Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter

Chaos, Solitons and Fractals 34 (2007) 1552–1559 www.elsevier.com/locate/chaos

Adaptive modified projective synchronization of a unified chaotic system with an uncertain parameter 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 28 April 2006

Abstract An adaptive modified projective synchronization (AMPS) is proposed to acquire a general kind of proportional relationship between the drive and response systems. Based on the Lyapunov stability theory, a nonlinear control scheme for the synchronization has been presented. The control performances are verified by numerical simulations. Ó 2006 Elsevier Ltd. All rights reserved.

1. Introduction Chaos synchronization is an important subject both theoretically and practically, for applications requiring oscillations out of chaos, machine and building structural stability analysis, and chaos generators design and so on. Chaos synchronization, first described by Fujisaka and Yamada [1] in 1983, did not received great attention until 1990 [2]. From then on, chaos synchronization has been developed extensively due to its various applications [3–29]. During the last decade, many techniques for handling chaos synchronization have been developed, such as PC method [2], OGY method [4,7], feedback approach [13,14,26], adaptive method [9–12,26,27], time-delay feedback approach [18], nonlinear control approach [8,15,25], backstepping design technique [19,22], etc. Most of research efforts mentioned above have concentrated on studying complete synchronization (CS), identical synchronization, or conventional synchronization, where two coupled chaotic systems exhibit identical oscillations. However, in the practical applications, CS only occurs at a certain point in the parameter space, and it is difficult to achieve CS except under ideal conditions. Recently, thus a more general form of synchronization scheme, called generalized synchronization (GS) has been extensively investigated [30–33], where the drive and response systems could be synchronized up to a scaling factor a. It suggests that one can achieve control of this synchronization in general classes of chaotic systems including non-partially-linear systems. More recently, Li [33] consider a new GS method, called modified projective synchronization (MPS), where the responses of the synchronized dynamical states synchronize up to a constant scaling matrix. In this paper, we consider the MPS problem of a unified chaotic system with a uncertain parameter. When the chaotic systems have some uncertain parameters, it is generally difficult to control the system. In this case, it is well-known *

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0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2006.04.047

J.H. Park / Chaos, Solitons and Fractals 34 (2007) 1552–1559

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that the adaptive control scheme is an effective method for the synchronization. Thus, we propose an adaptive modified projective synchronization (AMPS) method for uncertain unified chaotic systems. In this paper, AMPS of the system is proved by the Lyapunov stability theory. Now, consider the unified chaotic system [27] described by 8 > < x_ ¼ ð25b þ 10Þðy  xÞ; y_ ¼ ð28  35bÞx  xz þ ð29b  1Þy; ð1Þ > : z_ ¼ xy  8þb z; 3 where x, y, z are status variables and b is the unknown uncertain parameter. The system is chaotic for any b 2 [0, 1]. System (1) is called the general Lorenz, Lu¨, and Chen system, when b 2 [0, 0.8), when b = 0.8, and when b 2 (0.8, 1], respectively [27]. For instance, when b = 0 and an initial condition [x(0), y(0), z(0)]T = [5, 5, 5]T, the chaotic behavior of the system is given in Figs. 1 and 2. The organization of this paper is as follows. In Section 2, the problem statement and master-slave synchronization scheme are presented for a unified chaotic system. Also, we provide a numerical example to demonstrate the effectiveness of the proposed method. Finally concluding remarks are given in Section 3.

2. AMPS of a unified chaotic system The adaptive modified projective synchronization (AMPS) means that the state vectors of drive system with uncertain parameters and response systems with estimate parameters synchronize up to some scaling factor ai, that is, the state vectors of the systems become proportional. For AMPS of the unified chaotic systems (1), the drive (or master) and response (or slave) systems are defined below, respectively,

50 x(t) y(t) z(t) 40

30

20

10

0

–10

–20

–30

0

2

4

6

8

10

12

14

Fig. 1. Chaotic behavior of the Lorenz system.

