Solitons & Fractals 2004;22:221–7. Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of Lu¨ system with uncertain parameters. Chaos, Solitons & Fractals 2004;21:657–67. Park JH. Adaptive synchronization of a unified chaotic systems with an uncertain parameter. Int J Nonlin Sci Numer Simulat 2005;6:201–6. Genesio R, Tesi A. A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 1992;28:531–48. Ho MC, Hung YC. Synchronization of two different systems by using generalized active control. Phys Lett A 2002;301:424–8. Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons & Fractals 2005;23:131–40. Chen HK, Lee CI. Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals 2004;21:957–65.
Chaos synchronization between two different chaotic dynamical systems Ju H. Park
*
Robust Control and Nonlinear Dynamics Laboratory, Department of Electrical Engineering, Yeungnam University, 214-1 Dae-Dong, Kyongsan 712-749, Republic of Korea Accepted 31 March 2005
Abstract This work presents chaos synchronization between two different chaotic systems by nonlinear control laws. First, synchronization problem between Genesio system and Rossler system has been investigated, and then the similar approach is applied to the synchronization problem between Genesio system and a new chaotic system developed recently in the literature. The control performances are verified by two numerical examples. Ó 2005 Elsevier Ltd. All rights reserved.
1. Introduction During the last two decades, synchronization in chaotic dynamic systems has received a great deal of interest among scientists from various research fields [2–24] since Pecora and Carroll [1] introduced a method to synchronization two identical chaotic systems with different initial conditions. The idea of synchronization is to use the output of the master system to control the slave system so that the output of the response system follows the output of the master system asymptotically. A wide variety of approaches have been proposed for the synchronization of various chaotic systems which include PC method [1], OGY method [3,20], active control approach [12], adaptive control method [15,17–19], time-delay feedback approach [4], and backstepping design technique [9], etc. However, most of the methods mentioned above synchronize two identical chaotic systems. In fact, in many practical world such as laser array and biological systems, it is hardly the case that every component can be assumed to be identical. As a result, more and more applications of chaos synchronization in secure communications make it much more important to synchronize two different chaotic systems in recent years [23]. In this regard, some works on synchronization of two different chaotic systems have been performed [22,23]. This article considers the synchronization problems of two different chaotic systems. The Genesio system [6,21] is taken as drive system. The Rossler system and a new chaotic system devised by Chen and Lee [24] are selected as response system. In general, it is known that nonlinear control is an effective method for making two different chaotic systems be synchronized. Thus, a class of nonlinear control scheme is applied to solve the synchronization problems
*
Tel.: +82 53 8102491; fax: +82 53 8104629. E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.03.049
550
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
of this work. Then, the stability of error signals between drive and response system is easily proved by linear control theory. This article is organized as follows: In Section 2, the synchronization problem between Genesio system and Rossler system is considered. The result in Section 2 is extended to the problem between Genesio system and a new chaotic system in Section 3. In Section 4, we provide two numerical examples to demonstrate the effectiveness of the proposed method. Finally concluding remark is given in Section 5.
2. Synchronization between Genesio and Rossler systems In order to observe the chaos synchronization behavior in Genesio and Rossler systems, it is assumed that Genesio system drives the Rossler system. Thus, the drive and response systems are as follows: 8 x_ ¼ y; > > < y_ ¼ z; ð1Þ > > : 2 z_ ¼ ax by cz þ x ; and
8 x_ 1 ¼ y 1 z1 þ u1 ; > > < y_ 1 ¼ x1 þ b1 y 1 þ u2 ; > > : z_ 1 ¼ a1 þ z1 ðx1 c1 Þ þ u3 ;
ð2Þ
where a, b, c, a1, b1 and c1 are positive scalars, and u1, u2 and u3 are the control inputs to be designed. The aim of this section is to determine the control laws ui for the global synchronization of two different chaotic systems. The Genesio system is chaotic for the parameters, a = 6, b = 2.92, c = 1.2, and the Rossler system is chaotic for the parameters, a1 = 0.2, b1 = 0.2, c1 = 5.7. For chaotic synchronization of above two different systems, the error dynamical system is described by 8 e_ 1 ¼ y 1 z1 y þ u1 ; > > < e_ 2 ¼ x1 þ b1 y 1 z þ u2 ; ð3Þ > > : 2 e_ 3 ¼ a1 þ z1 ðx1 c1 Þ þ ax þ by þ cz x þ u3 ; where
8 e1 ðtÞ ¼ x1 ðtÞ xðtÞ; > > < e2 ðtÞ ¼ y 1 ðtÞ yðtÞ; > > : e3 ðtÞ ¼ z1 ðtÞ zðtÞ.
ð4Þ
Now we define the control functions u1,u2 and u3 as follows: 8 > < u1 ¼ 2y 1 þ z1 e1 ; u2 ¼ z x1 2b1 y 1 þ b1 y; > : u3 ¼ a1 þ x2 z1 x1 ax by cz1 þ c1 z1 . Hence the error system (3) becomes 8 > < e_ 1 ¼ e1 þ e2 ; e_ 2 ¼ b1 e2 ; > : e_ 3 ¼ ce3 .
