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

Abstract

This paper considers the complete synchronization problem for 𝑁 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.

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 of 𝑁 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 Section 2, 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 of 𝑁 Chaotic Systems and Controllers Design

Consider 𝑁 coupled chaotic systems with ring connections described by 𝑥̇₁=𝐴₁𝑥₁+𝑔₁𝑥₁+𝐷₁𝑥_𝑁−𝑥₁,𝑥̇₂=𝐴₂𝑥₂+𝑔₂𝑥₂+𝐷₂𝑥₁−𝑥₂,⋮𝑥̇_𝑁=𝐴_𝑁𝑥_𝑁+𝑔_𝑛𝑥_𝑁+𝐷_𝑁𝑥_𝑁−1−𝑥_𝑁, (1) where 𝑥₁,𝑥₂,…,𝑥_𝑁 are the state vectors of the chaotic systems; 𝑔_𝑖(𝑥_𝑖)  (𝑖=1,…,𝑁) is the continuous nonlinear function; 𝐴₁,𝐴₂,…,𝐴_𝑁 are constant matrices; 𝐷_𝑖=diag(𝑑_1𝑗,…,𝑑_𝑁𝑗) and 𝑑_𝑖𝑗≥0 are the diagonal matrices which represent the coupled parameters. If 𝐴_𝑖≠𝐴_𝑗𝑔,𝑖=1,…𝑁,𝑗=1,…𝑁,𝑖≠𝑗,_𝑖(⋅)≠𝑔_𝑗(⋅),𝑖=1,…𝑁,𝑗=1,…𝑁,𝑖≠𝑗, (2) 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 𝑁 chaotic systems. The systems are 𝑥̇₁=𝐴₁𝑥₁+𝑔₁𝑥₁+𝐷₁𝑥_𝑁−𝑥₁,𝑥̇₂=𝐴₂𝑥₂+𝑔₂𝑥₂+𝐷₂𝑥₁−𝑥₂+𝑢₁,⋮𝑥̇_𝑁=𝐴_𝑁𝑥_𝑁+𝑔_𝑛𝑥_𝑁+𝐷_𝑁𝑥_𝑁−1−𝑥_𝑁+𝑢_𝑁−1. (3)

Let the state error be 𝑒_𝑖−1=𝑥_𝑖−𝑥_𝑖−1, (𝑖=2,…,𝑁). It is not difficult to obtain the following dynamical error system:̇𝐴𝑒=²−𝐷₁−𝐷₂−𝐷₁−𝐷₁⋯−𝐷₁𝐷₂𝐴₃−𝐷₃0⋯00𝐷₃𝐴₄−𝐷₄⋯0⋮⋮⋮⋮⋮000⋯𝐴_𝑁−𝐷_𝑁𝑒₁𝑒₂𝑒₃⋮𝑒_𝑁−1+𝐴₂−𝐴₁𝑥₁+𝑔₂𝑥₂−𝑔₁𝑥₁+𝑢₁𝐴₃−𝐴₂𝑥₂+𝑔₃𝑥₃−𝑔₂𝑥₂+𝑢₂−𝑢₁⋮𝐴_𝑁−𝐴_𝑁−1𝑥_𝑁−1+𝑔_𝑁𝑥_𝑁−𝑔_𝑁−1𝑥_𝑁−1+𝑢_𝑁−1−𝑢_𝑁−2. (4)

Our purpose is to design the controllers 𝑢_𝑖  (𝑖=1,…,𝑁−1) such that the error system (4) is asymptotically stable at the origin. That is, lim_𝑡→∞𝑒_𝑖−1=lim_𝑡→∞𝑥_𝑖−𝑥_𝑖−1=0. (5) 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 𝑢_𝑖 to eliminate all known items that cannot be shown in the form of the error system 𝑒_𝑖. The controllers are given by 𝑢₁=𝑣₁−𝐴₂−𝐴₁𝑥₁−𝑔₂𝑥₂+𝑔₁𝑥₁,𝑢₂=𝑣₂−𝐴₃−𝐴₂𝑥₂−𝑔₃𝑥₃+𝑔₂𝑥₂+𝑢₁,⋮𝑢_𝑁−1=𝑣_𝑁−1−𝐴_𝑁−𝐴_𝑁−1𝑥_𝑁−1−𝑔_𝑁𝑥_𝑁+𝑔_𝑁−1𝑥_𝑁−1+𝑢_𝑁−2, (6) where 𝑣₁𝑣₂⋯𝑣_𝑁−1^𝑇=𝐻𝑒₁𝑒₂⋯𝑒_𝑁−1^𝑇 and 𝐻 is a coefficient matrix. So, the error systems (4) are rewritten by 𝑒̇=𝐿(𝑒)𝑒, (7) where𝐿(𝑒)=𝐴₂−𝐷₁−𝐷₂−𝐷₁−𝐷₁⋯−𝐷₁𝐷₂𝐴₃−𝐷₃0⋯00𝐷₃𝐴₄−𝐷₄⋯0⋮⋮⋮⋮⋮000⋯𝐴_𝑁−𝐷_𝑁+𝐻. (8)