16

18

20

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Fig. 2. The Lorenz chaotic attractor.

and

8 > < x_ m ¼ ð25b þ 10Þðy m  xm Þ; y_ m ¼ ð28  35bÞxm  xm zm þ ð29b  1Þy m ; > : z_ m ¼ xm y m  8þb zm ; 3

ð2Þ

8 > < x_ s ¼ ð25b1 þ 10Þðy s  xs Þ þ u1 ; y_ s ¼ ð28  35b1 Þxs  xs zs þ ð29b1  1Þy s þ u2 ; > : 1 z_ s ¼ xs y s  8þb zs þ u3 ; 3

ð3Þ

where the lower scripts m and s stand for the drive (or master) systems, the response (or slave) one, respectively, the parameter b of drive systems is uncertain, b1 is a parameter of the response system which estimates the parameter b, and u1, u2 and u3 are the nonlinear controllers such that two chaotic systems can be synchronized in the sense of MPS, i.e., 8 lim kxm  a1 xs k ¼ 0; > > > < t!1 lim ky m  a2 y s k ¼ 0; ð4Þ t!1 > > > : lim kz  a z k ¼ 0: t!1

m

3 s

In order to estimate the control input ui, i = 1, 2, 3, we subtract (3) from (2) e_ 1 ðtÞ ¼ ð25b þ 10Þðy m  xm Þ  a1 ð25b1 þ 10Þðy s  xs Þ  a1 u1 ; e_ 2 ðtÞ ¼ ð28  35bÞxm  a2 ð28  35b1 Þxs þ ð29b  1Þy m  a2 ð29b1  1Þy s  xm zm þ a2 xs zs  a2 u2 ; 8þb 8 þ b1 zm þ a3 e_ 3 ðtÞ ¼  zs þ xm y m  a3 xs y s  a3 u3 3 3 by defining state errors e1(t) = xm(t)a1xs(t), e2(t) = ym(t)a2ys(t), e3(t) = zm(t)a3zs(t).

ð5Þ

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The objective is to find a control law ui(i = 1, 2, 3) for stabilizing the error variables of system (5) at the origin. For this end, we propose the following control law and update algorithm for system (3): u1 ¼

1 b a2 1 ð38  10b1  a3 zs Þe2  25 1 e1 þ y s e3 þ ðð10 þ 25b1 Þða2  a1 ÞÞy s ; a1 a1 a1 a1

u2 ¼

29 ða2  a1 a3 Þ ða2  a1 Þ b1 e2 þ x s zs þ ð35b1  28Þxs ; a2 a2 a2

u3 ¼ 

ð6Þ

b1 1 e3 þ ða1 a2  a3 Þxs y s ; 3a3 a3

and 1 b_ 1 ¼ 25ðy m  xm Þe1 þ ð29y m  35xm Þe2  zm e3 : 3

ð7Þ

Then, we have the following theorem. Theorem. For given nonzero scalars ai (i = 1, 2, 3), AMPS between the two systems (2) and (3) will occur by the control input (6) and update law (7). Proof. Define a Lyapunov candidate 1 V ¼ ðe21 þ e22 þ e23 þ e2a Þ; 2

ð8Þ

where ea = b1  b.

e1 (t)

10 0

–10 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5 t (sec)

6

7

8

9

10

e2 (t)

10 0

–10

25

e3 (t)

20 15 10 5 0

Fig. 3. Synchronization errors.

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J.H. Park / Chaos, Solitons and Fractals 34 (2007) 1552–1559

The differential of the Lyapunov function along the trajectory of error system (5) is dV ¼ e_ 1 e1 þ e_ 2 e2 þ e_ 3 e3 þ e_ a ea dt ¼ e1 ½ð25b þ 10Þðy m  xm Þ  a1 ð25b1 þ 10Þðy s  xs Þ  a1 u1  þ e2 ½ð28  35bÞxm  a2 ð28  35b1 Þxs þ ð29b  1Þy m  a2 ð29b1  1Þy s   8þb 8 þ b1 zs þ xm y m  a3 xs y s  a3 u3 þ b_ 1 ðb1  bÞ:  xm zm þ a2 xs zs  a2 u2  þ e3  zm þ a3 3 3

