Transmission Synchronization Control of Multiple Non ...

Transmission Synchronization Control of Multiple Non-identical Coupled Chaotic Systems Xiangyong Chen1,2 , Jinde Cao1 , Jianlong Qiu2⋆ , and Chengdong Yang1,3 1. Department of Mathematics, Southeast University, Nanjing, 210096, China, [email protected],[email protected], 2. School of Sciences, Linyi University, Linyi, 276005, China [email protected] 3. School of informatics, Linyi University, Linyi, 276005, China [email protected]

Abstract. In this paper, we investigate the transmission projective synchronization control problem for multiple, non-identical, coupled chaotic systems. By considering the influence of the occurrence of a fault between a driving system and a responding system, we define our new transmission synchronization scheme. After that, control laws are designed to achieve transmission projective synchronization and a simple stability criteria is obtained for reaching the transmission synchronization among multi-systems. A numerical example is used to verify the effectiveness of the synchronization within a desired scaling factor. Keywords: Multiple coupled chaotic systems, transmission projective synchronization control, stability analysis.

1

Introduction

Increasing interest has been devoted to the study of chaos synchronization[1]. New synchronization schemes for multiple chaotic systems have been reported in the literature, such as combination synchronization [2, 3], hybrid Synchronization[4], targeting engineering synchronization[5], compound synchronization [6], and so on. These new synchronization schemes have advantages over conventional synchronization techniques for secure communication, information science, etc. Thus, the design of more effective synchronization schemes for an array of chaotic systems, has become a problem to be solved urgently. Recently, a new synchronization phenomenon called transmission synchronization[7] has been observed in multiple chaotic systems. With conventional mode, multiple response systems only synchronize to one drive system, however for transmission synchronization, every system is not only a drive system, but a response system as well. Transmission synchronization is completed among multiple systems according to a step by step transmission method. In a real system, ⋆

Corresponding author: Jianlong Qiu.

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X.Y. Chen, J.D. Cao , J.L. Qiu and C. D. Yang

the occurrence of a synchronization fault between two of those systems is inevitable. With previous synchronization models, the occurrence of a fault would disrupt synchronization. In our model, synchronization can be achieved through the remaining systems to overcome the trouble without affecting their synchronization and performance. This increased reliability of achieving and maintaining transmission projective synchronization among multiple chaotic systems in the presence of faults is the motivation of the present study. For these reasons, it is highly desirable to develop the effective controller that realizes the transmission projective synchronization synchronization for multiple chaotic systems. In recent years, the synchronization of an array of chaotic systems has became a hot topic in nonlinear research, especially for multiple coupled chaotic systems. Potential applications include multilateral communications, secret signaling and many other engineering areas. Much effort has been reported in the literature. In [4, 8–10], the authors investigated some synchronization control problems in an array of coupled chaotic systems,such as complete synchronization[8], antisynchronization[9], projective synchronization[10], hybrid synchronization[4]. Yu studied the global synchronization of three coupled chaotic systems with a ring connection [11]. Lv and Liu proposed the synchronization of N different coupled chaotic systems of with ring and chain connection[12]. Tang and Fang studied the synchronization of N-coupled fractional-order chaotic systems of with ring connection in Ref.[13]. Yang studied the synchronization of three identical systems and its application to secure communication with noise perturbations[14]. However, there has been very little effort on the transmission projective synchronization problem of an array of coupled chaotic systems. In [15],Cheng et al. only discussed the transmission projective synchronization problem of identical coupled chaotic systems. In [7], Sun et al. studied the transmission projective synchronization of multiple systems with non-delayed and delayed coupling via impulsive control. Our article extends the work reported in [7, 15], and provides a proper transmission projective synchronization criteria for multiple non-identical coupled chaotic systems into a nonlinear system with a special antisymmetric structure. In addition, we derive sufficient conditions to guarantee that the error systems asymptotically stabilize at the origin. Simulation results show the effectiveness of our presented synchronization control strategy.

