# Synchronization ofN Coupled Chaotic Systems with  Ring Connection Based on Special Antisymmetric Structure

Authors: Xiangyong Chen; Jianlong Qiu; Qiang Song; Ancai Zhang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper considers the complete synchronization problem forN coupled chaotic systems with ring connections. First, we use a direct design method to design a synchronization controller. It transforms the error system into a stable system with special antisymmetric structure. And then, we get some simple stability criteria of achieving the complete synchronization. These criteria are not only easily verified but also improve and generalize previous known results. Finally, numerical examples are provided to demonstrate the effectiveness of the theoretical analysis.


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## Body

## 1. Introduction

Since the pioneering work of Pecora and Carroll [1], chaotic synchronization has been intensively investigated due to its potential applications in many fields [2–4]. Several types of synchronization phenomena have been reported, such as complete synchronization [5], phase synchronization [6], antisynchronization [7], projective synchronization [8, 9], and generalized synchronization [10]. To solve the problem of chaotic synchronization, many approaches have been developed. That includes a sliding mode control method [11], an impulsive control method [11], an adaptive control method [7, 12], a pinning control method [13], and a sampled-data control method [14]. However, these control algorithms are just suitable to synchronize two identical or nonidentical chaotic systems.Nowadays, the synchronization of multiple chaotic systems has attracted increasing attention. It has been widely used in secure communication area in order to reduce the synchronizing cost of multiple chaotic communication and make simultaneous multiparty communications possible. Therefore, the synchronization in multiple chaotic systems has more advantages and deserves to be deeply investigated comparing with the conventional chaotic synchronization. Several types of synchronization in an array of chaotic systems have been investigated in the past few years, for example, the global synchronization in [15, 16], and the adaptive coupled synchronization in [17], the projective synchronization in [18]. In [19–21], the synchronization of N chaotic systems of with ring and chain connection was investigated. In addition, Yang and Zhang [22] studied the synchronization of an array of identical chaotic systems and discussed its application for secure communication with noise perturbation.However, it is notable that the realization of synchronization ofN coupled chaotic systems is much more difficult. So, it is necessary to find an easy method that realizes the synchronization of such multiple chaotic systems. Cai et al. [7, 8] investigated the generalized projective synchronization of two different chaotic systems based on a special antisymmetric structure. We studied the problem of synchronization of N different chaotic systems by using a special antisymmetric structure [21]. Inspired by the above discussions, we further discuss the synchronization of N coupled chaotic systems in this paper. First, a synchronization controller that is designed by using a direct design method transforms the error system into a stable system with special antisymmetric structure. And then, we derive some sufficient conditions in order to guarantee the asymptotical stabilization of the error system at the origin. It means that the synchronization of N coupled chaotic systems is realized.The paper is organized as follows. In Section2, the synchronization of N coupled chaotic systems with ring connection is theoretically analyzed. A stability theorem for such systems with special antisymmetric structure is given. In Section 3, the synchronization control schemes are applied to three identical and non-identical coupled chaotic systems. Simulation results demonstrate the effectiveness of proposed schemes. And finally some concluding remarks are given in Section 4.


