This paper investigates the adaptive cluster synchronization in the complex networks with different orders. By means of Lyapunov stability theory and the adaptive control technique, a novel adaptive synchronization controller is developed. To demonstrate the validity of the proposed method, the examples for the synchronization of systems with the chaotic and hyperchaotic node dynamics are presented.

In the past few years, complex networks have attracted more and more attention; examples of complex networks include the Internet, social networks, biological networks, organizational networks, neural networks, and many others [

A complex network can be seen as a large set of interconnected nodes and used to describe various systems with specific contents. Synchronization constitutes one of the most prevalent collective dynamics in complex networked systems. Until now, several types of synchronization have been investigated, such as phase synchronization and complete synchronization [

In general, there are two main factors devoting to network synchronization. One is the isolated node dynamics, and the other is the complicated network topology. In almost all the existing literature, a common assumption is that all nodes are identical. It should be noted that complex networks consisting of different nodes are ubiquitous in various fields. In the present paper, we focus on cluster synchronization where the local dynamics of the nodes in each group differ. Here by a group we mean a collection of systems that have the same dynamics, with any given group consisting of systems with dynamics that is different from the dynamics of systems in the other groups.

Specifically, in many technological, social, and biological networks, which can be divided naturally into several groups by certain rules, nodes in the same group often have the same type of function. The phenomenon of cluster synchronization is observed when an ensemble of oscillators splits into groups of synchronized elements. Up to now, much research effort has been devoted to investigating the cluster synchronization of complex dynamical networks [

The organization of this paper is organized as follows. Model of complex network with community structure is given in Section

Consider a dynamical network with

For better describing this kind of community network, a community network model with different orders is introduced. Figure

Topology structure of the complex networks with three communities.

A

Without loss of generality, the sets of subscripts of these clusters are

First, we will present some useful assumptions for deriving the main results.

Each block matrix

Suppose that there exist nonnegative constants

In the following, we introduce a scheme to achieve the cluster synchronization in colored network with community structure and adaptive coupling strengths. According to the above definition of the error variables and suppose that Assumption

Suppose that Assumption

Consider the following Lyapunov function:

Consider the community network shown in Figure

The node dynamics of the second community as chaotic Lorenz system

The node’s dynamic in the first community.

The node’s dynamic in the second community.

The node’s dynamic in the third community.

The node dynamics of the third community as hyperchaotic Lorenz system with

for nodes

In numerical simulations, choose feedback gains

The orbits of state variable and synchronization errors in the first community.

The orbits of state variable and synchronization errors in the second community.

The orbits of state variable and synchronization errors in the third community.

In this paper, synchronization of a community network with different order node dynamics is investigated. Both adaptive feedback control and stability theory are adopted to design proper controllers. Adaptive feedback controllers were designed for achieving cluster synchronization based on the Lyapunov stability theory. Finally, numerical examples were provided to illustrate the effectiveness of the theoretical results.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The work was partially supported by Tianshui Normal University “QingLan” Talent Engineering Funds.