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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 534743, 17 pages
http://dx.doi.org/10.1155/2012/534743
Research Article

Cluster Synchronization for Linearly Coupled Complex Networks with Identical and Nonidentical Nodes

College of Mathematics and Computational Science, Shenzhen University, Guangdong Shenzhen 518060, China

Received 11 April 2012; Accepted 7 July 2012

Academic Editor: Hai L. Liu

Copyright © 2012 Yi Zhao 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

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.

1. Introduction

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 [17].

There are many widely studied synchronization patterns, such as complete synchronization [8], lag synchronization [9], cluster synchronization [10], phase synchronization [11], and partial synchronization [12] Among them, the studies on cluster synchronization have received more and more attentions. The cluster synchronization requires that the coupled oscillators split into subgroups called clusters, such that the oscillators synchronize with one another in the same cluster, but there is no synchronization among different clusters, which could describe the behaviors of the complex network in the real world. For instance, the metabolic, neural, or software networks containing some different function communities. Thus, it is a natural idea to consider the cluster synchronization of such community networks.

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. [13] constructed a novel coupling scheme with cooperative and competitive weight couplings that guarantees the cluster synchronization of any connected networks with identical nodes. The authors also derived a sufficient condition for the global stability of cluster synchronization. In [14], Wu et al. have discussed the problem of driving linearly coupled networks to an arbitrarily selected cluster synchronization pattern via pinning control. They introduced a single negative feedback controller for each cluster to pin the coupled system to the assigned cluster synchronization pattern for any initial values. However, in most cases, couplings between nodes are not the same even if the diffusive condition is still satisfied. Nodes usually receive instantaneous information as well as delayed information from their neighbors. Thus, it is a nature idea to study the synchronization of the networks with both delayed and nondelayed coupling. In this paper, we would investigate cluster synchronization of LCODEs with both delayed and nondelayed coupling under pinning control scheme. First, we assume all the node in LCODEs are identical. By utilizing the Lyapunov stability method, the global stability of cluster synchronization in networks is investigated, and several sufficient conditions for the global stability are given. Furthermore, we propose an adaptive feedback algorithms to adjust the control strength for LCODEs.

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 [15]. One method to solve such problem is to consider the cluster synchronization of community networks with nonidentical nodes.There have been already some papers focused on sufficient conditions for the global stability of cluster synchronization of some related networks. A number of sufficient conditions were similarly obtained by Lu et al. [16] for the cluster synchronization of dynamical networks with community structure and nonidentical nodes in the presence of time delays by using a certain feedback control scheme. Lu et al. [17] studied the cluster synchronization of general bi-directed networks with nonidentical clusters and derived sufficient conditions for achieving local cluster synchronization of networks. The authors also discovered a relationship between the cluster synchronizability of a network and its intra-to-intercluster link ratio with the help of numerical examples. Recently, Wang et al. [18] 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. In this paper, we also investigate cluster synchronization of LCODEs with nonidentical nodes under pinning control scheme. By utilizing the Lyapunov-Krasovskii stability method, the global stability of cluster synchronization in networks is investigated, and several sufficient conditions for the global stability are given. Compared with [1618], the complex network model we considered in this paper is more general. And moreover, the coupling matrices with and without time delay are asymmetric.

The paper is organized as follows. In Section 2, some necessary and useful definitions and lemmas are given. In Section 3, we study the global cluster synchronization of LCODEs with identical nodes and give a sufficient condition for it. Then, the adaptive feedback algorithms on control strength are proposed to achieve cluster synchronization in the complex network. In Section 4 cluster synchronization of LCODEs with nonidentical nodes is investigated and sufficient conditions are derived to achieve cluster synchronization. And the adaptive feedback algorithms on control strength are also proposed. In Section 5, numerical simulation are presented. We conclude the paper in Section 6.

2. Preliminaries

First, we introduce the mathematical definition of cluster synchronization.

Definition 2.1 (see [14]). Let {𝑈1,,𝑈𝑚} be a partition of the set {1,2,,𝑁} into 𝑑 nonempty subsets, that is, 𝑈𝑙𝜙 and 𝑚𝑙=1={1,2,,𝑁}. For 𝑖{1,2,,𝑁}, let ̂𝑖 denote the subscript of the subset in which the number 𝑖 is, that is, ̂𝑖𝑖𝑈. A network with 𝑁 identical oscillators is said to realize 𝑚-cluster synchronization with the partition {𝑈1,,𝑈𝑚} if, for any initial values, the state variables of the oscillators satisfy lim𝑡||𝑥𝑖(𝑡)𝑥𝑗(𝑡)||=0 for ̂̂𝑗𝑖= and lim𝑡||𝑥𝑖(𝑡)𝑥𝑗(𝑡)||0 for ̂̂𝑗𝑖.
For convenience of the statement to our main results, we now make some definitions for a class of functions and a class of matrices.

