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We propose a mathematical model of a complex dynamical network consisting of two types of chaotic oscillators and investigate the schemes and corresponding criteria for cluster synchronization. The global asymptotically stable criteria for the linearly or adaptively coupled network are derived to ensure that each group of oscillators is synchronized to the same behavior. The cluster synchronization can be guaranteed by increasing the inner coupling strength in each cluster or enhancing the external excitation. Theoretical analysis and numerical simulation results show that the external excitation is more conducive to the cluster synchronization. All of the results are proved rigorously. Finally, a network with a scale-free subnetwork and a small-world subnetwork is illustrated, and the corresponding numerical simulations verify the theoretical analysis.

Since the pioneering works by Watts and Strogatz on the small-world network [

Synchronization of coupled chaotic oscillators is one of commonly collective coherent behaviors attracting a growing interest in physics, biology, communication, and other fields of science and technology. Synchronization of complex networks has attracted tremendous attention in recent years. Different synchronization phenomena in complex networks have been studied, such as global synchronization [

More recently, some new progress in cluster synchronization of complex dynamical networks have been reported [

In this paper, based on [

The rest of the paper is organized as follows. In Section

Let’s consider a dynamical network with two clusters, each cluster contains a number of identical dynamical systems, however, the subsystems composing the two clusters can be different, that is, the individual dynamical system in one cluster can differ from that in the other cluster. Suppose that two clusters are composed of

Equivalently, (

A matrix

Note that, we suppose that all network models throughout this paper satisfy “same-input” condition. From this condition, we have

Denote

Then, we have

The set

Therefore, the synchronous errors are denoted as

Obviously, the stability problem of cluster synchronous manifold

In order to achieve cluster synchronization, a useful assumption and a lemma are introduced as follows.

Note that we assume that

For the above matrices

Since

Take transformation

Moreover,

Similarly, one can obtain

In this section, we propose linear coupling schemes to achieve the cluster synchronization, and derive the corresponding criteria for the coupling strength in the network (

For the linearly coupled network (

For the linearly coupled network (

Consider the function

Its time derivative along the trajectory of (

since

Similarly, one has

One has

By Lemma

Notice that

According to Theorem

For the following network:

In model (

Note that, the Lipschitz constants

For the network (

Consider the Lyapunov function

Its time derivative along the trajectory of (

By the LaSalle-Yoshizawa theorem [

Similarly, we can further obtain the following theorem.

For the network (

Consider the Lyapunov function

For the network (

In this section, illustrative examples are provided to verify the above theoretical analysis.

We consider a network which consists of two clusters with a scale-free sub-network with 50 Lorenz chaotic oscillators [

In the simulation, we take initial values

Figures

Cluster synchronization in the linearly coupled network (

The synchrony state

The synchrony states

Figures

Cluster synchronization of the linearly coupled network (

The synchrony state

The synchrony states

Cluster synchronization of the adaptively coupled network (

The synchrony state

The synchrony states

The estimation of coupling strength

The estimation of coupling strength

Cluster synchronization of the adaptively coupled network (

The synchrony state

The synchrony states

The estimation of coupling strength

The estimation of coupling strength

Figures

In this paper, we have further investigated the cluster synchronization of a complex dynamical network with given configuration which is connected by two groups of different oscillators. we present a linear coupling scheme and the corresponding sufficient condition is derived for the cluster synchronization. Moreover, an adaptive coupling scheme to lead the cluster synchronization is proposed based on adaptive control technique. Our study shows that the global stability of the cluster synchronization can be guaranteed by increasing coupling strength in each cluster or enhancing the external excitation even if there are no connections insider a cluster. Chaos synchronization of delay systems [

This work was jointly supported by the National Natural Science Foundation of China under Grants 61164020, 61004101, the Natural Science Foundation of Guangxi under Grant 2011GXNSFA018147, and Guangxi Key Laboratory of Spatial Information and Geomatics under Grant 1103108-24.