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: 𝐿^𝑇₁(𝑒)=−𝐿₁(𝑒),𝐿₂=diag𝑙₁,…,𝑙_𝑛,𝑙_𝑖<0,(𝑖=1,…,𝑛), (9) 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 1𝑉=2𝑒^𝑇𝑒. (10)
The derivative of 𝑉 is 𝑉̇=12𝑒^𝑇̇𝑒+𝑒^𝑇𝑒̇=12𝑒^𝑇𝐿(𝑒)^𝑇+𝐿(𝑒)𝑒. (11)
Since 𝐿(𝑒)=𝐿₁(𝑒)+𝐿₂, 𝐿^𝑇₁(𝑒)=−𝐿₁(𝑒) and 𝐿₂=diag(𝑙₁,…,𝑙_𝑛), 𝑙_𝑖<0, (𝑖=1,…,𝑛), it is easy to get 𝑉̇=𝑒^𝑇𝐿₂𝑒<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 𝑒̇=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 for 𝐻 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 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.

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: 𝑥̇₁₁=−10𝑥₁₁+10𝑥₁₂+𝑑₁₁𝑥₃₁−𝑥₁₁,𝑥̇₁₂=28𝑥₁₁−𝑥₁₂−𝑥₁₁𝑥₁₃+𝑑₁₂𝑥₃₂−𝑥₁₂,𝑥̇₁₃8=−3𝑥₁₃+𝑥₁₁𝑥₁₂+𝑑₁₃𝑥₃₃−𝑥₁₃,𝑥̇₂₁=−10𝑥₂₁+10𝑥₂₂+𝑑₂₁𝑥₁₁−𝑥₂₁+𝑢₁₁,𝑥̇₂₂=28𝑥₂₁−𝑥₂₂−𝑥₂₁𝑥₂₃+𝑑₂₂𝑥₁₂−𝑥₂₂+𝑢₁₂,𝑥̇₂₃8=−3𝑥₂₃+𝑥₂₁𝑥₂₂+𝑑₂₃𝑥₁₃−𝑥₂₃+𝑢₁₃,𝑥̇₃₁=−10𝑥₃₁+10𝑥₃₂+𝑑₃₁𝑥₂₁−𝑥₃₁+𝑢₂₁,𝑥̇₃₂=28𝑥₃₁−𝑥₃₂−𝑥₃₁𝑥₃₃+𝑑₃₂𝑥₂₂−𝑥₃₂+𝑢₂₂,𝑥̇₃₃8=−3𝑥₃₃+𝑥₃₁𝑥₃₂+𝑑₃₃𝑥₂₃−𝑥₃₃+𝑢₂₃, (13) where 𝐴₁=𝐴₂=𝐴₃8=𝐴=−1010028−1000−3,𝑔₁𝑥₁=0−𝑥₁₁𝑥₁₃𝑥₁₁𝑥₁₂,𝑔₂𝑥₂=0−𝑥₂₁𝑥₂₃𝑥₂₁𝑥₂₂,𝑔₃𝑥₃=0−𝑥₃₁𝑥₃₃𝑥₃₁𝑥₃₂, (14) 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 be 𝑒̇_𝑖−1=𝑥̇_𝑖−𝑥̇_𝑖−1, (𝑖=2,3). The error dynamical equations are ̇Γ𝑒=¹Γ₂Γ₃Γ₄𝑢𝑒+¹¹−𝑥₂₁𝑥₂₃+𝑥₁₁𝑥₁₃+𝑢₁₂𝑥₂₁𝑥₂₂−𝑥₁₁𝑥₁₂+𝑢₁₃𝑢₂₁−𝑢₁₁−𝑥₃₁𝑥₃₃+𝑥₂₁𝑥₂₃+𝑢₂₂−𝑢₁₂𝑥₃₁𝑥₃₂−𝑥₂₁𝑥₂₂+𝑢₂₃−𝑢₁₃, (15) where Γ₁=−10−𝑑₁₁−𝑑₂₁10028−1−𝑑₁₂−𝑑₂₂0800−3−𝑑₁₃−𝑑₂₃,Γ₂=−𝑑₁₁000−𝑑₁₂000−𝑑₁₃,Γ₃=𝑑₂₁000𝑑₂₂000𝑑₂₃,Γ₄=−10−𝑑₃₁10028−1−𝑑₃₂0800−3−𝑑₃₃. (16)