2

Preliminaries and Problem Statement

In the section, we formulate a model of multiple coupled chaotic systems described as follows,    x˙ 1 = A1 x1 + g1 (x1 ) + D1 (xN − x1 ),   x˙ 2 = A2 x2 + g2 (x2 ) + D2 (x1 − x2 ), ..  .    x˙ N = AN xN + gn (xN ) + DN (xN −1 − xN ),

(1)

Transmission Synchronization of Multiple Chaotic Systems

3

where x1 , x2 , · · · , xN are the state vectors of the chaotic systems; gi (xi )(i = 1, · · · , N ) is the continuous nonlinear function; A1 , A2 , · · · , AN are constant matrices; Di = diag(di1 , · · · , diN ), and dij ≥ 0 are the diagonal matrices which represent the coupled parameters. If the coefficient matrices Ai 6= Aj (i, j = 1, · · · N, i 6= j) and the functions gi (·) 6= gj (·) then the system (1) is an array of non-identical chaotic systems. Now the above simple coupling form is applied to investigate the transmission projective synchronization among multiple systems. The control terms are of the following form:    x˙ 1 = A1 x1 + g1 (x1 ) + D1 (xN − x1 ) + u1   x˙ 2 = A2 x2 + g2 (x2 ) + D2 (x1 − x2 ) + u2 , (2) ..  .    x˙ N = AN xN + gn (xN ) + DN (xN −1 − xN ) + uN .

Before showing the main results of this paper, we first define transmission projective synchronization in an array of non-identical coupled chaotic systems. Definition 1. For N non-identical coupled chaotic systems as described by (2), we say that they are in transmission projective synchronization if there exist controllersu1(t), · · · , uN (t) such that all trajectories x1 (t), · · · , xN (t) in (2) with any initial condition (x1 (0), · · · , xN (0)) satisfy the following conditions. lim kei (t)k = lim kxi+1 (t) − λi xi (t)k = 0, i = 1, · · · , N − 1,

(3)

lim keN (t)k = lim kx1 (t) − λN xN (t)k = 0,

(4)

t→∞

t→∞

t→∞

t→∞

and the desired scaling factor λi (i = 1, · · · , N ) satisfy λ1 λ2 · · · λN = 1.

(5)

Next, we introduce a lemma which is needed in the proof of the main theorem. Lemma 1 [16, 17]. Consider the systems with the state dependent coefficient: y˙ = L(y)y,

(6)

where y = [y1 , · · · , yn ]T is the state variable, L(y) is the coefficient matrix. If L(y) = L1 (y) + L2 with LT1 (y) = −L1 (y) and L2 = diag(l1 , · · · , ln ), li < 0, (i = 1, · · · , n), then the system (6) is asymptotically stable.

3

Transmission Synchronization of Multiple Non-identical Coupled Chaotic Systems

In this section, our objective is to design the appropriate controllers ui (t),(i = 1, · · · , N ) such that the state errors e1 (t), · · · , eN (t) convergence to 0 as time t approaches to infinity, which implies the transmission projective synchronization of (2) is attained starting with arbitrary initial conditions.

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X.Y. Chen, J.D. Cao , J.L. Qiu and C. D. Yang

Firstly, the following errors dynamic systems are obtained as,      (λ1 − λ1 /λN )D1 + (1 − λ1 )D2 + x1 +     +λ1 (A2 − A1 )     +g (x ) − λ g (x ) + u − λ u 1 1 1 2 1 1   2 2        (λ − λ /λ )D − (λ − 1)D + 2 2 1 2 2 3   e1 x2 +     +λ2 (A3 − A2 )   e2       +g (x ) − λ g (x ) + u − λ u 3 3 2 2 2 3 2 2   e3       . .. e˙ = Γ  e4  +   (7)         ..   (λN −1 − λN −1 /λN −2 )DN −1 −   .     −(λN −1 − 1)DN + λN −1 (AN − AN −1 ) xN −1 +     eN  +gN (xN ) − λN −1 gN −1 (xN −1 ) + uN − λN −1 uN −1         λN (A1 − AN ) + (1 − λN )D1 +   x + N    +(λN − λN /λN −1 )DN λN  +g1 (x1 ) − λN gN (xN ) + u1 − λN uN where 

A2 − D2 − λ1N D1 − λN1λ2 D1 − λN λ12 λ3 D1  (λ2 /λ1 ) D2 A3 − D3 0   0 (λ3 /λ2 ) D3 A4 − D4 Γ =  .. .. ..  . . . 0 0 0

··· ··· ...

−λ1 D1 0 0 .. .



