Generalized projective synchronization of different chaotic systems based on antisymmetric structure

Chaos, Solitons and Fractals 42 (2009) 1190–1196

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Generalized projective synchronization of different chaotic systems based on antisymmetric structure Na Cai *, Yuanwei Jing, Siying Zhang College of Information Science and Engineering, Northeastern University, Shenyang 110004, China

a r t i c l e

i n f o

Article history: Accepted 9 March 2009

a b s t r a c t The problem of generalized projective synchronization of different chaotic systems is investigated. By using the direct design method, a controller is designed to transform the error system into a nonlinear system with the special antisymmetric structure. The sufficient stability conditions are presented for such systems. Then the state of error system is asymptotically stable at the origin. That is to say, the generalized projective synchronization between response system and drive system is realized. Finally, the corresponding numerical simulation results demonstrate the effectiveness of the proposed schemes. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Chaos synchronization has gained a lot of attention among scientists from a variety of research fields in recent years since the pioneering work of Pecora and Carroll [1]. It has become an active research subject in nonlinear science because of its potential applications in secure communication [2], neuron systems [3] and the study of laser dynamics [4]. And most of research efforts have been done about complete synchronization (CS) [5–7], where two coupled chaotic systems exhibit identical oscillations. In practical applications, however, CS only occurs at a certain point in the parameter space, and it is difficult to achieve CS without ideal conditions. So a wide variety of synchronization phenomena besides CS have been reported, such as phase synchronization [8], lag synchronization [9], generalized synchronization [10], anti-synchronization [11], projective synchronization [12], etc. Recently, a new synchronization phenomenon called generalized projective synchronization is observed in chaotic systems [13]. This synchronization has the same topological invariants as those of projective one. In recent years, a little research achievements on the problem have been obtained [13,14]. Linear control method is adopted to realize the generalized projective synchronization in Ref. [13]. Adaptive control method is proposed to achieve the synchronization by Lyapunov stability theory in Ref. [14]. The generalized projective synchronization is an important synchronization phenomenon. Complete synchronization and anti-synchronization are the special cases of generalized projective synchronization. A typical character of the chaotic system is its extreme sensitivity to initial conditions. Slight errors occurring in initial states of two identical oscillators will lead to completely different trajectories after enough transient time. Then the realization of generalized projective synchronization of different chaotic systems is much more difficult. So it is necessary to find an easy method to realize such synchronization. In this paper, an appropriate controller for the response system is designed by direct design method [15,16]. The error system is transformed into a system with the special antisymmetric structure under the controller. Then the states of error system with this special structure are asymptotically stable at the origin according to the stability theorems are proposed. That is, the generalized projective synchronization between response system and drive system is realized. The paper is organized as follows. In Section 2, the generalized projective synchronization of different chaotic systems is theoretically analyzed. Two stability theorems for nonlinear systems with a special antisymmetric structure are given. In Section 3, the proposed synchronization schemes with the direct design method are applied to several chaotic systems such * Corresponding author. E-mail address: [email protected] (N. Cai). 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.015

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as hyper-chaotic Chen system, hyper-chaotic Lorenz system and Chua’s circuit. The simulation results demonstrate the effectiveness of proposed schemes. And finally some concluding remarks are given in Section 4. 2. Generalized projective synchronization of chaotic systems and controllers design Generally speaking, dynamics of chaotic systems are described by a set of nonlinear differential equations with respect to state variables. Moreover, most dynamical equations can be divided into two parts: a linear part and a nonlinear part. Therefore, the drive system and controlled response system are described as follows:

x_ ¼ A1 xðtÞ þ f ðxÞ: y_ ¼ A2 yðtÞ þ gðyÞ þ u;

ð1Þ ð2Þ

where x = [x1,x2,    ,xn]T and y = [y1,y2,    ,yn]T are the state variables. f(x) and g(y) are the nonlinear functions. u is the control input. A1 and A2 are coefficient matrices. If the coefficient matrix A1 – A2 and the function f() – g(), the response system is different from the drive system. Let us define the state errors between the response system and the drive system as e = y  ax, where a is a scaling factor. Then the error system can be described as follows:

e_ ¼ A2 e þ ðA2  A1 Þax þ gðyÞ  af ðxÞ þ u:

ð3Þ

Definition 1 [14]. For two different systems described by Eqs. (1) and (2), we say that they are generalized projective synchronous with respect to the scaling matrix K = aInn (a is a nonzero constant and I is the identity matrix) if there exists a vector controller u such that all trajectories (x(t), y(t)) in (1) and (2) with any initial condition (x(0), y(0)) approach the manifold M = {x(t), y(t):y(t) = ax(t)} as time t goes to infinity, that is,

lim keðtÞk ¼ lim kyðtÞ  axðtÞk ¼ 0:

t!1

ð4Þ

t!1

It implies that the error dynamical system between the response system and drive system is globally asymptotically stable. From the definition of generalized projective synchronization, we know that the synchronization between two different chaotic systems is transformed into a problem to choose a control law u to make error system (3) asymptotically converge to zero. Here the direct design method is used to achieve the objective. The direct design method is proposed for synchronization of chaotic systems in Ref. [15]. It has two advantages: (i) it presents an easy procedure for selecting proper controllers in chaos synchronization; (ii) it constructs simple controllers easily to implement. So we adopt this method to transform the error system into a stable system with a special antisymmetric structure. The concrete form for the structure and the main results are given as follows. Theorem 1. Consider the systems with state dependent coefficients:

x_ ¼ BðxÞx;

ð5Þ T

where x = [x1,x2,    ,xn] is the state variable, B(x) is the coefficient matrix. If B(x) = B1(x) + B2 with diag(b1,b2,    ,bn), bi < 0, (i = 1,2,    ,n), then the system (5) is asymptotically stable.

BT1 ðxÞ

¼ B1 ðxÞ and B2 =

Proof. We choose the Lyapunov function as follows:

V ¼ 0:5xT x:



Then the derivative of V is given by

_ ¼ 0:5ðxT BT ðxÞx þ xT BðxÞxÞ ¼ 0:5½xT ðBT ðxÞ þ BðxÞÞx V_ ¼ 0:5ðx_ T x þ xT xÞ ¼ 0:5½xT ðBT1 ðxÞ þ BT2 þ B1 ðxÞ þ B2 Þx;

ð6Þ

where, BT1 ðxÞ ¼ B1 ðxÞ, BT2 ¼ B2 . So we get

V_ ¼ xT B2 x; where B2 is negative definite. Therefore, V_ is negative definite. According to the Lyapunov asymptotical stability theory, the generalized projective synchronization of the different chaotic systems is achieved. Theorem 2. Consider the system (5) with the coefficient matrix B(x) = B1(x) + B2. If the matrices B1(x) and B2 satisfy the following assumptions: i) BT1 ðxÞ ¼ B1 ðxÞ; ii) B2 = diag(b1,b2,    ,bn), bi 6 0, (i = 1,2,    ,n), and the invariant set of the system x_ ¼ BðxÞx only includes the origin, then the system (5) is asymptotically stable.

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Proof. We choose the Lyapunov function as follows:

V ¼ 0:5xT x:



Then the derivative of V is given by

_ ¼ xT B2 x; V_ ¼ 0:5ðx_ T x þ xT xÞ where B2 is semi-negative definite. So the V_ is semi-negative definite. At the same time, because the invariant set of the system x_ ¼ BðxÞx only includes the origin, the equilibrium x = 0 of the system (5) is asymptotically stable. Here, our purpose is to design a controller for the error system by the direct design method. The system (3) is transformed into the system e˙ = Be under the control law u, where B possesses the antisymmetric structure. Then the error system (3) is asymptotically stable at the origin according to the Theorems 1 and 2. Furthermore, the generalized projective synchronization between the response system (2) and the drive system (1) is realized. According to the aforementioned discussion, we have

A2 e þ ðA2  A1 Þax þ gðyÞ  af ðxÞ þ uðtÞ ¼ Be: Then the controller is

uðtÞ ¼ ðB  A2 Þe  ðA2  A1 Þax  gðyÞ þ af ðxÞ:

ð7Þ

Remark 1. If the response system and drive system are identical systems, we have A1 = A2 = A, f() = g(). Then the controller (7) can be rewritten as follows:

uðtÞ ¼ ðB  AÞe  f ðyÞ þ af ðxÞ:

ð8Þ

The identical chaotic systems can achieve the generalized projective synchronization under the controller (8). Remark 2. The antisymmetric structures in the Theorems 1 and 2 are the generalization of the tridiagonal structures. The antisymmetric structures of this paper degenerate to the tridiagonal structures, if the coefficient matrix is B(x) = (bij)nn, where bij = 0 for ji  jj > 1. Remark 3. The error system constructed with the antisymmetric structure is more convenient than the one with tridiagonal structure, if the original system has some zero elements at the tridiagonal position and nonzero elements at other positions. The selecting of the coefficient matrix with antisymmetric structure is an important and difficult problem, because the antisymmetric structure is related to the coefficient matrix and the states of the original system. In the next section, we will demonstrate the proposed approaches basing on the direct design method for the special structure through two numerical examples.

