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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 958405, 12 pages
http://dx.doi.org/10.1155/2012/958405
Research Article

Cluster Synchronization of Time-Varying Delays Coupled Complex Networks with Nonidentical Dynamical Nodes

1Department of Mathematics and Physics, Changzhou Campus, Hohai University, Jiangsu, Changzhou 213022, China
2Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 26 August 2011; Accepted 7 December 2011

Academic Editor: J. Biazar

Copyright © 2012 Shuguo Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper investigates a new cluster synchronization scheme in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.

1. Introduction

Complex networks model is used to describe various interconnected systems of real world, which have become a focal research topic and have drawn much attention from researchers working in different fields; one of the most important reasons is that most practical systems can be modeled by complex dynamical networks. Recently, the research on synchronization and dynamical behavior analysis of complex network systems has become a new and important direction in this field [113]; many control approaches have been developed to synchronize complex networks such as feedback control, adaptive control, pinning control, impulsive control, and intermittent control [1421].

Cluster synchronization means that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups [2225]; Belykh et al. [26] investigated systems of diffusively coupled identical chaotic oscillators; an effective method to determine the possible states of cluster synchronization and ensure their stability is presented. The method, which may find applications in communication engineering and other fields of science and technology, is illustrated through concrete examples of coupled biological cell models. Wu and Lu [27] investigated cluster synchronization in the adaptive complex dynamical networks with nonidentical nodes by a local control method and a novel adaptive strategy for the coupling strengths of the networks. Ma et al. [28] proposed cluster synchronization scheme via dominant intracouplings and common intercluster couplings. Sorrentino and Ott [29] studied local cluster synchronization for bipartite systems, where no intracluster couplings (driving scheme) exist. Chen and Lu [30] investigated global cluster synchronization in networks of two clusters with inter- and intracluster couplings. Belykh et al. [31, 26] studied this problem in 1D and 2D lattices of coupled identical dynamical systems. Lu et al. [32] studied the cluster synchronization of general networks with nonidentical clusters and derived sufficient conditions for achieving local cluster synchronization of networks. Recently, Wang et al. [33] considered the cluster synchronization of dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community by using pinning control schemes. However, there is few theoretical result on the cluster synchronization of nonlinear coupled complex networks with time-varying delays coupling and time-varying delays in nonidentical dynamical nodes.

Motivated by the above discussions, this paper investigates cluster synchronization in the nonlinear coupled complex dynamical networks with nonidentical nodes. The controllers are designed based on the community structure of the networks; some sufficient criteria are derived to ensure cluster synchronization in nonlinear coupled complex dynamical networks with time-varying delays coupling and time-varying delays in dynamic nodes. Particularly the weight configuration matrix is not assumed to be symmetric, irreducible.

The paper is organized as follows: the network model is introduced followed by some definitions, lemmas, and hypotheses in Section 2. The cluster synchronization of the complex coupled networks is discussed in Section 3. Simulations are obtained in Section 4. Finally, in Section 5 the various conclusions are discussed.

2. Model and Preliminaries

The network with nondelayed and time-varying delays coupling and adaptive coupling strengths can be described bẏ𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡),𝑖=1,2,,𝑁,(2.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),,𝑥𝑖𝑛(𝑡))𝑇𝑅𝑛 is the state vector of node 𝑖;𝑓𝜙𝑖𝑅𝑛𝑅𝑛 describes the local dynamics of nodes in the 𝜙𝑖th community. For any pair of nodes 𝑖 and 𝑗, if 𝜙𝑖𝜙𝑗, that is, nodes 𝑖 and 𝑗 belong to different communities, then 𝑓𝜙𝑖𝑓𝜙𝑗𝜂𝜙𝑖(𝑡),𝜏𝜙𝑖(𝑡), is a time-varying delay. 𝐻1() and 𝐻2() are nonlinear functions. 𝑐 is coupling strength. 𝐴=(𝑎𝑖𝑗)𝑁×𝑁,𝐵=(𝑏𝑖𝑗)𝑁×𝑁 are the weight configuration matrices. If there is a connection from node 𝑖 to node 𝑗(𝑗𝑖), then the 𝑎𝑖𝑗>0,𝑏𝑖𝑗>0 otherwise, 𝑎𝑖𝑗=𝑎𝑗𝑖=0,𝑏𝑖𝑗=𝑏𝑗𝑖=0, and the diagonal elements of matrix 𝐴,𝐵 are defined as 𝑎𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝑎𝑗𝑖,𝑏𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝑏𝑗𝑖,𝑖=1,2,,𝑁.(2.2) Particularly, the weight configuration matrix is not assumed to be symmetric, irreducible.

