Synchronization of N Coupled Chaotic Systems with Ring Connection Based on Special Antisymmetric Structure

and Applied Analysis 3 Since L(e) = L 1 (e) + L 2 , L 1 (e) = −L 1 (e) and L 2 = diag(l 1 , . . . , l n ), l i < 0, (i = 1, . . . , n), it is easy to get ?̇? = e T L 2 e < 0. (12) From Lyapunov stability theory, we know that the equilibrium e = 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 u i , 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 defineH 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. Theantisymmetric structures inTheorem 1 are the generalization of the tridiagonal structures. The error system constructed with the antisymmetric structure is more convenient than the onewith tridiagonal structurewhen the original systemhas 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: ?̇? 11 = − 10x 11 + 10x 12 + d 11 (x 31 − x 11 ) , ?̇? 12 = 28x 11 − x 12 − x 11 x 13 + d 12 (x 32 − x 12 ) ,


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][3][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][20][21], the synchronization of  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 of  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  different chaotic systems by using a special antisymmetric structure [21].Inspired by the above discussions, we further discuss the synchronization of  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  coupled chaotic systems is realized.
The paper is organized as follows.In Section 2, the synchronization of  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.

Synchronization of 𝑁 Chaotic Systems and Controllers Design
Consider  coupled chaotic systems with ring connections described by where  1 , then the system (1) is an array of non-identical chaotic systems.Now the above simple coupling form is applied to investigate the synchronization of  chaotic systems.The systems are Let the state error be  −1 =   −  −1 , ( = 2, . . ., ).It is not difficult to obtain the following dynamical error system: Our purpose is to design the controllers   ( = 1, . . ., − 1) such that the error system (4) is asymptotically stable at the origin.That is, lim 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 input   to eliminate all known items that cannot be shown in the form of the error system   .The controllers are given by where and  is a coefficient matrix.So, the error systems (4) are rewritten by where Theorem 1.Consider the systems (7) with the state dependent coefficient () =  1 () +  2 .If  1 () and  2 satisfy the following assumptions: then the system (7) is asymptotically stable, which means that the complete synchronization of  coupled chaotic systems (3) are achieved.
Remark 2. The error dynamic system (4) is transformed into the systems ė = () under the control law   , where () 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 for  as long as it guarantees the error dynamic system (7) to be asymptotically stable at the origin.Without loss of generality, we define  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 Theorem 1 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.

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: where and where We design the controllers  1 and  2 to be where The error systems (15) become where are satisfied, then Theorem 1 tells us that the error systems (19) are asymptotically stable at the origin under the controllers  1 and  2 .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 ( 11 (0),  12 (0),  13 (0)) = (4, 5, −3), ( 21 (0),  22 (0),  23 (0)) = (5, 2, −5), and ( 31 (0),  32 (0),  33 (0)) = (11,15,10).respectively.And we choose that  12 =  22 =  32 =  13 =  23 =  33 = 0 and  11 =  21 =  31 = 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.
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: where The controllers  1 and  2 are designed as where The error systems (24) become where From Theorem 1, we know that the conditions ensure that the error systems (28) are asymptotically stable at the origin under the controllers  1 and  2 .Thus, the synchronization between the response systems and the drive systems is realized.Similar to Case 1, let the initial conditions of the drive system and the response systems be ( 11 (0),  12 (0),  13 (0)) = (10, 20, 30), ( 21 (0),  22 (0),  23 (0)) = (−5.8,8, 10), and ( 31 (0),  32 (0),  33 (0)) = (11,15,26), respectively.And we choose that  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.

Conclusions
This paper concerns the synchronization of  coupled chaotic systems with ring connection.A direct design control method is firstly used to design the controllers.The synchronization of  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  coupled chaotic systems.This will lead to a rapid development in multilateral communications.

Figure 3 :
Figure 3: The state trajectories  12 ,  22 , and  32 of the Lorenz systems with time .

Figure 4 :
Figure 4: The state trajectories  13 ,  23 , and  33 of the Lorenz systems with time .

Figure 7 :Figure 8 :
Figure 7: The state trajectories  12 ,  22 , and  32 of the chaotic systems with time .