Synchronization of 𝑁 Different Chaotic Systems Based on Antisymmetric Structure

Abstract

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 system with 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.

1. 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 [2–4]. 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 𝑁 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 𝑁 chaotic discrete-time drive systems and 𝑁 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 𝑁 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 𝑁 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 𝑁 different chaotic systems is theoretically analyzed. A stability theorem for 𝑁 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.

2. Synchronization of 𝑁 Different Chaotic Systems and Controllers Design

Consider the following chaotic systems: 𝑥̇₁=𝐴₁𝑥₁+𝑔₁𝑥₁,𝑥̇₂=𝐴₂𝑥₂+𝑔₂𝑥₂,⋮𝑥̇_𝑁=𝐴_𝑁𝑥_𝑁+𝑔_𝑛𝑥_𝑁, (1) where 𝑥₁,𝑥₂,…,𝑥_𝑁∈ℝ^𝑛(𝑁>2) represent state vectors of the chaotic systems; 𝑔∶ℝ^𝑛→ℝ^𝑛 is a continuous nonlinear function; 𝐴₁,𝐴₂,…,𝐴_𝑁 are constant matrices. For 𝑖,𝑗=1,…,𝑁, and 𝑖≠𝑗, if 𝐴_𝑖≠𝐴_𝑗,𝑔_𝑖(⋅)≠𝑔_𝑗(⋅), (2) then the systems (1) are the different chaotic systems.

We consider 𝑁 different chaotic systems. The drive systems and the controled response systems are described as follows: 𝑥̇₁=𝐴₁𝑥₁+𝑔₁𝑥₁,𝑥̇₂=𝐴₂𝑥₂+𝑔₂𝑥₂+𝑢𝑥₁,𝑥₂,⋮𝑥̇_𝑁=𝐴_𝑁𝑥_𝑁+𝑔_𝑛𝑥_𝑁+𝑢𝑥₁,𝑥_𝑁. (3)

Let us define the errors systems between response systems and drive systems as 𝑒_𝑖̇=𝑥̇_𝑖−𝑥̇₁,(𝑖=2,…,𝑁), then the dynamics of the synchronization errors can be expressed as ̇𝑒̇𝑒=²𝑒₃̇⋮𝑒_𝑁̇=𝐴₂𝑒₂+𝐴₂−𝐴₁𝑥₁+𝑔₂𝑥₂−𝑔₁𝑥₁+𝑢𝑥₂,𝑥₁𝐴₃𝑒₃+𝐴₃−𝐴₁𝑥₁+𝑔₃𝑥₃−𝑔₁𝑥₁+𝑢𝑥₃,𝑥₁⋮𝐴_𝑁𝑒_𝑁+𝐴_𝑁−𝐴₁𝑥₁+𝑔_𝑁𝑥_𝑁−𝑔₁𝑥₁+𝑢𝑥_𝑁,𝑥₁. (4)

Our purpose is to design the appropriate controllers 𝑢(𝑥_𝑖,𝑥₁) for the response systems such that the error systems (4) are asymptotically stable, which implies that the complete synchronization of 𝑁 different chaotic systems (3) is realized; that is, lim_𝑡→∞𝑒_𝑖=lim_𝑡→∞𝑥_𝑖−𝑥₁=0. (5)