When the control inputs 𝑢𝑖(𝑡)𝑅𝑛 and 𝑣𝑖(𝑡)𝑅𝑛(𝑖=1,2,,𝑁) are introduced, the controlled dynamical network with respect to network (2.1) can be written aṡ𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,(2.3) where 𝐽𝜙𝑖 denotes all the nodes in the 𝜙𝑖th community and 𝐽𝜙𝑖 represents the nodes in the 𝜙𝑖th community which have direct links with the nodes in other communities.

The study presents the mathematical definition of the cluster synchronization.

Let {𝐶1,𝐶2,,𝐶𝑚} denote 𝑚(2𝑚𝑁) communities of the networks and 𝑚𝑖=1𝐶𝑖={1,2,𝑁}. If node 𝑖 belongs to the 𝑗th community, then we denote 𝜙𝑖=𝑗. We employ 𝑓𝑖() to represent the local dynamics of all nodes in the 𝑖th community. Let 𝑠𝑖(𝑡) be the solution of the system ̇𝑠𝑖(𝑡)=𝑓𝜙𝑖(𝑡,𝑠𝑖(𝑡),𝑠𝑖(𝑡𝜏𝜙𝑖(𝑡))),(𝑖=1,2,,𝑚) where lim𝑡𝑠𝑖(𝑡)𝑠𝑗(𝑡)0(𝑖𝑗); the set 𝑆={𝑠1(𝑡),𝑠2(𝑡),,𝑠𝑚(𝑡)} is used as the cluster synchronization manifold for network (2.3). Cluster synchronization can be realized if and only if the manifold 𝑆 is stable.

Definition 2.1 (see [19]). The error variables as 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)𝑠𝜙𝑖(𝑡) for 𝑖=1,2,,𝑁, where 𝑠𝜙𝑖(𝑡) satisfies ̇𝑠𝜙𝑖(𝑡)=𝑓𝜙𝑖(𝑡,𝑠𝜙𝑖(𝑡),𝑠𝜙𝑖(𝑡𝜏𝜙𝑖(𝑡))).

Definition 2.2 (see [19]). Let {1,2,,𝑁} be the 𝑁 nodes of the network and {𝐶1,𝐶2,,𝐶𝑚} be the 𝑚 communities, respectively. A network with 𝑚 communities is said to realize cluster synchronization if lim𝑡𝑒𝑖(𝑡)=0 and lim𝑡𝑥𝑖(𝑡)𝑥𝑗(𝑡)0 for 𝜙𝑖𝜙𝑗.

Lemma 2.3. For any two vectors 𝑥 and 𝑦, a matrix Q>0 with compatible dimensions, one has 2𝑥𝑇𝑦𝑥𝑇𝑄𝑥+𝑦𝑇𝑄1𝑦.

Assumption 2.4. For the vector valued function 𝑓𝜙𝑖(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏𝜙𝑖)), assuming that there exist positive constants 𝛼𝜙𝑖>0,𝛾𝜙𝑖>0 such that f satisfies the semi-Lipschitz condition 𝑥𝑖(𝑡)𝑦𝑖(𝑡)𝑇𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖𝑓𝜙𝑖𝑡,𝑦𝑖(𝑡),𝑦𝑖𝑡𝜏𝜙𝑖𝛼𝜙𝑖𝑥𝑖(𝑡)𝑦𝑖(𝑡)𝑇𝑥𝑖(𝑡)𝑦𝑖(𝑡)+𝛾𝜙𝑖𝑥𝑖𝑡𝜏𝜙𝑖𝑦𝑖𝑡𝜏𝜙𝑖𝑇𝑥𝑖𝑡𝜏𝜙𝑖𝑦𝑖𝑡𝜏𝜙𝑖,(2.4) for all 𝑥,𝑦𝑅𝑛 and 𝜏𝜙𝑖(𝑡)0.𝑖=1,2,,𝑁.

Assumption 2.5. 𝜂𝜙𝑖(𝑡) and 𝜏𝜙𝑖(𝑡) is a differential function with 0̇𝜂𝜙𝑖(𝑡)𝜀1 and .𝜏0𝜙𝑖(𝑡)𝜀1. Clearly, this assumption is certainly ensured if the delay 𝜂𝜙𝑖(𝑡) and 𝜏𝜙𝑖(𝑡) is constant.