## 2. Synchronization ofN Chaotic Systems and Controllers Design

ConsiderN coupled chaotic systems with ring connections described by
(1)x˙1=A1x1+g1(x1)+D1(xN-x1),x˙2=A2x2+g2(x2)+D2(x1-x2),⋮x˙N=ANxN+gn(xN)+DN(xN-1-xN),
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(d1j,…,dNj) and dij≥0 are the diagonal matrices which represent the coupled parameters. If
(2)Ai≠Aj,i=1,…N,j=1,…N,i≠j,gi(·)≠gj(·),i=1,…N,j=1,…N,i≠j,
then the system (1) is an array of non-identical chaotic systems.Now the above simple coupling form is applied to investigate the synchronization ofN chaotic systems. The systems are
(3)x˙1=A1x1+g1(x1)+D1(xN-x1),x˙2=A2x2+g2(x2)+D2(x1-x2)+u1,⋮x˙N=ANxN+gn(xN)+DN(xN-1-xN)+uN-1.Let the state error beei-1=xi-xi-1, (i=2,…,N). It is not difficult to obtain the following dynamical error system:(4)e˙=[A2-D1-D2-D1-D1⋯-D1D2A3-D30⋯00D3A4-D4⋯0⋮⋮⋮⋮⋮000⋯AN-DN][e1e2e3⋮eN-1]+[(A2-A1)x1+g2(x2)-g1(x1)+u1(A3-A2)x2+g3(x3)-g2(x2)+u2-u1⋮(AN-AN-1)xN-1+gN(xN)-gN-1(xN-1)+uN-1-uN-2].Our purpose is to design the controllersui  (i=1,…,N-1) such that the error system (4) is asymptotically stable at the origin. That is,
(5)limt→∞∥ei-1∥=limt→∞∥xi-xi-1∥=0.
This implies that the synchronization of chaotic systems (3) is realized.Here a direct design control method [7, 8, 21] is used to achieve the objective. This method presents an easy procedure of selecting proper controllers in chaos synchronization. So we adopt this method to transform the error system into a stable system with a special antisymmetric structure. The main results are given below.First, we choose the control inputui to eliminate all known items that cannot be shown in the form of the error system ei. The controllers are given by
(6)u1=v1-(A2-A1)x1-g2(x2)+g1(x1),u2=v2-(A3-A2)x2-g3(x3)+g2(x2)+u1,⋮uN-1=vN-1-(AN-AN-1)xN-1-gN(xN)+gN-1(xN-1)+uN-2,
where [v1v2⋯vN-1]T=H[e1e2⋯eN-1]T and H is a coefficient matrix. So, the error systems (4) are rewritten by
(7)e˙=L(e)e,
where(8)L(e)=[A2-D1-D2-D1-D1⋯-D1D2A3-D30⋯00D3A4-D4⋯0⋮⋮⋮⋮⋮000⋯AN-DN]+H.Theorem 1.
Consider the systems (7) with the state dependent coefficient L(e)=L1(e)+L2. If L1(e) and L2 satisfy the following assumptions:
(9)L1T(e)=-L1(e),L2=diag(l1,…,ln),li<0,(i=1,…,n),
then the system (7) is asymptotically stable, which means that the complete synchronization of N coupled chaotic systems (3) are achieved.Proof.
Choose a Lyapunov function to be(10)V=12eTe.
The derivative ofV is
(11)V˙=12(e˙Te+eTe˙)=12eT(L(e)T+L(e))e.
SinceL(e)=L1(e)+L2, L1T(e)=-L1(e)  and L2=diag(l1,…,ln), li<0, (i=1,…,n), it is easy to get
(12)V˙=eTL2e<0.
From Lyapunov stability theory, we know that the equilibriume=0 of the system (7) is global asymptotically stable. Then the complete synchronization of N chaotic systems (3) is achieved.Remark 2.
The error dynamic system (4) is transformed into the systems e˙=L(e)e under the control law ui, where L(e) possesses the antisymmetric structure. Theorem 1 ensures that the error system (4) is asymptotically stable at the origin.Remark 3.
There are many possible choices forH as long as it guarantees the error dynamic system (7) to be asymptotically stable at the origin. Without loss of generality, we define H to be a state dependent coefficient matrix. As a result, the sufficient stability conditions of the systems (7) are given by transforming it into a stable system with a special antisymmetric structure.Remark 4.
The antisymmetric structures in Theorem1 are the generalization of the tridiagonal structures. The error system constructed with the antisymmetric structure is more convenient than the one with tridiagonal structure when the original system has some zero elements at the tridiagonal position and nonzero elements at other positions.Since the antisymmetric structure is related to the coefficient matrices and the states of the original system, the selecting of the coefficient matrices with antisymmetric structure is an important and difficult task. In the next section, we will demonstrate the proposed approaches for the special structure through numerical examples.