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 𝑢(𝑥_𝑖,𝑥₁) to eliminate all known items that cannot be shown in the form of the error system 𝑒_𝑖. The controller can be given by 𝑢𝑥_𝑖,𝑥₁=𝑢_𝑖−𝐴_𝑖−𝐴₁𝑥₁−𝑔_𝑖𝑥_𝑖+𝑔₁𝑥₁,𝑖=2,…,𝑁. (6) The error systems (4) can be rewritten as ̇𝑒̇𝑒=²𝑒₃̇⋮𝑒_𝑁̇=𝐴₂0⋯00𝐴₃⋯0⋮⋮⋮⋮00⋯𝐴_𝑁𝑒₂𝑒₃⋮𝑒_𝑁+𝑢₂𝑢₃⋮𝑢_𝑁. (7) Defining 𝑢_𝑖=𝑝_𝑖𝑒_𝑖, we can get the error systems (8) as follows: ̇𝑒̇𝑒=²𝑒₃̇⋮𝑒_𝑁̇=𝐴₂+𝑝₂0⋯00𝐴₃+𝑝₃⋯0⋮⋮⋮⋮00⋯𝐴_𝑁+𝑝_𝑁𝑒₂𝑒₃⋮𝑒_𝑁. (8) There are many possible choices for 𝑝_𝑖, as long as it assures that the error dynamic system (8) is asymptotically stable at the origin. Without loss of generality, let us define 𝑝_𝑖 as a state-dependent coefficient matrix, then the error systems (8) can be rewritten as ̇𝐵𝑒=²𝑒₂0⋯00𝐵₃𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁𝑒_𝑁𝑒₂𝑒₃⋮𝑒_𝑁. (9) 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.

Theorem 1. Consider the systems (9) with the state-dependent coefficient matrices 𝐵_𝑖(𝑒_𝑖)=𝐵_𝑖1(𝑒_𝑖)+𝐵_𝑖2. If the matrices 𝐵_𝑖1(𝑒_𝑖) and 𝐵_𝑖2 satisfy the assumptions that 𝐵^𝑇_𝑖1𝑒_𝑖=−𝐵_𝑖1𝑒_𝑖,(),𝐵𝑖=2,…,𝑁_𝑖2=diag𝑏_𝑖1,…,𝑏_𝑖𝑛,𝑏_𝑖𝑗<0,𝑗=1,…,𝑛, (10) and the invariant set of the system (9) only includes the origin, then the system (9) is asymptotically stable.

Proof. Choose Lyapunov function as follows: 1𝑉=2𝑒^𝑇𝑒. (11) Then𝑉̇=12𝑒^𝑇̇𝑒+𝑒^𝑇𝑒̇=12𝑒^𝑇𝐵₂(𝑒₂)0⋯00𝐵₃𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁𝑒_𝑁^𝑇+𝐵₂𝑒₂0⋯00𝐵₃𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁𝑒_𝑁𝑒=12𝑒^𝑇𝐵₂₁𝑒₂+𝐵₂₂𝑒₂0⋯00𝐵₃₁𝑒₃+𝐵₃₂𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁1𝑒_𝑁+𝐵_𝑁2𝑒_𝑁^𝑇+𝐵₂₁𝑒₂+𝐵₂₂𝑒₂0⋯00𝐵₃₁𝑒₃+𝐵₃₂𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁1𝑒_𝑁+𝐵_𝑁2𝑒_𝑁𝑒, (12)where 𝐵^𝑇_𝑖1(𝑒_𝑖)=−𝐵_𝑖1(𝑒_𝑖),(𝑖=2,…,𝑁), and 𝐵^𝑇_𝑖2(𝑒_𝑖)=𝐵_𝑖2(𝑒_𝑖). So we get that 𝑉̇=𝑒^𝑇𝐵₂₂𝑒₂0⋯00𝐵₃₂𝑒₃⋯0⋮⋮⋮⋮00⋯𝐵_𝑁2𝑒_𝑁𝑒, (13) where, for 𝑗=1,…,𝑛 and 𝑏_𝑖𝑗<0, 𝐵_𝑖2=diag(𝑏_𝑖1,…,𝑏_𝑖𝑛) is negative definite. Therefore, 𝑉 is negative definite. According to Lyapunov asymptotical stability theory, the complete synchronization of the different chaotic systems is achieved.

Remark 2. The system (9) is transformed into the system 𝑒_𝑖̇=𝐵_𝑖(𝑒_𝑖)𝑒_𝑖 under the control law 𝑢(𝑥_𝑖,𝑥₁), where 𝐵_𝑖(𝑒_𝑖) 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.