Definition 2.2 (see [1921]). Suppose that 𝑓(𝑥,𝑡) is a class of continuous functions 𝑓𝑅𝑛×[0,)𝑅𝑛. Let 𝑃=diag{𝑝1,𝑝2,,𝑝𝑛} be a positive definite diagonal matrix and Δ=diag{𝛿1,𝛿2,,𝛿𝑛} be a diagonal matrix. 𝑓(𝑥,𝑡)QUAD(𝑃,Δ,𝜂) if and only if (𝑥𝑦)𝑇𝑃((𝑓(𝑥,𝑡)𝑓(𝑦,𝑡))Δ(𝑥𝑦))𝜂(𝑥𝑦)𝑇(𝑥𝑦),(2.1) holds for some 𝜂>0, where 𝑥,𝑦𝑅𝑛 and 𝑡>0.

Definition 2.3 (see [14]). For 𝑁×𝑁 matrix 𝐴𝐴=11𝐴11𝐴1𝑑𝐴21𝐴22𝐴2𝑑𝐴𝑑1𝐴𝑑2𝐴𝑑𝑑,(2.2) where 𝐴𝑢𝑣𝑅𝑘𝑢×𝑘𝑣, 𝑢,𝑣=1,2,,𝑑. If each block 𝐴𝑢𝑣 is a zero-row-sum matrix, then we say that 𝐴𝐌𝟏(𝐤).
For an asymmetric matrix with zero-row-sums, we have the following.

Lemma 2.4 (see [19]). Let 𝑄 and 𝑅 be two symmetric matrices, and matrix 𝑆 has suitable dimension. Then 𝑆𝑄𝑆𝑇𝑅<0,(2.3) if and only if both 𝑅<0 and 𝑄𝑆𝑅1𝑆𝑇<0.

3. Cluster Synchronization of LCODEs with Identical Nodes

The complex network we considered in this section can be described as ̇𝑥𝑖𝑥(𝑡)=𝑓𝑖(𝑡),𝑡+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗(𝑡𝜏),𝑖=1,2,,𝑁,(3.1) where 𝑁 is the networks size, 𝑥𝑖=(𝑥1𝑖,𝑥2𝑖,,𝑥𝑛𝑖)𝑇𝑅𝑛 is the state vector of the 𝑖th oscillator, 𝑓𝑅𝑛×[0,)𝑅𝑛 is a continues map, and 𝑐>0 is the coupling strength. 𝜏 is the time delay. 𝐴=(𝑎𝑖𝑗)𝑅𝑛×𝑛 and 𝐵=(𝑏𝑖𝑗)𝑅𝑛×𝑛 are the coupling configuration matrix with zero-sum rows. It represents the topological structure of the network, in which 𝑎𝑖𝑗>0 if there is a connection from node 𝑗 to node 𝑖 (𝑖𝑗) and is zero otherwise. 𝐵 has the same properties. Here, 𝐴 and 𝐵 need not be symmetric since asymmetric topological structures are most common in the real world.

Without loss of generality, we set the partition of nodes 𝑈1={1,2,,𝑘1}, 𝑈2={𝑘1+1,,𝑘1+𝑘2}, , 𝑈𝑚={𝑘1++𝑘𝑚1+1,,𝑘1++𝑘𝑚1+𝑘𝑚}, where 1<𝑚<𝑁, 1<𝑘𝑙<𝑁 and 𝑚𝑙=1𝑘𝑙=𝑁. Let 𝑠1(𝑡),,𝑠𝑚(𝑡) be the 𝑚 special solutions of the homogenous system ̇𝑠(𝑡)=𝑓(𝑠(𝑡),𝑡), which satisfy lim𝑡||𝑠𝑖(𝑡)𝑠𝑗(𝑡)||0 for 𝑖𝑗. By using similar pinning control method in [14], we let the controlled oscillator set 𝐽 be 𝐽={𝑘1,𝑘1+𝑘2,,𝑘1+𝑘2++𝑘𝑚}, which means we only put the control on the last node of each partition 𝑈𝑙. We use linear negative feedback controllers and the LCODEs (3.1) become ̇𝑥𝑖𝑥(𝑡)=𝑓𝑖(𝑡),𝑡+𝑐𝑚𝑗=1𝑎𝑖𝑗𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗(𝑡𝜏)𝜉𝑖𝑥𝑖(𝑡)𝑠𝑖(𝑡),𝑖𝐽,̇𝑥𝑖𝑥(𝑡)=𝑓𝑖(𝑡),𝑡+𝑐𝑚𝑗=1𝑎𝑖𝑗𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗(𝑡𝜏),𝑖𝐽,(3.2) where 𝜉𝑢>0 with 𝑢𝐽 are the control strengths.

