Global synchronization in adaptive coupling networks is studied in this paper. A new simple adaptive controller is proposed based on a concept of asymptotically stable led by partial state variables. Under the proposed adaptive update law, the network can achieve global synchronization without calculating the eigenvalues of the outer coupling matrix. The update law is only dependent on partial state variables of individual oscillators. Numerical simulations are given to show the effectiveness of the proposed method, in which the unified chaotic system is chosen as the nodes of the network with different topologies.

Synchronization in complex networks of identical chaotic oscillators has been studied extensively and deeply in various fields of science and engineering in the last few years [

In [

In [

Another active research named consensus or agreement considers the information exchange between the connected agents [

Consider an undirected complex dynamical network consisting of

The outer coupling matrix

An equilibrium point

The

Letting

Now we can drive the following main result.

Suppose that system (

Nonlinear function

Equilibrium point

The nonzero weight strength

Let the Lyapunov function be

Condition (I) is easily satisfied if

In practice, we can set

By (

Our synchronization strategy is setting adaptive coupling strengths between two nodes to achieve global synchronization, and our adaptive update law of

To demonstrate the theoretical result in Section

The unified chaotic [

We consider

For

If

If we set

According to Remark

We consider two types of regular network: star coupled network and ring coupled network. Note that our adaptive strategy is also valid for other type of complex networks.

Star coupled network

Figure

Synchronization errors

Weight strength

Ring coupled network

Figure

Synchronization errors

Weight strength

In this paper, we presented a new concept of asymptotically stable led by partial state variables and designed a simple adaptive controller for global synchronization in weight complex dynamical networks. For this network, we proved by using Lyapunov’s direct method that the states of such complex network can achieve global synchronization finally under our adaptive update law. Our synchronization strategy is to set adaptive coupling strength, and our adaptive update law of

The authors are grateful to the anonymous referees for constructive comments. This work is supported by the Foundation of Zhangzhou Normal University under Grants no. SK09017 and the Foundation for Supporting Universities in Fujian Province of China under Grant no. JK2009020.