We design the controllers 𝑢₁ and 𝑢₂ to be 𝑢₁=𝑣₁−𝑔₂𝑥₂+𝑔₁𝑥₁,𝑢₂=𝑣₂+𝑣₁−𝑔₃𝑥₃+𝑔₁𝑥₁, (17) where 𝑣₁=𝑣000000−3800000000000𝑒,₂=𝑑₁₁−𝑑₂₁0000038𝑑₁₂−𝑑₂₂0−380000𝑑₁₃−𝑑₂₃000𝑒. (18)

The error systems (15) become ̇Γ𝑒=^∗1Γ^∗₂Γ^∗₃Γ^∗₄𝑒, (19) where Γ^∗₁=−10−𝑑₁₁−𝑑₂₁100−10−1−𝑑₁₂−𝑑₂₂0800−3−𝑑₁₃−𝑑₂₃,Γ^∗₂=−𝑑₁₁000−𝑑₁₂000−𝑑₁₃,Γ^∗₃=𝑑₁₁000𝑑₁₂000𝑑₁₃,Γ^∗₄=−10−𝑑₃₁100−10−1−𝑑₃₂0800−3−𝑑₃₃. (20)

If the conditions −10−𝑑₁₁−𝑑₂₁<0,−1−𝑑₁₂−𝑑₂₂−8<0,3−𝑑₁₃−𝑑₂₃<0,−10−𝑑₃₁<0,−1−𝑑₃₂8<0,−3−𝑑₃₃<0 (21) 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 variables 𝑒11, 𝑒12, 𝑒13, 𝑒21, and 𝑒22 and 𝑒23 of the error systems 𝑒1 and 𝑒2 with time 𝑡.

Figure 2 

The state trajectories 𝑥11, 𝑥21, and 𝑥31 of the Lorenz systems with time 𝑡.

Figure 3 

The state trajectories 𝑥12,  𝑥22, and 𝑥32 of the Lorenz systems with time 𝑡.