3. Applications of generalized projective synchronization schemes 3.1. Anti-synchronization between hyper-chaotic Chen system and hyper-chaotic Lorenz system To further illustrate the effectiveness of the proposed schemes in this section, we select the hyper-chaotic Chen system [17] and hyper-chaotic Lorenz system [12] as the drive system and the response system, respectively.

2

a

6 d 6 x_ ¼ 6 4 0 0

a

0

1

2

3

3

0

6 x x 7 07 6 1 37 7 7 7x þ 6 4 x1 x2 5 5 0 b 0 c

0

0

0

ð9Þ

x2 x3

r

and

2

a1

6 b 6 1 y_ ¼ 6 4 0 0

3

2

0

3

2

u1

3

a1

0

1

1

0

0

c1

6 y y 7 6 u 7 07 7 6 1 37 6 27 7y þ 6 7 þ 6 7: 4 y1 y2 5 4 u3 5 05

0

0

d1

y1 y3

ð10Þ

u4

where x = [x1,x2,x3,x4]T and y = [y1,y2,y3,y4]T are the state variables of the two systems, a, b, c, d, r and a1, b1, c1, d1 are real constants. When a = 35, b = 3, c = 12 d = 7 and 0 6 r 6 0.085, the system (9) is chaotic. The chaotic attractor of system (9) is shown in Fig. 1. When a1 = 10, b1 = 28, c1 = 8/3 and d1 = 1.3, the system (10) is a hyper-chaotic system. The chaotic attractor of system (10) is shown in Fig. 2.

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Let us define the state errors as ei = yi  axi, (i = 1,2,3,4). We can get the error system between (9) and (10),

2

a1

0

1

ðax1 þ e1 Þ

ax1 þ e1

c1

a1

6 b  ax 3 6 1 e_ ¼ 6 4 ax2 ax3

1

3

2

2

u1

3

6 ðb  dÞax  ðc þ 1Þax þ ða  a2 Þx x 7 6 u 7 07 1 2 1 37 7 6 1 6 27 7e þ 6 7 þ 6 7; 5 4 5 4 u3 5 0 ða2  aÞx1 x2 þ ðb  c1 Þax3 ðd1  rÞax4  a2 x1 x3  ax2 x3

ðax1 þ e1 Þ d1

0

3

ða  a1 Þðax1  ax2 Þ

ð11Þ

u4

T

where e = [e1,e2,e3,e4] . The error system (11) can be changed into the following form, by designing a proper controller.

e_ ¼ BðeÞe;

ð12Þ

where

2

a1

6 b  ax 3 6 1 BðeÞ ¼ 6 4 ax2

ðb1  ax3 Þ

ax2

1

ðax1 þ e1 Þ

ax1 þ e1

c1

0

ðax1 þ e1 Þ

ax3

3

ax3

7 7 7: ax1 þ e1 5 0

1

According to the system (9) and (10) we can get

2

a

6 d 6 A1 ¼ 6 4 0 0

a

0

1

3

2

07 7 7; 0 b 0 5 0 0 r c

0

a1

6 b 6 1 A2 ¼ 6 4 0 0

3

a1

0

1

1

0

0 0

c1 0

07 7 7; 05

2

0

3

6 x x 7 6 1 37 f ðxÞ ¼ 6 7; 4 x1 x2 5

2

3

6 y y 7 6 1 37 gðyÞ ¼ 6 7: 4 y1 y2 5

x2 x3

d1

0

y1 y3

Based on Eq. (7), we design the controller as follows:

3 3 2 u1 ðtÞ ðax3  a1  b1 Þe2  ax2 e3 þ ðax3  1Þe4 þ aða1  aÞðx1  x2 Þ 6 u ðtÞ 7 6 ax e  ðe þ ax Þe  ðb  dÞax þ ðc þ 1Þax þ y y  ax x 7 3 1 1 1 3 1 1 2 1 37 6 2 7 6 1 3 7: 6 7¼6 4 u3 ðtÞ 5 4 ax2 e1 þ ðe1 þ ax1 Þe2 þ ðe1 þ ax1 Þe4  ðb  c1 Þax3  y1 y2 þ ax1 x2 5 2