ð5Þ
The error system (5) is a linear system of the form, w_ ¼ Aw. Thus by linear control theory, if the system matrix A is Hurwitz, the system is asymptotically stable. Hence the error system (5) with 2 3 1 1 0 6 7 A ¼ 4 0 b1 0 5 ð6Þ 0 0 c
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
551
has all eigenvalues with negative real parts. This guarantees the asymptotic stability of the system (5), which implies that the Rossler system (2) synchronize the Genesio system (1). 3. Synchronization between Genesio and a new chaotic system Recently, on studying anti-control of chaos, Chen and Lee [24] introduced a new chaotic system, which is described by the following nonlinear differential equation, 8 < x_ 1 ¼ y 1 z1 þ a1 x1 ; y_ ¼ x1 z1 b1 y 1 ; ð7Þ : 1 z_ 1 ¼ ð1=3Þx1 y 1 c1 z1 ; where a1, b1 and c1 are positive scalars. The system (7) is chaotic for the parameters, a1 = 5.0, b1 = 10, and c1 = 3.8. In order to observe the chaos synchronization behavior in Genesio and the new systems (7), it is assumed that Genesio system drives the new chaotic system. Thus, the drive and response systems are as follows: 8 < x_ ¼ y; y_ ¼ z; ð8Þ : z_ ¼ ax by cz þ x2 ; and 8 < x_ 1 ¼ y 1 z1 þ a1 x1 þ u1 ; y_ ¼ x1 z1 b1 y 1 þ u2 ; : 1 z_ 1 ¼ ð1=3Þx1 y 1 c1 z1 þ u3 ;
ð9Þ
where u1, u2 and u3 are the control laws to be designed. The aim of this section is to determine the control laws ui for the global synchronization of two different chaotic systems. For chaotic synchronization of two different systems, the error dynamical system is expressed by 8 > < e_ 1 ¼ y 1 z1 þ a1 x1 y þ u1 ; e_ 2 ¼ x1 z1 b1 y 1 z þ u2 ; ð10Þ > : e_ 3 ¼ ð1=3Þx1 y 1 c1 z1 þ ax þ by þ cz x2 þ u3 ; where
8 > < e1 ðtÞ ¼ x1 ðtÞ xðtÞ; e2 ðtÞ ¼ y 1 ðtÞ yðtÞ; > : e3 ðtÞ ¼ z1 ðtÞ zðtÞ.
ð11Þ
Now we define the control functions u1, u2 and u3 as follows: 8 > < u1 ¼ y 1 þ y 1 z1 2a1 x1 þ a1 x; u2 ¼ z x1 z1 þ b1 y; > : u3 ¼ ð1=3Þx1 y 1 þ c1 z þ x2 ax by cz. Hence the error system (10) becomes 8 > < e_ 1 ¼ a1 e1 þ e2 ; e_ 2 ¼ b1 e2 ; > : e_ 3 ¼ c1 e3 . Hence, the error system (12) with 3 2 1 0 a1 7 6 b1 0 5 A¼4 0 0 0 c1
ð12Þ
ð13Þ
has all eigenvalues with negative real parts. Thus, one can conclude that the new chaotic system (9) synchronize the Genesio system (8) by linear control theory mentioned in Section 2.
552
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
4. Numerical examples In this section, to verify and demonstrate the effectiveness of the proposed method, we consider two numerical examples. In the numerical simulations, the fourth-order Runge–Kutta method is used to solve the systems with time step size 0.001. Example 1. Consider the systems given in (1) and (2). The system parameters of two systems in simulation are a = 6, b = 2.92, c = 1.2, a1 = 0.2, b1 = 0.2 and c1 = 5.7 so that the two systems exhibit the chaotic behavior. The initial values of the drive system and response system are taken as (x(0), y(0), z(0)) = (3, 4, 2) and (x1(0), y1(0), z1(0)) = (3, 3, 3),
10 x x1
5 0 -5
0
5
10
15
20 25 Time (sec)
30
35
10
40
y y1
5 0 -5 -10 0
5
10
15
20 25 Time (sec)
30
35
40
20 z z1
10 0 -10 -20
0
5
10
15
20 25 Time (sec)
30
35
40
Fig. 1. State trajectories.
e1 (t)
5 0 -5 -10 0
5
10
15
20 25 Time (sec)
30
35
40
5
10
15
20 25 Time (sec)
30
35
40
5
10
15
20 25 Time (sec)
30
35
40
8 e2 (t)
6 4 2 0
0
e3 (t)
1 0.5 0 -0.5
0
Fig. 2. Synchronization errors.