3. 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: 𝑥̇₁₁=5𝑥₁₁−𝑥₁₂𝑥₁₃,𝑥̇₁₂=−10𝑥₁₂+𝑥₁₁𝑥₁₃,𝑥̇₁₃=−3.8𝑥₁₃+13𝑥₁₁𝑥₁₂,𝑥̇₂₁=−10𝑥₂₁+10𝑥₂₂+𝑢₂₁,𝑥̇₂₂=28𝑥₂₁−𝑥₂₂−𝑥₂₁𝑥₂₃+𝑢₂₂,𝑥̇₂₃8=−3𝑥₂₃+𝑥₂₁𝑥₂₂+𝑢₂₃,𝑥̇₃₁=−𝑥₃₂−𝑥₃₃+𝑢₃₁,𝑥̇₃₂=𝑥₃₁+0.2𝑥₃₂+𝑢₃₂,𝑥̇₃₃=−5.7𝑥₃₃+0.2+𝑥₃₁𝑥₃₃+𝑢₃₃, (14)(15)(16) where 𝐴₁=5000−10000−3.8,𝐴₂=8−1010028−1000−3,𝐴₃=0−1−110.2000−5.7,𝑔₁𝑥₁=−𝑥₁₂𝑥₁₃𝑥₁₁𝑥₁₃13𝑥₁₁𝑥₁₂,𝑔₂𝑥₂=0−𝑥₂₁𝑥₂₃𝑥₂₁𝑥₂₂,𝑔₃𝑥₃=000.2+𝑥₃₁𝑥₃₃. (17)

Let the synchronization error state be 𝑒_𝑖̇=𝑥̇_𝑖−𝑥̇₁, (𝑖=2,3), then the error dynamical states can be written as 𝑒̇₂₁=−10𝑒₂₁+10𝑒₂₂+𝑢₂₁+𝑥₁₂𝑥₁₃−15𝑥₁₁+10𝑥₁₂,𝑒̇₂₂=28𝑒₂₁−𝑒₂₂+𝑢₂₂−𝑥₂₁𝑥₂₃−𝑥₁₁𝑥₁₃+28𝑥₁₁+9𝑥₁₂,𝑒̇₂₃8=−3𝑒₂₃+𝑢₂₃+𝑥₂₁𝑥₂₂−13𝑥₁₁𝑥₁₂+83.8−3𝑥₁₃,𝑒̇₃₁=−𝑒₃₂−𝑒₃₃+𝑢₃₁+𝑥₁₂𝑥₁₃−5𝑥₁₁−𝑥₁₂−𝑥₁₃,𝑒̇₃₂=𝑒₃₁+0.2𝑒₃₂+𝑢₃₂+𝑥₁₁+10.2𝑥₁₂−𝑥₁₁𝑥₁₃,𝑒̇₃₃=−5.7𝑒₃₃+𝑢₃₃−1.9𝑥₁₃+0.2+𝑥₃₁𝑥₃₃−13𝑥₁₁𝑥₁₂. (18)(19) Then the controllers are designed as follows, 𝑢𝑥₂,𝑥₁=𝑢₂₁𝑢₂₂𝑢₂₃=𝑝₁𝑒₂₁𝑒₂₂𝑒₂₃−815−1002890003.8−3𝑥₁₁𝑥₁₂𝑥₁₃−𝑔₂𝑥₂+𝑔₁𝑥₁,𝑢𝑥₃,𝑥₁=𝑢₃₁𝑢₃₂𝑢₃₃=𝑝₂𝑒₃₁𝑒₃₂𝑒₃₃−−5−1−1𝑥110.2000−1.9¹¹𝑥₁₂𝑥₁₃−𝑔₃𝑥₃+𝑔₁𝑥₁, (20) where 𝑝₁=00−𝑥₁₂−180−𝑥₁₁+𝑒₂₁𝑥₁₂𝑥₁₁+𝑒₂₁0,𝑝₂=𝑥−1000−1.2−₁₁+𝑒₃₁1𝑥₁₁+𝑒₃₁0. (21) Then the error systems (18) can be rewritten aṡ𝑒̇𝑒=²𝑒₃̇=𝑒̇₂₁𝑒̇₂₂𝑒̇₂₃𝑒̇₃₁𝑒̇₃₂𝑒̇₃₃^𝑇=−10−10−𝑥₁₂𝑥00010−1−₁₁+𝑒₂₁𝑥000₁₂𝑥₁₁+𝑒₂₁−83𝑥000000−1−1−10001−1−₁₁+𝑒₃₁0001𝑥₁₁+𝑒₃₁𝑒−5.7²¹𝑒₂₂𝑒₂₃𝑒₃₁𝑒₃₂𝑒₃₃. (22)

