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We investigate chaos control nodes of the complex network synchronization. The structure of the coupling functions between the connected nodes is obtained based on the chaos control method and Lyapunov stability theory. Moreover a complex network with nodes of the new unified Loren-Chen-Lü system, Coullet system, Chee-Lee system, and the New system is taken as an example; numerical simulations are used to verify the effectiveness of the method.

Since the most famous random graph model was proposed by Erdös and Rényi [

Recently, network synchronization has been an important part of the dynamic study of complex network. It has aroused great interest of scholars both domestically and abroad to build weighted network models and study the characteristics of them. Lü et al. [

The motivation in this paper lies in the complex network synchronization and chaos control importance. Network synchronization is one of the most practical and valuable issues. A synchronization of network means the situation in which the output of all nodes in the study of the complex network is consistent with any given external input signal under a certain condition. The working mechanism of a single chaotic system to track any given external input signal is relatively interesting and significance. Numerical simulations are used to verify the effectiveness of the proposed techniques.

It is organized as follows. Firstly, the theory and the method are presented in Section

We summarized the main steps for the complex network synchronization based on Lü et al. [

Assume the state of node

Considering the coupling of network, the dynamic function for node

The errors between the state variables of the network are defined as

Choose

Constructing the Lyapunov function according to the weighted complex network with different nodes,

The unified chaotic system, the New system, Coullet system, and Chee-Lee system are taken as nodes of the network to show the synchronization mechanism mentioned above. Simulation is made as the number of the nodes is

The dynamic equation of the unified chaotic system [

Generlized chaotic attractors: (a) Lorenz, (b) Lü, (c) Chen.

The new chaotic system of three-dimensional quadratic autonomous ordinary differential equations [

The chaos attractor of new chaotic system (a) and the Coullet system (b).

The dynamic equation of Coullet system [

The dynamic equation of new system [

The chaos attractor of (

The dynamic equation of the Chen-Lee system [

Choose the unified chaotic system (

Network synchronization errors between node 1 and node 2.

Network synchronization errors between node 2 and node 3.

Network synchronization errors between node 3 and node 4.

Network synchronization errors between node 4 and node 5.

For further details, the dynamic attractors for each node with coupling network synchronization are shown in Figures

The chaos attractors between node 1 and node 2 (a) and between node 2 and node 3 (b).

The chaos attractors between node 3 and node 4 (a) and between node 4 and node 5 (b).

The temporal evolution of variables between node 1 (“

The temporal evolution of variables between node 2 (“

The temporal evolution of variables between node 3 (“

The temporal evolution of variables between node 4 (“

In this paper, the complex network synchronization is investigated. With the help of symbolic computation, different order chaotic systems are adopted as the nodes of this complex network; the structure of the coupling functions among the connected nodes is obtained based on Lyapunov stability theory. Being of network and physical interests, the temporal evolution of variables and node interaction of the dynamic equation are discussed and simulated by computer. This method has universal significance for network synchronization, and the weight value of the coupling strength between the nodes and the number of the nodes does not affect synchronization of the whole network.

This work is supported by the NSF of China under Grant nos. 11172181, Guangdong Provincial NSF of China under Grant no. 10151200501000008 and 94512001002983, Guangdong Provincial UNYIF of China under Grant, and Science Foundation of Shaoguan University.