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
553
respectively. Thus, the initial errors are e1(0) = 6, e2(0) = 7, and e3(0) = 1. The simulation results are illustrated in Figs. 1 and 2. As we expect, one can observe that response system starts to trace drive system and finally becomes the same after t P 20. Example 2. Consider the systems given in (8) and (9). The parameters of two systems in simulation are a = 6, b = 2.92, c = 1.2, a1 = 5.0, b1 = 10 and c1 = 3.8 so that the two systems exhibit the chaotic behavior. The initial conditions of the drive and response system are taken as (x(0), y(0), z(0)) = (3, 5, 5) and (x1(0), y1(0), z1(0)) = (5, 5, 5), respectively. The simulation results are also illustrated in Figs. 3 and 4. From the figures, it can be seen that the synchronization error converge to zero and two different systems are indeed achieving chaos synchronization. 10
x x1
5 0 -5 0
2
4
6
8
10
12
14
16
18
Time (sec) 10
y y1
5 0 -5 -10 0
2
4
6
8 10 Time (sec)
12
14
16
18
20 z z1
10 0 -10 -20
0
2
4
6
8
10
12
14
16
18
Time (sec)
Fig. 3. State trajectories.
e1 (t)
5 0 -5 -10 0
2
4
6
8 10 12 Time (sec)
14
16
18
2
4
6
8 10 Time (sec)
12
14
16
18
2
4
6
8 10 Time (sec)
12
14
16
18
e2 (t)
5 0 -5 -10 0
10 e3 (t)
5 -0 -5 0
Fig. 4. Synchronization errors.
554
J.H. Park / Chaos, Solitons and Fractals 27 (2006) 549–554
5. Concluding remark This article demonstrates that chaos in two different systems can be easily controlled using nonlinear control techniques. The stability of the resulting closed-loop error signals is easily proved by linear control theory. Two numerical simulations have been done to show the effectiveness of control scheme proposed.
Acknowledgement The author would like to thank E.K. Park, J.H. Park and J.Y. Park for their enthusiastic guidance and encouragement.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. Chen G, Dong X. From chaos to order. Singapore: World Scientific; 1998. Ott E, Grebogi C, Yorke JA. Controlling chaos. Phys Rev Lett 1990;64:1196–9. Park JH, Kwon OM. A novel criterion for delayed feedback control of time-delay chaotic systems. Chaos, Solitons & Fractals 2005;23:495–501. Chua LO, Itah M, Kosarev L, Eckert K. Chaos synchronization in ChuaÕs circuits. J Circuit Syst Comput 1993;3:93–108. Chen M, Han Z. Controlling and synchronizing chaotic Genesio system via nonlinear feedback control. Chaos, Solitons & Fractals 2003;17:709–16. Lu¨ J, Chen G, Cheng D, Celikovsky S. Bridge the gap between the Lorenz system and the Chen system. Int J Bifurcat Chaos 2002;12:2917–26. Lu J, Wu X, Lu¨ J. Synchronization of a unified chaotic system and the application in secure communication. Phys Lett A 2002;305:365–70. Wu X, Lu J. Parameter identification and backstepping control of uncertain Lu¨ system. Chaos, Solitons & Fractals 2003;18:721–9. Li D, Lu JA, Wu X. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos, Solitons & Fractals 2005;23:79–85. Lu¨ J, Zhou T, Zhang S. Chaos synchronization between linearly coupled chaotic systems. Chaos, Solitons & Fractals 2002;14:529–41. Agiza HN, Yassen MT. Synchronization of Rossler and Chen dynamical systems using active control. Phys Lett A 2001;278:191–7. Park JH. Stability criterion for synchronization of linearly coupled unified chaotic systems. Chaos, Solitons & Fractals 2005;23:1319–25. Chen HK. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;23:1245–51. Wang Y, Guan ZH, Wang HO. Feedback an adaptive control for the synchronization of Chen system via a single variable. Phys Lett A 2003;312:34–40. Park JH. On synchronization of unified chaotic systems via nonlinear control. Chaos, Solitons & Fractals 2005;25:699–704. Lu J, Wu X, Han X, Lu¨ J. Adaptive feedback synchronization of a unified chaotic system. Phys Lett A 2004;329:327–33. Han X, Lu JA, Wu X. Adaptive feedback synchronization of Lu¨ systems. Chaos, Solitons & Fractals 2004;22:221–7. Elabbasy EM, Agiza HN, El-Dessoky MM. Adaptive synchronization of Lu¨ system with uncertain parameters. Chaos, Solitons & Fractals 2004;21:657–67. Park JH. Adaptive synchronization of a unified chaotic systems with an uncertain parameter. Int J Nonlin Sci Numer Simulat 2005;6:201–6. Genesio R, Tesi A. A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems. Automatica 1992;28:531–48. Ho MC, Hung YC. Synchronization of two different systems by using generalized active control. Phys Lett A 2002;301:424–8. Yassen MT. Chaos synchronization between two different chaotic systems using active control. Chaos, Solitons & Fractals 2005;23:131–40. Chen HK, Lee CI. Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals 2004;21:957–65.