Assumption 2.6 (34) (Global Lipschitz Condition). Suppose that there exist nonnegative constants 𝜗,𝛽, for all 𝑡𝑅+, such that for any time-varying vectors 𝑥(𝑡),𝑦(𝑡)𝑅𝑛𝐻1(𝑥)𝐻1𝐻(𝑦)𝜗𝑥𝑦,2(𝑥)𝐻2(𝑦)𝛽𝑥𝑦,(2.5) where denotes the 2-norm throughout the paper.

3. Main Results

In this section, a control scheme is developed to synchronize a delayed complex network with nonidentical nodes to any smooth dynamics 𝑠𝜙𝑖(𝑡). Let synchronization errors 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)𝑠𝜙𝑖(𝑡) for 𝑖=1,2,,𝑁, according to system (2.1), the error dynamical system can be derived aṡ𝑒𝑖𝑓(𝑡)=𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)𝐻1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖+(𝑡)𝑁𝑖=1𝑎𝑖𝑗𝐻1𝑠𝜙𝑖+(𝑡)𝑁𝑖=1𝑏𝑖𝑗𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,̇𝑒𝑖𝑓(𝑡)=𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)𝐻1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,(3.1) where 𝑓𝜙𝑖(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏𝜙𝑖(𝑡)))=𝑓𝜙𝑖(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏𝜙𝑖(𝑡)))𝑓𝜙𝑖(𝑡,𝑠𝜙𝑖(𝑡),𝑠𝜙𝑖(𝑡𝜏𝜙𝑖(𝑡))) for 𝑖=1,2,,𝑁.

According to the diffusive coupling condition (2.2) of the matrix 𝐴,𝐵 we have𝑐𝑁𝑖=1𝑎𝑖𝑗𝐻1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑖=1𝑏𝑖𝑗𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)=0,𝑖𝐽𝜙𝑖𝐽𝜙𝑖.(3.2) On the basis of this property, for achieving cluster synchronization, we design controllers as follows:𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑎𝑖𝑗𝐻1𝑠𝜙𝑖(𝑡)𝑐𝑁𝑖=1𝑏𝑖𝑗𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖,(3.3) where ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡).

Theorem 3.1. Suppose assumptions 2.42.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold: 𝛼+𝜗𝑐𝜆max1(𝑄)+2𝛽2𝑐2𝜆max𝑃𝑃𝑇+111𝜀𝛾+2<𝑑,(3.4) where 𝛼=max(𝛼𝜙1,𝛼𝜙2,,𝛼𝜙𝑚),𝛾=max(𝛾𝜙1,𝛾𝜙2,,𝛾𝜙𝑚). Then, the systems (2.3) is cluster synchronization.

