The cluster synchronization of linearly coupled complex networks with identical and nonidentical nodes is studied. Without assuming symmetry, we proved that these linearly coupled complex networks could achieve cluster synchronization under certain pinning control schemes. Sufficient conditions guaranteeing cluster synchronization for any initial values are derived by using Lyapunov function methods. Moreover, the adaptive feedback algorithms are proposed to adjust the control strength. Several numerical examples are given to illustrate our theoretical results.

Recently, an increasing interest has been devoted to the study of complex networks. Among them, synchronization is the most interesting. In fact, synchronization of complex networks has been found to be a universal phenomenon in nature and has important potential applications in real-world dynamical systems. Great interests and attentions have been received for the synchronization of complex networks in many research and application fields including secure communication, seismology, parallel image processing, chemical reaction, and others [

There are many widely studied synchronization patterns, such as complete synchronization [

The complex network we considered in this paper is the linearly coupled ordinary differential equations (LCODEs). In fact, LCODEs are a large class of dynamical systems with continuous time and state, as well as discrete space, which are widely used to describe coupling oscillators. Nowadays, cluster synchronization of different kinds of LCODEs has been widely studied, and many results have already exist on the various properties of such problem. For instance, Ma et al. [

Since in the real world, many networks contain some different function communities and the local dynamics between two function communities are different. For instances, in metabolic, neural, or software community networks, the individual nodes in each community can be viewed as the identical functional units, whereas the nodes in different communities are different since they have different functions [

The paper is organized as follows. In Section

First, we introduce the mathematical definition of cluster synchronization.

Let

For convenience of the statement to our main results, we now make some definitions for a class of functions and a class of matrices.

Suppose that

For

For an asymmetric matrix with zero-row-sums, we have the following.

Let

The complex network we considered in this section can be described as

Without loss of generality, we set the partition of nodes

In this section, sufficient conditions are derived for the attainment of cluster synchronization for any initial value by control pinning, that is, by making

Suppose that the coupling matrices

We define

Choose a Lyapunov function as

Note that

If (

By using adaptive adjustments, we can find relatively small control strength to realize cluster synchronization. We regard the control strength of the network functions varying with time. Then, we could design the adaptive control strength. Then, we have the following result.

Suppose that the coupling matrix

Choose a Lyapunov function as

The complex network considered in this section is

Let

According to the above definition of error variables, we can write the corresponding error system with respect to (

Since

Let

Denote

Differentiating (

Noticing the inequalities of (

Thus, we get

In this section, we give numerical simulations to verify the theorems obtained in Section

In this simulation, we consider a network with 30 nodes and 3 communities. It is too high and we do not show it out. We show the topology structure in Figure

The controlled complex network (

Let

For the controlled dynamic network (

Figures

The controlled complex network (

For the controlled dynamic network (

Figure

The controlled complex network (

In this simulation, we consider a network with 30 nodes and 3 communities. It is too high and we do not show it out. We give the topology structure Figure

The controlled complex network (

Let

For the controlled dynamic network (

Figures

The controlled complex network (

In the paper, we have investigated the cluster synchronization on pinning control of LCODEs with identical and nonidentical nodes. We give a sufficient condition to make the complex network achieve cluster synchronization. Moreover, adaptive feedback control techniques are used to adjust control strength. Finally, some numerical examples are given, which is essential to verify our theoretical analysis.

The authors are grateful to the editors and the reviewers for their valuable suggestions and comments. This work was supported by Guangdong Education University Industry Cooperation Projects (2009B090300355), Shenzhen Basic Research Project (JC201006010743A), and the 2011 Foundation for Distinguished Young Talents in Higher Education of Guangdong (LYM11115).