ð9Þ

Substituting the update rule (7) into (9) gives that dV ¼ 10e21  25b1 e21 þ 10e1 e2 þ 10ða2  a1 Þy s e1 þ 25b1 e1 e2 þ 25b1 ða2  a1 Þy s e1  a1 e1 u1  e22 dt þ 29b1 e22  35b1 e1 e2 þ 35b1 ða2  a1 Þxs e2 þ 28e1 e2  28ða2  a1 Þxs e2 8 b þ ðxm zm þ a2 xs zs Þe2  a2 u2 e2  e23  1 e23 þ ðxm y m  a3 xs y s Þe3  a3 e3 u3 ; 3 3 ¼ ð10 þ 25b1 Þe21 þ ð38  10b1 Þe1 e2 þ ðð10 þ 25b1 Þða2  a1 ÞÞy s e1  a1 e1 u1  ð1  29b1 Þe22 1 þ ðð35b1  28Þða2  a1 ÞÞxs e2 þ ðxm zm þ a2 xs zs Þe2  a2 u2 e2  ð8 þ b1 Þe23 þ ðxm y m  a3 xs y s Þe3  a3 e3 u3 3 ¼ ð10 þ 25b1 Þe21 þ ð38  10b1  a3 z3 Þe1 e2 þ ðð10 þ 25b1 Þða2  a1 ÞÞy s e1 þ a2 y s e3 e1  a1 e1 u1  ð1  29b1 Þe22 1 þ ðð35b1  28Þða2  a1 ÞÞxs e2 þ ða2  a1 a3 Þxs zs e2  a2 u2 e2  ð8 þ b1 Þe23 þ ða1 a2  a3 Þxs y s e3  a3 e3 u3 : 3 ð10Þ Utilizing the control input (6), we have dV ¼ eT Pe; dt

ð11Þ

15

10

β1

5

0

–5

–10

–15

0

1

2

3

4

5

6

Fig. 4. Estimated parameter b1(t).

7

8

9

10

J.H. Park / Chaos, Solitons and Fractals 34 (2007) 1552–1559

where

2

3 e1 6 7 e ¼ 4 e2 5; e3

1557

2

3 10 0 0 60 1 07 P ¼4 5: 0 0 83

Since V_ is negative semidefinite, we cannot immediately obtain that the origin of error system (5) is asymptotically stable. In fact, as V_ 6 0, then e1 ; e2 ; e3 ; ea 2 L1 . From the error system (5), we have e_ 1 ; e_ 2 ; e_ 3 2 L1 . Since V_ ¼ eT Pe and P is a positive-definite matrix, then we have Z t Z t Z t kmin ðP Þkek2 dt 6 eT Pedt 6 V_ dt ¼ V ð0Þ  V ðtÞ 6 V ð0Þ; 0

0

0

where kmin(P) is the minimum eigenvalue of positive-definite matrix P. Thus e1 ; e2 ; e3 2 L2 . According to the Barbalat’s lemma, we have e1(t), e2(t), e3(t) ! 0 as t ! 1. Therefore, the response system (3) synchronize the drive system (2) in the sense of MPS (4) by the controller (6). This completes the proof. h Example. Now, we give numerical experiments to illustrate the effectiveness of the proposed approach. In the numerical simulations, fourth-order Runge–Kutta method is used to solve the systems with a time step size of 0.001. Case I: Lorenz system. When b = 0, systems (2) and (3) are Lorenz’s system. For this numerical simulation, we assume that the initial condition, (xm(0), y m(0), zm(0)) = (10, 2.5, 5), and (xs(0), ys(0), zs(0)) = (10, 5.5, 9) are employed. Let a1 = 1, a2 = 2, a3 = 3. Synchronization of systems (2) and (3) via adaptive control law (6) and (7) with the initial estimated parameter b1(0) = 0.5 are shown in Figs. 3 and 4. Fig. 3 shows, synchronization errors of systems (2) and (3). Fig. 4 illustrates that the estimate b1(t) of the parameter b converges to b = 0 as t ! 1.

e1 (t)

5 0 –5 0

1

2

3

4

5

6

7

8

9

10

0

1

2

3

4

5

6

7

8

9

10

1

2

3

4

5 t (sec)

6

7

8

9

10

e2 (t)

5 0 –5

e3 (t)

0 –5

–10 0

Fig. 5. Synchronization errors.

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J.H. Park / Chaos, Solitons and Fractals 34 (2007) 1552–1559

8

6

4

β1

2

0

–2

–4

–6

–8 0

1

2

3

4

5

6

7

8

9

10

Fig. 6. Estimated parameter b1(t).

Case II: Lu¨ system. When b = 0.8, Eqs. (2) and (3) are Lu¨’s system. For given datas, (xm(0), ym(0), z m(0)) = (5, 3, 1), (xs(0), ys(0), zs(0)) = (1, 5, 9), b1(0) = 1, and a1 = 3, a2 = 2, a3 = 1, the simulation results are given in Figs. 5 and 6. 3. Concluding remark In this paper, we proposed a new adaptive MPS scheme for a unified chaotic systems with a uncertain parameter. A sufficient condition is attained for the stability of the error dynamics between drive and response systems using the Lyapunov stability theory. Finally, a numerical simulation is provided to show the effectiveness of our method.

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