Synchronization of N Different Chaotic Systems Based on Antisymmetric Structure

The problem of synchronization of N different chaotic systems is investigated. By using the direct design control method, the synchronization controler is designed to transform the error system into a nonlinear systemwith a special antisymmetric structure. The sufficient stability conditions are presented for such systems, and the complete synchronization of chaotic systems is realized. Finally, the corresponding numerical simulations demonstrate the effectiveness of the proposed schemes.


Introduction
Since Pecora and Carroll have synchronizes two identical chaotic systems with different initial conditions [1], chaotic synchronization has been intensively and extensively investigated due to its potential applications in many fields [2][3][4].And most of research efforts have been done about some synchronization phenomena, such as phase synchronization [5], antisynchronization [6], and projective synchronization [7].Many techniques have been developed to realize chaos synchronization which includes sliding mode control [8], direct design control method [9], and backstepping method [10].However, most of researches mentioned above mainly concentrated on synchronizing two identical or different chaotic systems.
In fact, more and more applications of chaos synchronization in secure communication make it much more important to synchronize multiple chaotic systems.It can satisfy the synchronization of multiple chaotic communication systems with a lower cost, and it can make it possible to realize multiparty communications simultaneously.Therefore, the synchronization of multiple chaotic systems has become a hot topic.It is more essential and useful in real-life applications.For example, Yu and Zhang studied the global synchronization of three coupled chaotic systems with ring connection in [11].The adaptive coupled synchronization among multi-Lorenz systems family is investigated in [12].
The synchronization of  different coupled chaotic systems with ring and chain connection was proposed in [13].The synchronization of N-coupled fractional-order chaotic systems with ring connection was investigated using the stability criteria of fractional-order system in [14].Zhang studied the synchronization of multi-chaotic systems based on the impulsive control theory in [15].Grassi researched the propagation of projective synchronization in a series connection of  chaotic discrete-time drive systems and  response systems in [16].Yang and Zhang studied the synchronization of three identical systems and its application for secure communication with noise perturbation in [17].
However, the realization of synchronization of  identical or nonidentical chaotic systems is much more difficult, so it is necessary to find an easy method to realize such synchronization of multiple chaotic systems.Motivated by the above discussions, in this paper, we consider the problem of synchronization of  different chaotic systems.With the active control method, the synchronization controller is designed to transform the error system into a nonlinear system with the special antisymmetric structure.The complete synchronization of multiple chaotic systems is realized.
This paper is organized as follows.In Section 2, the synchronization of  different chaotic systems is theoretically analyzed.A stability theorem for  different chaotic systems with a special antisymmetric structure is given.In Section 3, the proposed synchronization schemes with the direct design control method are applied to three different chaotic systems, that is, New system, Lorenz system and Rössler system.The simulations demonstrate the effectiveness of proposed schemes.And finally some concluding remarks are given in Section 4.

Synchronization of 𝑁 Different Chaotic Systems and Controllers Design
Consider the following chaotic systems: . . .
where  1 , then the systems (1) are the different chaotic systems.We consider  different chaotic systems.The drive systems and the controled response systems are described as follows: Let us define the errors systems between response systems and drive systems as ė  = ẋ  − ẋ 1 , ( = 2, . . ., ), then the dynamics of the synchronization errors can be expressed as Our purpose is to design the appropriate controllers (  ,  1 ) for the response systems such that the error systems (4) are asymptotically stable, which implies that the complete synchronization of  different chaotic systems (3) is realized; that is, lim Here, the direct design control method is used to achieve the objective.In accordance to the active control design strategy, we choose the control input (  ,  1 ) to eliminate all known items that cannot be shown in the form of the error system   .The controller can be given by The error systems (4) can be rewritten as Defining   =     , we can get the error systems (8) as follows: There are many possible choices for   , as long as it assures that the error dynamic system ( 8) is asymptotically stable at the origin.Without loss of generality, let us define   as a statedependent coefficient matrix, then the error systems (8) can be rewritten as The sufficient stability conditions of the systems (9) will be given by transforming it into a stable system with a special antisymmetric structure.The main result is described as follows.
and the invariant set of the system (9) only includes the origin, then the system (9) is asymptotically stable.
Remark 2. The system ( 9) is transformed into the system ė  =   (  )  under the control law (  ,  1 ), where   (  ) possesses the antisymmetric structure; then the error system (4) is asymptotically stable at the origin according to the Theorem 1.
Remark 3. 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, if the original system has some zero elements at the tridiagonal position and nonzero elements at other positions.
The selecting of the coefficient matrices with antisymmetric structure is an important and difficult technique, which relates to the coefficient matrices and the states of the original system.In the following section, we will demonstrate the proposed approaches for the special structure with an example.

Applications of Synchronization Control Schemes
In this section, we employ a simulation example to illustrate the effectiveness of the proposed schemes.The synchronization is simulated for three different chaotic systems.We choose New system as drive system, and we consider Lorenz system and Rössler system as the response systems.They are described as follows: Then the controllers are designed as follows, ] where Then the error systems (18) can be rewritten as Let us define  2 ( 2 ) and  3 ( 3 ) as follows: Then we declare that the error systems (18) and (19) are asymptotically stable at the origin under the controllers (20) according to Theorem 1. Then the synchronization between the response systems and the drive system is realized.

Conclusions
In this paper, the synchronization problem of  different chaotic systems is investigated.The direct design method is adopted to realize the complete synchronization of  different chaotic systems according to the proposed theorems.And then the stability theorems about the error systems with the special antisymmetric structure are presented.Numerical simulations of the synchronization about the different chaotic systems, respectively, illustrate the validity of the proposed schemes.How to realize other types of synchronization of N-coupled chaotic systems is our further work.Inspired by the studies [18][19][20], how to extend the current results to chaotic systems with discontinuous functions is also our future research directions.

Figure 3 :Figure 4 :
Figure3: Dynamics of the variables of the error system between the response systems (15) and (16) with time .

Figure 5 :Figure 6 :
Figure 5: The state trajectories  12 ,  22 , and  32 of the chaotic systems with time .