Let us define 𝐵₂(𝑒₂) and 𝐵₃(𝑒₃) as follows: 𝐵₂₁𝑒₂=0−10−𝑥₁₂100−𝑥₁₁+𝑒₂₁𝑥₁₂𝑥₁₁+𝑒₂₁0+8−10000−1000−3,𝐵₃𝑒₃=𝑥0−1−110−₁₁+𝑒₃₁1𝑥₁₁+𝑒₃₁0+.−1000−1000−5.7 (23) 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.

The fourth-order Runge-Kutta integration method is used to numerical simulation with time step size 0.001s. Let the initial conditions of the drive system and the response systems be (𝑥₁₁(0),𝑥₁₂(0),𝑥₁₃(0))^𝑇=(11,15,26)^𝑇, (𝑥₂₁(0),𝑥₂₂(0),𝑥₂₃(0))^𝑇=(1,6,8)^𝑇, and (𝑥₃₁(0),𝑥₃₂(0),𝑥₃₃(0))^𝑇=(4.8,0.8,0.2)^𝑇, respectively. Then the initial values of the error states are (𝑒₂₁(0),𝑥₂₂(0),𝑥₂₃(0))^𝑇=(−10,−9,−18)^𝑇 and (𝑒₃₁(0),𝑥₃₂(0),𝑥₃₃(0))^𝑇=(−6.2,−14.2,−25.8)^𝑇. The state trajectories of the error systems are shown in Figures 1, 2, and 3, and the state trajectories of 3 different chaotic systems are shown in Figures 4, 5, and 6. We can make out that, from Figures 1–6, the state trajectories asymptotically converge to zero near 2.5 s, 4.5 s, and 5 s under the controllers, and the state vectors of the different chaotic systems achieve the complete synchronization. The numerical simulations demonstrate that the proposed design method is feasible and effective to realize the complete synchronization of 𝑁 different chaotic systems which satisfy Theorem 1.

Figure 1 

Dynamics of the variables , , and of the error system (18) with time .

Figure 2 

Dynamics of the variables , , and of the error system (19) with time .

Figure 3 

Dynamics of the variables of the error system between the response systems (15) and (16) with time .

Figure 4 

The state trajectories , , and of the chaotic systems with time .

Figure 5 

The state trajectories , , and of the chaotic systems with time .

Figure 6 

The state trajectories , , and of the chaotic systems with time .

4. Conclusions

In this paper, the synchronization problem of 𝑁 different chaotic systems is investigated. The direct design method is adopted to realize the complete synchronization of 𝑁 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 [18–20], how to extend the current results to chaotic systems with discontinuous functions is also our future research directions.

Acknowledgments

The authors’ work was supported in part by the National Natural Science Foundation of China, nos. 61273012, 11201212, and a Project of Shandong Province Higher Educational Science and Technology Program, nos. J13LI11, J12LI58. The authors would like to express their thanks to the gratefully acknowledged suggestions and comments made by the referees.