Proof. Construct the following Lyapunov functional: 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝛾(𝑡)+1𝜀𝑡𝑡𝜏𝜙𝑖𝑁(𝑡)𝑖=1𝑒𝑇𝑖(𝜃)𝑒𝑖+1(𝜃)𝑑𝜃2(1𝜀)𝑡𝑡𝜂𝜙𝑖𝑁(𝑡)𝑖=1𝑒𝑇𝑖(𝜃)𝑒𝑖1(𝜃)𝑑𝜃+2𝑁𝑖=1𝑑𝑖𝑑2𝑘𝑖.(3.5) Calculating the derivative of 𝑉(𝑡), we have ̇𝑉(𝑡)=𝑁𝑖=1𝑒𝑇𝑖(𝑡)̇𝑒𝑖1(𝑡)+11𝜀𝛾+2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝛾.𝜏(𝑡)1𝜙𝑖(𝑡)1𝜀𝑁𝑖=1𝑒𝑇𝑖𝑡𝜏𝜙𝑖𝑒(𝑡)𝑖𝑡𝜏𝜙𝑖(𝑡)1̇𝜂𝜙𝑖(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖+(𝑡)𝑁𝑖=1𝑑𝑖𝑒𝑑𝑇𝑖(𝑡)𝑒𝑖=(𝑡)𝑁𝑖=1𝑒𝑇𝑖𝑓(𝑡)𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)𝐻1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖+1(𝑡)11𝜀𝛾+2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝛾.𝜏(𝑡)1𝜙𝑖(𝑡)1𝜀𝑁𝑖=1𝑒𝑇𝑖𝑡𝜏𝜙𝑖𝑒(𝑡)𝑖𝑡𝜏𝜙𝑖(𝑡)1̇𝜂𝜙𝑖(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖+(𝑡)𝑁𝑖=1𝑑𝑖𝑒𝑑𝑇𝑖(𝑡)𝑒𝑖(𝑡).(3.6) By assumptions 2.42.6, we obtain 𝛼𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)+𝛾𝑁𝑖=1𝑒𝑇𝑖𝑡𝜏𝜙𝑖𝑒(𝑡)𝑖𝑡𝜏𝜙𝑖(𝑡)+𝜗𝑐𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝛽𝑐𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖+1(𝑡)11𝜀𝛾+2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)𝛾𝑁𝑖=1𝑒𝑇𝑖𝑡𝜏𝜙𝑖𝑒(𝑡)𝑖𝑡𝜏𝜙𝑖1(𝑡)2𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖+(𝑡)𝑁𝑖=1𝑑𝑖𝑒𝑑𝑇𝑖(𝑡)𝑒𝑖(𝑡)𝛼𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)+𝜗𝑐𝑒𝑇(𝐴𝐼)𝑒+𝛽𝑐𝑒𝑇(𝐵𝐼)𝑒𝑡𝜂𝜙𝑖+1(𝑡)11𝜀𝛾+2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖1(𝑡)2𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑒𝑇(𝑡)𝑒(𝑡).(3.7) Let 𝑒(𝑡)=(𝑒𝑇1(𝑡),𝑒𝑇2(𝑡),,𝑒𝑇𝑁(𝑡))𝑇𝑅𝑛𝑁,𝑄=(𝐴𝐼),𝑃=(𝐵𝐼), where represents the Kronecker product. Then ̇𝑉(𝑡)𝛼𝑒𝑇(𝑡)𝑒(𝑡)+𝜗𝑐𝑒𝑇(𝑡)𝑄𝑒(𝑡)+𝛽𝑐𝑒𝑇(𝑡)𝑃𝑒𝑡𝜂𝜙𝑖+1(𝑡)11𝜀𝛾+2𝑒𝑇1(𝑡)𝑒(𝑡)2𝑒𝑇𝑡𝜂𝜙𝑖𝑒(𝑡)𝑡𝜂𝜙𝑖(𝑡)𝑑𝑒𝑇(𝑡)𝑒(𝑡).(3.8) By the Lemma 2.3, we have 𝛼𝑒𝑇(𝑡)𝑒(𝑡)+𝜗𝑐𝑒𝑇1(𝑡)𝑄𝑒(𝑡)+2(𝛽𝑐)2𝑒𝑇(𝑡)𝑃𝑃𝑇1𝑒(𝑡)+11𝜀𝛾+2𝑒𝑇(𝑡)𝑒(𝑡)𝑑𝑒𝑇(𝑡)𝑒(𝑡)𝛼+𝜗𝑐𝜆max1(𝑄)+2𝛽2𝑐2𝜆max𝑃𝑃𝑇+111𝜀𝛾+2𝑒𝑑𝑇(𝑡)𝑒(𝑡).(3.9) Therefore, if we have 𝛼+𝜗𝑐𝜆max(𝑄)+(1/2)𝛽2𝑐2𝜆max(𝑃𝑃𝑇)+(1/(1𝜀))(𝛾+(1/2))<𝑑 then ̇𝑉(𝑡)0.(3.10) Theorem 3.1 is proved completely.
We can conclude that, for any initial values, the solutions 𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑁(𝑡) of the system (2.3) satisfy lim𝑡𝑚𝑘=1𝑖𝐶𝑘𝑥𝑖(𝑡)𝑠𝑘(𝑡)=0, that is, we get the global stability of the cluster synchronization manifold 𝑆. Therefore, cluster synchronization in the network (2.3) is achieved under the local controllers (3.3). This completes the proof.

Corollary 3.2. When 𝐴=0, network (2.1) is translated into ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡),𝑖=1,2,,𝑁.(3.11)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑏𝑖𝑗𝐻2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖.(3.12)

Corollary 3.3. When 𝐵=0, network (2.1) is translated into ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡),𝑖=1,2,,𝑁.(3.13)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑎𝑖𝑗𝐻1𝑠𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖,(3.14)