0 0 0

··· 0 · · · (λN /λN −1 ) DN A1 − D1

     

And then, we choose the control inputs ui to eliminate all known items that cannot be shown in the form of the error system ei . The controller ui can be given by    (λ1 − λ1 /λN )D1 +   u2 = v1 − x1 − g2 (x2 ) + λ1 g1 (x1 ) + λ1 u1      +(1 − λ1 )D2 + λ1 (A2 − A1 )    (λ2 − λ2 /λ1 )D2 −   u3 = v2 − x − g3 (x3 ) + λ2 g2 (x2 ) + λ2 u2   −(λ2 − 1)D3 + λ2 (A3 − A2 ) 2     .. .     (λ − λ /λ )DN −1 − (λN −1 − 1)DN +  N −1 N −1 N −2  x −  N −1   +λN −1 (AN − AN −1 ) uN = vN −1 −      −gN (xN ) + λN −1 gN −1 (xN −1 ) − λN −1 uN −1       [λ (A − A  N 1 N ) + (1 − λN )D1 + (λN − λN /λN −1 )DN λN ] xN +  u = v −  1 N −g1 (x1 ) + λN gN (xN ) − λN uN (8)  T  T where v1 v2 v3 · · · vN = H e1 e2 e3 · · · eN , H is a coefficient matrix. Then the error systems (7) with the controllers ui can be rewritten by e˙ = L(e)e, where L(e) = Γ + H.

(9)

Transmission Synchronization of Multiple Chaotic Systems

5

According to Lemma 1, we should choose the proper H to transform the error systems (9) into a nonlinear stable system, which conforms to the structure of system (6). The concrete form for the structure and the main results are given as follows. Theorem 1. Consider the error dynamic system (9) with the state dependent coefficient L(e) = L1 (e) + L2 , if L1 (e) and L2 satisfy the following assumptions: LT1 (e) = −L1 (e), ; L2 = diag(l1 , · · · , ln ), li < 0, (i = 1, · · · , n) then the system (9) is asymptotically stable, which means that N coupled chaotic systems (7) achieves the transmission projective synchronization. Proof: Choose the Lyapunov function to be V =

1 T e e. 2

The derivative of V is  1 1  V˙ = (e˙ T e + eT e) ˙ = eT L(e)T + L(e) e, 2 2 where LT1 (e) = −L1 (e) and L2 = diag(l1 , · · · , ln ), li < 0, (i = 1, · · · , n). Then we get that V˙ = eT L2 e < 0. From Lyapunov stability theory, the equilibrium x = 0 of the system (9) is global asymptotically stable. Then the transmission projective synchronization of N chaotic systems (7) is achieved. According to Theorem 1, we can design the controllers by choosing the proper coefficient matrix H, which can guarantee L(e) to be a special antisymmetric structure. We can find that there are many possible choices for H as long as it guarantees the error dynamic system (9) to be a stable system with a special antisymmetric structure. However, the selection of the coefficient matrix H is an important and difficult problem, because antisymmetric structure is related to the coefficient matrix H and the states of the original system. In the next section, we will demonstrate the proposed approaches for the special structure through a numerical examples.

4

Numerical Example and Simulation

In order to observe the chaos synchronization behavior for an array of nonidentical coupled chaotic systems using the synchronous scheme in this paper, we consider the Chen system, L¨ u system and Lorenz system as drive system and response systems.   x˙ 11 = −35x11 + 35x12 + d11 (x31 − x11 ) + u11 , x˙ 12 = −7x11 + 28x12 − x11 x13 + d12 (x32 − x12 ) + u12 , (10)  x˙ 13 = −3x13 + x11 x12 + d13 (x33 − x13 ) + u13 .

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X.Y. Chen, J.D. Cao , J.L. Qiu and C. D. Yang

  x˙ 21 = −36x21 + 36x22 + d21 (x11 − x21 ) + u21 , x˙ 22 = 20x22 − x21 x23 + d22 (x12 − x22 ) + u22 ,  x˙ 23 = −3x23 + x21 x22 + d23 (x13 − x23 ) + u23 .

  x˙ 31 = −10x31 + 10x32 + d31 (x21 − x31 ) + u31 , x˙ 32 = 28x31 − x32 − x31 x33 + d32 (x22 − x32 ) + u32 ,  x˙ 33 = − 83 x33 + x31 x32 + d33 (x23 − x33 ) + u33 .

where

(11)

(12)



     −35 35 0 −36 36 0 −10 10 0 A1 =  −7 28 0  , A2 =  0 20 0  , A3 =  28 −1 0  , 0 0 −3  0  0 −3  0 0 − 83  0 0 0 g1 (x1 ) =  −x11 x13  , g2 (x2 ) =  −x21 x23  , g3 (x3 ) =  −x31 x33  x11 x12 x21 x22 x31 x32 and D1 = diag(d11 , d12 , d13 ),D2 = diag(d21 , d22 , d23 ) and D3 = diag(d31 , d32 , d33 ) are the coupled matrices, u1 = [u11 , u12 , u13 ]T and u2 = [u21 , u22 , u23 ]T are the control inputs. Here we choose the scaling factor λ1 = λ2 = 3 and λ3 = 1/9. Then, the synchronization error state be e˙ i = x˙ i+1 − 3x˙ i , (i = 1, 2), and e˙ 3 = x˙ 1 − 1/9x˙ 3 , the errors dynamic systems are written as, 