## 3. Applications of Synchronization Control Schemes

In this section, we use two simulation examples to illustrate the effectiveness of the proposed schemes. The synchronization is simulated for the non-identical and identical chaotic systems, respectively.Case 1.
When the drive system and response systems are identical chaotic systems, the drive system and response systems are all the Lorenz chaotic system. They are described as follows:(13)x˙11=-10x11+10x12+d11(x31-x11),x˙12=28x11-x12-x11x13+d12(x32-x12),x˙13=-83x13+x11x12+d13(x33-x13),x˙21=-10x21+10x22+d21(x11-x21)+u11,x˙22=28x21-x22-x21x23+d22(x12-x22)+u12,x˙23=-83x23+x21x22+d23(x13-x23)+u13,x˙31=-10x31+10x32+d31(x21-x31)+u21,x˙32=28x31-x32-x31x33+d32(x22-x32)+u22,x˙33=-83x33+x31x32+d33(x23-x33)+u23,
where
(14)A1=A2=A3=A=[-1010028-1000-83],g1(x1)=[0-x11x13x11x12],g2(x2)=[0-x21x23x21x22],g3(x3)=[0-x31x33x31x32],
and D1=diag(d11,d12,d13), D2=diag(d21,d22,d23), D3=diag(d31,d32,d33) are the coupled matrices, and u1=[u11,u12,u13]T and u2=[u21,u22,u23]T are the control inputs.Let the synchronization error state bee˙i-1=x˙i-x˙i-1, (i=2,3). The error dynamical equations are
(15)e˙=[Γ1Γ2Γ3Γ4]e+[u11-x21x23+x11x13+u12x21x22-x11x12+u13u21-u11-x31x33+x21x23+u22-u12x31x32-x21x22+u23-u13],
where
(16)Γ1=[-10-d11-d2110028-1-d12-d22000-83-d13-d23],Γ2=[-d11000-d12000-d13],Γ3=[d21000d22000d23],Γ4=[-10-d3110028-1-d32000-83-d33].We design the controllersu1 and u2 to be
(17)u1=v1-g2(x2)+g1(x1),u2=v2+v1-g3(x3)+g1(x1),
where
(18)v1=[000000-3800000000000]e,v2=[d11-d210000038d12-d220-380000d13-d23000]e.The error systems (15) become
(19)e˙=[Γ1*Γ2*Γ3*Γ4*]e,
where
(20)Γ1*=[-10-d11-d21100-10-1-d12-d22000-83-d13-d23],Γ2*=[-d11000-d12000-d13],Γ3*=[d11000d12000d13],Γ4*=[-10-d31100-10-1-d32000-83-d33].If the conditions(21)-10-d11-d21<0,-1-d12-d22<0,-83-d13-d23<0,-10-d31<0,-1-d32<0,-83-d33<0
are satisfied, then Theorem 1 tells us that the error systems (19) are asymptotically stable at the origin under the controllers u1 and u2. It means that the synchronization of (13) is realized.Fourth order Runge-Kutta integration method is used to numerical simulation with time step size 0.001. 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. And we choose that d12=d22=d32=d13=d23=d33=0 and d11=d21=d31=1. The state trajectories of the error systems and chaotic systems are shown in Figures 1, 2, 3, and 4, respectively. It is easy to see that the state variables and the error variables all tend towards to less than 2 s under the controllers. Simulation results demonstrate that the convergence rates are much faster than the earlier research results proposed in the literature [19, 21]; then the effectiveness of the synchronization control schemes is confirmed.Figure 1
Dynamics of the variablese11, e12, e13, e21, and e22 and e23 of the error systems e1 and e2 with time t.Figure 2
The state trajectoriesx11, x21, and x31 of the Lorenz systems with time t.Figure 3
The state trajectoriesx12,  x22, and x32 of the Lorenz systems with time t.Figure 4
The state trajectoriesx13,  x23, and x33 of the Lorenz systems with time t.Case 2.
When the drive system and response systems are non-identical chaotic systems, the Chen system, Lü system, and Lorenz system are considered as drive system and response systems, respectively. They are described as follows:(22)x˙11=-35x11+35x12+d11(x31-x11),x˙12=-7x11+28x12-x11x13+d12(x32-x12),x˙13=-3x13+x11x12+d13(x33-x13),x˙21=-36x21+36x22+d21(x11-x21)+u11,x˙22=20x22-x21x23+d22(x12-x22)+u12,x˙23=-3x23+x21x22+d23(x13-x23)+u13,x˙31=-10x31+10x32+d31(x21-x31)+u21,x˙32=28x31-x32-x31x33+d32(x22-x32)+u22,x˙33=-83x33+x31x32+d33(x23-x33)+u23,