u4 ðtÞ

ð13Þ

ax3 e1  ðe1 þ ax1 Þe3  ðd1 þ 1Þe4  ðd1  rÞax4 þ y1 y3 þ ax2 x3

The error system (11) is asymptotically stable at the origin under the controller (13) according to Theorem 1. Then the anti-synchronization between the response system and the drive system is realized. Simulation results are presented to demonstrate and verify the performance of the present design. Parameters of the hyper-chaotic Chen system are a = 35, b = 3, c = 12, d = 7, r = 0.08, and parameters of the hyper-chaotic Lorenz system are a1 = 10, b1 = 28, c1 = 8/3, d1 = 1.3. The initial states of drive system (9) are (x1(0), x2(0), x3(0), x4(0)) = (3,5,7,8) and the initial states of response system (10) are (y1(0), y2(0), y3(0), y4(0)) = (12,2,3,  5). Here we choose the scaling factor a =  1. Then, the generalized projective synchronization is degraded into the anti-synchronization. The anti-synchronized state trajectories of hyper-chaotic systems and error states of controlled system are shown in the Fig. 3. From the simulation results, the anti-synchronization of the response system (10) and drive system (9) is observed when t P 2.5, which verifies the validity of the proposed control technique. 3.2. Generalized projective synchronization of the Chua’s circuit In order to observe the chaos synchronization behavior for the Chua’s circuit using the synchronous scheme in this paper, we assume that the drive system and the controlled response system are all Chua’s circuit.

30 20

x2

10 0 -10 -20 -30 40 30

200

x3

100

20 10 0

-100 -200

0

x4

Fig. 1. The chaotic attractor of hyper-chaotic Chen system.

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N. Cai et al. / Chaos, Solitons and Fractals 42 (2009) 1190–1196 30 20 10

x2

0 -10 -20 -30 60 40

x3

20 0

-100

-200

0

300

200

100

x4

Fig. 2. The chaotic attractor of hyper-chaotic Lorenz system.

They are described as follows:

2

1:43

6 x_ ¼ 4 1 0

3 10x31 6 7 7 1 1 5x þ 4 0 5 16 0 0 10

0

2

3

ð14Þ

and

2

3 2 3 1:43 10 0 10y31 þ u1 6 7 6 7 y_ ¼ 4 1 1 1 5y þ 4 u2 5: 0 16 0 u3

ð15Þ

Let us define the state errors as ei = yi  axi, (i = 1,2,3). We can get the error system between (14) and (15)

2 6 e_ ¼ 4

1:43  10e21  30ax1 e1  30a2 x21

10

1

1

0

16 0

0

2

3

6 7 1 5e þ 4

ð10a  10a3 Þx31 0 0

3

2

u1

3

7 6 7 5 þ 4 u2 5: u3

ð16Þ

The error system (16) can be changed into the following form, by designing a proper controller.

a

b

15 e1 10

15

c

10

e2 e3

5

e4

15 10 5

0

0

-x2,y2

-x1,y1

0 -5 -10

-5 -10

-5 -15

-15

-10

-x 1

-20

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

-25

5

0

0.5

1

1.5

2

time(s)

d

2.5

3

3.5

4

4.5

-x 2

-20

y1

y2 -25

5

0

0.5

1

1.5

2

time(s)

5

e

-x 3 0

y3

-5

2.5

3

3.5

time(s)

100 -x 4 80

y4

60

-10 40 -15

-x4,y4

-15

-x3,y3

e1,e2,e3,e4

5

-20

20 0

-25 -20

-30

-40

-35 -40

0

0.5

1

1.5

2

2.5

time(s)

3

3.5

4

4.5

5

-60

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

time(s)

Fig. 3. The anti-synchronized state trajectories between hyper-chaotic systems and error states of controlled system.

4

4.5

5

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N. Cai et al. / Chaos, Solitons and Fractals 42 (2009) 1190–1196

2 6 e_ ¼ 4

10e21  30a2 x21

1

3

0

7 1 16 5e: 16 0

1 0

ð17Þ

Based on Eq. (8), we design the controller as follows:

2

u1

3

2

6 7 6 4 u2 5 ¼ 4 u3

11e2  ð1:43  30ax1 e1 Þe1  ð10a  10a3 Þx31 15e3

3 7 5:

ð18Þ

0

The system (16) is asymptotically stable at the origin under the controller (18) according to Theorem 2. Simulation results are presented to demonstrate and verify the performance of the presented design method. The initial states of drive system (14) are (x1(0), x2 (0), x3(0)) = (1,2,1) and the initial states of response system (15) are (y1(0), y2(0), y3(0)) = (10,5,5). Here we choose the scaling factor a = 2. The generalized projective synchronized state trajectories of Chua’s circuit system and error states of controlled system are shown in the Fig. 4. We can make out that, from the Fig. 4, the states of error system is unstable before the controller is employed, and the error states asymptotically converge to the origin near the 8 s under the controller. The numerical simulation results demonstrate that the proposed design method is feasible and effective to realize the generalized projective synchronization between the chaotic systems which satisfy Theorem 2. 4. Conclusions In this paper, a special antisymmetric structure is proposed. The stability theorems about nonlinear systems with the antisymmetric structure are presented. And then the direct design method is adopted to realize the generalized projective synchronization between different chaotic systems according to the proposed theorems. Numerical simulation results of the

a

10

10

b

e1

2x 1

e2

y1

8

e3 6

2x1,y!

e1,e2,e3

5

0

4 2 0 -2

-5

0

1

2

3

4

5

6

7

8

9

-4 0

10

1

2

3

4

time(s)

c

8 y2

6

5

2

0

0

-10

-4

-15

3

4

5

time(s)

8

9

10

-5

-2

2

7

15

4

1

6

10

2x3,y3

2x2,y2

d

2x 2

-6 0

5

time(s)

6

7

8

9

10

-20

2x 3 y3

0

1

2

3

4

5

6

7

8

time(s)

Fig. 4. Generalized projective synchronized state trajectories of Chua’s circuit system and error states of controlled system.

9

10

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N. Cai et al. / Chaos, Solitons and Fractals 42 (2009) 1190–1196

synchronization about the different and same chaotic systems, respectively, illustrate the validity of proposed schemes. How to realize the generalized projective synchronization of different order chaotic systems with disturbance and structure uncertainty is our further work. Acknowledgements The authors thank the support from the National Nature Science Foundation of China (Grant No. 60274009). The authors gratefully acknowledge this support. References [1] Pecora LM, Carroll TL. Synchronization in chaotic systems. Phys Rev Lett 1990;64:821–4. [2] Miliou AN, Antoniades IP, Stavrinides SG, Anagnostopoulos AN. Secure communication by chaotic synchronization robustness under noisy conditions. Nonlinear Anal 2007;8:1003–12. [3] Aguirre C, Campos D, Pascual P, Serrano E. Synchronization effects using a piecewise linear map-based spiking-bursting neuron model. Neurocomputing 2006;69:1116–9. [4] Gross N, Kinzel W, Kanter I, Rosenbluh M, Khaykovich L. Synchronization of mutually versus unidirectionally coupled chaotic semiconductor lasers. Optics Commun 2006;267:464–8. [5] Yan JJ, Yang YS, Chiang TY, Chen CY. Robust synchronization of unified chaotic systems via sliding mode control. Chaos Solitons & Fractals 2007;34:947–54. [6] Lü JH, Zhou TS, Zhang SC. Chaos synchronization between linearly coupled chaotic systems. Chaos Solitons & Fractals 2002;14:529–41. [7] Li DM, Lu JA, Wu XQ. Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems. Chaos Solitons & Fractals 2005;23:79–85. [8] Guan SG, Lai CH, Wei GW. Phase synchronization between two essentially different chaotic systems. Phys Rev E 2005;72:016205-1-8. [9] Chen Y, Chen XX, Gu SS. Lag synchronization of structurally nonequivalent chaotic systems with time delays. Nonlinear Anal 2007;66:1929–37. [10] Zhang G, Liu ZR, Ma ZJ. Generalized synchronization of different dimensional chaotic dynamical systems. Chaos Solitons & Fractals 2007;32:773–9. [11] Li GH, Zhou SP. Anti-synchronization in different chaotic systems. Chaos Solitons & Fractals 2007;32:516–20. [12] Jia Q. Projective synchronization of a new hyperchaotic Lorenz system. Phys Lett A 2007;370:40–5. [13] Yan JP, Li CP. Generalized projective synchronization of a unified chaotic system. Chaos Solitons & Fractals 2005;26:1119–24. [14] Li RH, Xu W, Li S. Adaptive generalized projective synchronization in different chaotic systems based on parameter identification. Phys Lett A 2007;367:199–206. [15] Liu B, Zhou YM, Jiang M, Zhang ZK. Synchronizing chaotic systems using control based on tridiagonal structure. Chaos Solitons & Fractals. doi:10.1016/ j.chaos.2007.06.099. [16] Liu B, Zhang ZK. Stability of nonlinear systems with tridiagonal structure and its applications. Acta Automatica Sinica 2007;33:442–5. [17] Yan ZY. Controlling hyperchaos in the new hyperchaotic Chen system. Appl Math Comput 2005;168:1239–50.