  A2 − D2 − 9D1 −3D1 0 e1   e2  + D2 A3 − D3 0 e˙ =  0 (1/27) D3 A1 − D1 e3   (−24D1 −2D2 + 3(A2 − A1 )) x1 + g2 (x2 ) − 3g1 (x1 ) + u2 − 3u1  [2D2 − 2D3 + 3(A3 − A2 )] x2 + g3 (x3 ) − 3g2 (x2 ) + u3 − 3u2     +   [(1/9) (A1 − A3 ) + (8/9) D1 + (2/243) D3 ] x3 + +g1 (x1 ) − (1/9) g3 (x3 ) + u1 − (1/9) u3

(13)

We consider that there exists a fault between the system (10) and (12), then it is easy to get that u1 = 0, and we can write the controllers as follows,   u2 − 3u1 =v1 + (24D1 +2D2 − 3(A2 − A1 )) x1 − g2 (x2 ) + 3g1 (x1 ) u3 − 3u2 = v2 − [2D2 − 2D3 + 3(A3 − A2 )]
x2 − g3 (x3 ) + 3g2 (x2 ) (14)  2 D3 x3 − g1 (x1 ) + 19 g3 (x3 ) u1 − 19 u3 =v3 − 19 (A1 − A3 ) + 89 D1 + 243 Design v1 , v2 and v3 to be     1 0 00000000 0 0 0 − 27 d31 0 0 0 00 1 v1 =  −36 0 0 0 0 0 0 0 0  e, v3 =  0 0 0 0 − 27 d32 0 −28 0 0  e. 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − d 27 33 0 0 0   3d11 − d21 0 0 0 00000 0 3d12 − d22 0 −38 0 0 0 0 0  e, v2 =  0 0 3d13 − d23 0 0 0 0 0 0 From Theorem 1, we get the conditions −36 − d21 − 9d11 < 0, 20 − d22 − 9d12 < 0, −3 − d23 − 9d13 < 0, −10 − d31 < 0, −1 − d32 < 0, − 38 − d33 < 0, −35 − d11 < 0, 28 − d12 < 0, −3 − d13 < 0

Transmission Synchronization of Multiple Chaotic Systems

7

and then, let the initial conditions of the drive system and the response systems be (x11 (0), x12 (0), x13 (0)) = (4, 5, −3), (x21 (0), x22 (0), x23 (0)) = (5, 2, −5) and (x31 (0), x32 (0), x33 (0)) = (11, 15, 10) respectively. The initial value of the error states are (e11 (0), e12 (0), e13 (0)) = (−7, −13, 4), (e21 (0), e22 (0), e23 (0)) = (−4, 9, 25) and (e31 (0), e32 (0), e33 (0)) = (25/9, 10/3, −37/9). In order to reduce the control cost, we choose that d12 = 30, d11 = d22 = d32 = d13 = d23 = d33 = 0, d21 = 2 and d31 = 2. The error state trajectories of the error dynamic systems are shown in Fig.1. Fig.1 show that the error state trajectories have asymptotically converged to zero under the controllers (13). This implies that the transmission projective synchronization is realized.

30

25 e11 e12

20

e13

e11,e12,e13,e21,e22,e23,e31,e32,e33

e21 e22

15

e

23

e31 10

e32 e33

5

0

−5

−10

−15

0

0.5

1

1.5 Time (s)

2

2.5

3

Fig. 1. Time behavior for the error state variables e11 ,e12 ,e13 ,e21 , e22 ,e23 ,e31 ,e32 and e33 with the control strategies ui .

5

Conclusions

In this paper, we attained the transmission projective synchronization between multiple non-identical coupled chaotic systems. We realized the transmission projective synchronization between the different coupled chaotic systems by designing the controllers, and a new synchronization criteria is given for the synchronization of the chaotic systems. This technology will possess better theory and application value in engineering practice. Furthermore, our synchronization control strategy can ensure the strict synchronization of such chaotic systems. Acknowledgement This work was supported in part by the Applied Mathematics Enhancement Program (AMEP) of Linyi University and the National Natural Science Foundation of China (No.61403179, 61273012), by a Project of the Postdoctoral Sustentation Fund of Jiangsu Province under Grant 1402042B.

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