Figure 4 

The state trajectories 𝑥13,  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: 𝑥̇₁₁=−35𝑥₁₁+35𝑥₁₂+𝑑₁₁𝑥₃₁−𝑥₁₁,𝑥̇₁₂=−7𝑥₁₁+28𝑥₁₂−𝑥₁₁𝑥₁₃+𝑑₁₂𝑥₃₂−𝑥₁₂,𝑥̇₁₃=−3𝑥₁₃+𝑥₁₁𝑥₁₂+𝑑₁₃𝑥₃₃−𝑥₁₃,𝑥̇₂₁=−36𝑥₂₁+36𝑥₂₂+𝑑₂₁𝑥₁₁−𝑥₂₁+𝑢₁₁,𝑥̇₂₂=20𝑥₂₂−𝑥₂₁𝑥₂₃+𝑑₂₂𝑥₁₂−𝑥₂₂+𝑢₁₂,𝑥̇₂₃=−3𝑥₂₃+𝑥₂₁𝑥₂₂+𝑑₂₃𝑥₁₃−𝑥₂₃+𝑢₁₃,𝑥̇₃₁=−10𝑥₃₁+10𝑥₃₂+𝑑₃₁𝑥₂₁−𝑥₃₁+𝑢₂₁,𝑥̇₃₂=28𝑥₃₁−𝑥₃₂−𝑥₃₁𝑥₃₃+𝑑₃₂𝑥₂₂−𝑥₃₂+𝑢₂₂,𝑥̇₃₃8=−3𝑥₃₃+𝑥₃₁𝑥₃₂+𝑑₃₃𝑥₂₃−𝑥₃₃+𝑢₂₃, (22) where 𝐴₁=−35350−728000−3,𝐴₂=,𝐴−36360020000−3₃=8−1010028−1000−3,𝑔₁𝑥₁=0−𝑥₁₁𝑥₁₃𝑥₁₁𝑥₁₂,𝑔₂𝑥₂=0−𝑥₂₁𝑥₂₃𝑥₂₁𝑥₂₂,𝑔₃𝑥₃=0−𝑥₃₁𝑥₃₃𝑥₃₁𝑥₃₂, (23) 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 be 𝑒̇_𝑖−1=𝑥̇_𝑖−𝑥̇_𝑖−1, (𝑖=2,3). The error dynamical equations are ̇Δ𝑒=¹Δ₂Δ₃Δ₄𝑒+−𝑥₁₁−𝑥₁₂+𝑢₁₁7𝑥₁₁−8𝑥₁₂−𝑥₂₁𝑥₂₃+𝑥₁₁𝑥₁₃+𝑢₁₂𝑥₂₁𝑥₂₂−𝑥₁₁𝑥₁₂+𝑢₁₃26𝑥₂₁+26𝑥₂₂+𝑢₂₁−𝑢₁₁28𝑥₂₁−21𝑥₂₂−𝑥₃₁𝑥₃₃+𝑥₂₁𝑥₂₃+𝑢₂₂−𝑢₁₂13𝑥₂₃+𝑥₃₁𝑥₃₂−𝑥₂₁𝑥₂₂+𝑢₂₃−𝑢₁₃, (24) where Δ₁=−36−𝑑₁₁−𝑑₂₁360020−𝑑₁₂−𝑑₂₂000−3−𝑑₁₃−𝑑₂₃,Δ₂=−𝑑₁₁000−𝑑₁₂000−𝑑₁₃,Δ₃=𝑑₂₁000𝑑₂₂000𝑑₂₃,Δ₄=−10−𝑑₃₁10028−1−𝑑₃₂0800−3−𝑑₃₃. (25)

The controllers 𝑢₁ and 𝑢₂ are designed as 𝑢₁=𝑣₁−𝐴₂−𝐴₁𝑥₁−𝑔₂𝑥₂+𝑔₁𝑥₁,𝑢₂=𝑣₂+𝑣₁−𝐴₃−𝐴₁𝑥₁−𝑔₃𝑥₃+𝑔₁𝑥₁, (26) where 𝑣₁=𝑣000000−3600000000000𝑒,₂=𝑑₁₁−𝑑₂₁0000036𝑑₁₂−𝑑₂₂0−380000𝑑₁₃−𝑑₂₃000𝑒. (27)

The error systems (24) become ̇Δ𝑒=^∗1Δ^∗₂Δ^∗₃Δ^∗₄𝑒, (28) where Δ^∗₁=−36−𝑑₁₁−𝑑₂₁360−3620−𝑑₁₂−𝑑₂₂000−3−𝑑₁₃−𝑑₂₃,Δ^∗₂=−𝑑₁₁000−𝑑₁₂000−𝑑₁₃,Δ^∗₃=𝑑₁₁000𝑑₁₂000𝑑₁₃,Δ^∗₄=−10−𝑑₃₁100−10−1−𝑑₃₂0800−3−𝑑₃₃. (29)

From Theorem 1, we know that the conditions −36−𝑑₁₁−𝑑₂₁<0,20−𝑑₁₂−𝑑₂₂<0,−3−𝑑₁₃−𝑑₂₃<0,−10−𝑑₃₁<0,−1−𝑑₃₂8<0,−3−𝑑₃₃<0 (30) 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 Case 1, 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 variables 𝑒11, 𝑒12, and 𝑒13 of the error system 𝑒1 with time 𝑡.

Figure 6 

The state trajectories 𝑥11, 𝑥21, and 𝑥31 of the chaotic systems with time 𝑡.

Figure 7 

The state trajectories 𝑥12, 𝑥22, and 𝑥32 of the chaotic systems with time 𝑡.

Figure 8 

The state trajectories 𝑥13, 𝑥23, and 𝑥33 of the chaotic systems with time 𝑡.

4. Conclusions

This paper concerns the synchronization of 𝑁 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.

Acknowledgments

The author’s work was supported in part by the Applied Mathematics Enhancement Program (AMEP) of Linyi University and the National Natural Science Foundation of China, under Grants 61273012 and 61273218, by a project supported by the Scientific Research Fund of Zhejiang Provincial Education Department under Grant Y201223649, by a Project of Shandong Province Higher Educational Science and Technology Program under Grants J13LI11 and J12LI58.