4. Illustrative Examples

In this section, a numerical example will be given to demonstrate the validity of the synchronization criteria obtained in the previous sections. Considering the following network:̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡),𝑥𝑖𝑡𝜏𝜙𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝐻1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝐻2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,𝑖=1,2,,𝑁,(4.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),𝑥𝑖3(𝑡))𝑇,𝑓1(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏1(𝑡)))=𝐷1𝑥𝑖(𝑡)+11(𝑥𝑖(𝑡))+12(𝑥𝑖(𝑡𝜏1(𝑡))),𝑓2(𝑡,𝑥𝑖(𝑡),𝑥𝑖𝜏(𝑡2(𝑡)))=𝐷2𝑥𝑖(𝑡)+21(𝑥𝑖(𝑡))+22(𝑥𝑖(𝑡𝜏2(𝑡)))+𝑉,𝑓3(𝑡,𝑥𝑖(𝑡),𝑥𝑖(𝑡𝜏3(𝑡)))=𝐷3𝑥𝑖(𝑡)+31(𝑥𝑖(𝑡))+32(𝑥𝑖𝜏(𝑡3(𝑡))).𝑘1=𝑘2==𝑘𝑁=10,𝑐=1,𝐻1(𝑥)=sin𝑥,𝐻2(𝑥)=cos𝑥.

In simulation, we choose 11(𝑥𝑖)=(0,𝑥𝑖1𝑥𝑖3,𝑥𝑖1𝑥𝑖2)𝑇,12(𝑥𝑖)=(0,5𝑥𝑖2,0)𝑇,21(𝑥𝑖)=(0,0,𝑥𝑖1𝑥𝑖3)𝑇,22(𝑥𝑖)=(𝑥𝑖1,0,0)𝑇,𝑉=[0,0,0.2]𝑇,31(𝑥𝑖)=(3.247(|𝑥𝑖1+1||𝑥𝑖11|),0,0)𝑇,32(𝑥𝑖)=(0,0,3.906sin(0.5𝑥𝑖1))𝑇,𝜏1(𝑡)=𝑒𝑡𝑒/(1+𝑡),𝜏2(𝑡)=2𝑒𝑡/(1+𝑒𝑡),𝜏3(𝑡)=0.5𝑒𝑡/(1+𝑒𝑡),𝐷1=8101002840003,𝐷2=01110.20001.2,𝐷3=2.169100111019.530.1636.(4.2) Taking the weight configuration coupling matrices𝐴=𝐵=210001121000012100001210000121100012.(4.3) The following quantities are utilized to measure the process of cluster synchronization𝐸(𝑡)=𝑁𝑖=1𝑥𝑖(𝑡)𝑠𝜙𝑖,𝐸(𝑡)12𝑥(𝑡)=𝑢(𝑡)𝑥𝑣(𝑡),𝑢𝐶1,𝑣𝐶2,𝐸13𝑥(𝑡)=𝑢(𝑡)𝑥𝑣(𝑡),𝑢𝐶1,𝑣𝐶3,𝐸23(𝑥𝑡)=𝑢(𝑡)𝑥𝑣(𝑡),𝑢𝐶2,𝑣𝐶3,(4.4) where 𝐸(𝑡) is the error of cluster synchronization for this controlled network (2.2); 𝐸12(𝑡), 𝐸13(𝑡), and 𝐸23(𝑡) are the errors between two communities; cluster synchronization is achieved if the synchronization error 𝐸(𝑡) converges to zero and 𝐸12(𝑡),𝐸13(𝑡) and 𝐸23(𝑡) do not as 𝑡. Simulation results are given in Figures 1, 2, 3, and 4. From the Figures 14, we see the time evolution of the synchronization errors. The numerical results show that Theorem 3.1 is effective.

958405.fig.001
Figure 1: Time evolution of the synchronization errors 𝐸(𝑡).
958405.fig.002
Figure 2: Time evolution of the synchronization errors 𝐸12(𝑡).
958405.fig.003
Figure 3: Time evolution of the synchronization errors 𝐸13(𝑡).
958405.fig.004
Figure 4: Time evolution of the synchronization errors 𝐸23(𝑡).

5. Conclusions

The problems of cluster synchronization and adaptive feedback controller for the nonlinear coupled complex networks are investigated. The weight configuration matrix is not assumed to be symmetric, irreducible. It is shown that cluster synchronization can be realized via adaptive feedback controller. The study showed that the use of simple control law helps to derive sufficient criteria which ensure that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups is derived. Particularly the synchronization criteria are independent of time delay. The developed techniques are applied three complex community networks which are synchronized to different chaotic trajectories. Finally, the numerical simulations were performed to verify the effectiveness of the theoretical results.

Acknowledgments

This research is partially supported by the National Nature Science Foundation of China (no. 70871056) and by the Six Talents Peak Foundation of Jiangsu Province.

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