where
(23)A1=[-35350-728000-3],A2=[-36360020000-3],A3=[-1010028-1000-83],g1(x1)=[0-x11x13x11x12],g2(x2)=[0-x21x23x21x22],g3(x3)=[0-x31x33x31x32],
and D1=diag(d11,d12,d13), D2=diag(d21,d22,d23), and D3=diag(d31,d32,d33) are the coupled matrices, and u1=[u11,u12,u13]T and u2=[u21,u22,u23]T are the control inputs.Let the synchronization error state bee˙i-1=x˙i-x˙i-1, (i=2,3). The error dynamical equations are
(24)e˙=[Δ1Δ2Δ3Δ4]e+[-x11-x12+u117x11-8x12-x21x23+x11x13+u12x21x22-x11x12+u1326x21+26x22+u21-u1128x21-21x22-x31x33+x21x23+u22-u1213x23+x31x32-x21x22+u23-u13],
where
(25)Δ1=[-36-d11-d21360020-d12-d22000-3-d13-d23],Δ2=[-d11000-d12000-d13],Δ3=[d21000d22000d23],Δ4=[-10-d3110028-1-d32000-83-d33].The controllersu1 and u2 are designed as
(26)u1=v1-(A2-A1)x1-g2(x2)+g1(x1),u2=v2+v1-(A3-A1)x1-g3(x3)+g1(x1),
where
(27)v1=[000000-3600000000000]e,v2=[d11-d210000036d12-d220-380000d13-d23000]e.The error systems (24) become
(28)e˙=[Δ1*Δ2*Δ3*Δ4*]e,
where
(29)Δ1*=[-36-d11-d21360-3620-d12-d22000-3-d13-d23],Δ2*=[-d11000-d12000-d13],Δ3*=[d11000d12000d13],Δ4*=[-10-d31100-10-1-d32000-83-d33].From Theorem1, we know that the conditions
(30)-36-d11-d21<0,20-d12-d22<0,-3-d13-d23<0,-10-d31<0,-1-d32<0,-83-d33<0
ensure that the error systems (28) are asymptotically stable at the origin under the controllers u1 and u2. Thus, the synchronization between the response systems and the drive systems is realized.Similar to Case1, let the initial conditions of the drive system and the response systems be (x11(0),x12(0),x13(0))=(10,20,30), (x21(0),x22(0),x23(0))=(-5.8,8,10), and (x31(0),x32(0),x33(0))=(11,15,26), respectively. And we choose that d11=d21=d13=d23=d31=d33=0, d12=10, d22=11, and d32=1. The state trajectories of the error systems and chaotic systems are shown in Figures 5, 6, 7, and 8, respectively. It is easy to see that the state variables and the error variables all tend towards to less than 1.5 s under the controllers. Simulation results demonstrate that the convergence rates are much faster than the earlier research results proposed in the literature [19, 21]; then the effectiveness of the synchronization control schemes is confirmed.Figure 5
Dynamics of the variablese11, e12, and e13 of the error system e1 with time t.Figure 6
The state trajectoriesx11, x21, and x31 of the chaotic systems with time t.Figure 7
The state trajectoriesx12, x22, and x32 of the chaotic systems with time t.Figure 8
The state trajectoriesx13, x23, and x33 of the chaotic systems with time t.


## 4. Conclusions

This paper concerns the synchronization ofN coupled chaotic systems with ring connection. A direct design control method is firstly used to design the controllers. The synchronization of N coupled chaotic systems is realized by coupling the state variables. This technology will undoubtedly improve performance of secret signaling and possess better application value in practice. Furthermore, our presented strategy can ensure strict synchronization between N coupled chaotic systems. This will lead to a rapid development in multilateral communications.



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*Source: 680604-2013-10-03.xml*