In this section, sufficient conditions are derived for the attainment of cluster synchronization for any initial value by control pinning, that is, by making lim𝑡||||||𝑥𝑖̂𝑖||||||(𝑡)𝑠(𝑡)=0for𝑖=1,2,,𝑁.(3.3)

Theorem 3.1. Suppose that the coupling matrices 𝐴 and 𝐵 in (3.2) satisfy 𝐴𝐌𝟏(𝐤) and 𝐵𝐌𝟏(𝐤). Let 𝑃=diag{𝑝1,𝑝2,,𝑝𝑛} be a positive definite diagonal matrix and Δ=diag{𝛿1,𝛿2,,𝛿𝑛} be a diagonal matrix such that 𝑓(𝑥,𝑡)QUAD(𝑃,Δ,𝜂). Define 𝐴𝑆=(𝐴+𝐴𝑇)/2. For 𝑘=1,2,,𝑛, if Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝑝𝑘𝑐Ξ+2𝑝4Λ2𝑘𝐵𝐵𝑇<0,(3.4) where ̃𝜉Ξ=diag{1,̃𝜉2̃𝜉,,𝑁} satisfies ̃𝜉𝑖=𝜉𝑖,𝑖𝐽0,𝑖𝐽.(3.5) Then, for any initial values, the solution 𝑥1(𝑡), 𝑥2(𝑡), , 𝑥𝑁(𝑡) of the system (3.1) under the control (3.2) can achieve cluster synchronization and satisfies (3.3).

Proof. We define 𝑒𝑖(𝑡)=𝑥𝑖̂𝑖(𝑡)𝑠(𝑡), where 𝑖=1,2,,𝑁. Denote 𝑒𝑘(𝑡)=(𝑒𝑘1(𝑡),𝑒𝑘2(𝑡),,𝑒𝑘𝑁(𝑡))𝑇 for 𝑘=1,2,𝑛. Since 𝐴𝐌𝟏(𝐤), which means that 𝑖𝑈𝑙𝑎𝑖𝑗=0 hold for all 𝑖=1,2,,𝑁 and 𝑙=1,2,,𝑚. It is readily seen that 𝑁𝑗=1𝑎𝑖𝑗𝑥𝑗=𝑚𝑙=1𝑗𝑈𝑙𝑎𝑖𝑗𝑥𝑗=𝑚𝑙=1𝑗𝑈𝑙𝑎𝑖𝑗𝑥𝑗𝑠𝑙+𝑠𝑙=𝑚𝑙=1𝑗𝑈𝑙𝑎𝑖𝑗𝑒𝑗+𝑚𝑙=1𝑗𝑈𝑙𝑎𝑖𝑗𝑠𝑙=𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗.(3.6) And similarly, 𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗=𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗. Thus, 𝑒𝑖(𝑡) for 𝑖=1,𝑁 satisfies the following differential equation: ̇𝑒𝑖𝑥(𝑡)=𝑓𝑖𝑠̂𝑖(𝑡),𝑡𝑓(𝑡),𝑡+𝑐𝑚𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝜉𝑖𝑒𝑖(𝑡),𝑖𝐽,̇𝑒𝑖𝑥(𝑡)=𝑓𝑖𝑠̂𝑖(𝑡),𝑡𝑓(𝑡),𝑡+𝑐𝑚𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏),𝑖𝐽,(3.7)
Choose a Lyapunov function as 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+2Λ𝑡𝑡𝜏𝑒𝑖(𝜃)𝑇𝑒𝑖.(𝜃)𝑑𝜃(3.8)
Note that 𝑓(𝑥,𝑡)QUAD(𝑃,Δ,𝜂). We differentiate (3.8) along (3.7) and have 𝑑𝑉=𝑑𝑡𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃̇𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏)𝑒𝑖=(𝑡𝜏)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝑥𝑖𝑠̂𝑖(𝑡),𝑡𝑓(𝑡),𝑡+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝑖𝐽𝑒𝑖(𝑡)𝑇𝑃𝜉𝑖𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏)𝑒𝑖(𝑡𝜏)𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃Δ𝑒𝑖(𝑡)+𝑐𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝑖𝐽𝜉𝑖𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏)𝑒𝑖(𝑡𝜏)=𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑝𝑘𝛿𝑘𝑒𝑘(𝑡)𝑇𝑒𝑘(𝑡)+𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐴𝑆𝑒𝑘(𝑡)𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇Ξ𝑒𝑘(𝑡)+𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐵𝑒𝑘(𝑡𝜏)+Λ𝑛𝑘=1𝑒𝑘(𝑡)𝑇𝑒𝑘(𝑡)Λ𝑛𝑖=1𝑒𝑘(𝑡𝜏)𝑇𝑒𝑘(𝑡𝜏)=𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑒𝑘(𝑡)𝑇,𝑒𝑘(𝑡𝜏)𝑇×Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝑝𝑘Ξ𝑐2𝑝𝑘𝐵𝑐2𝑝𝑘𝐵𝑇𝑒Λ𝐼𝑘(𝑡),𝑒𝑘,(𝑡𝜏)(3.9) where Ξ is defined in (3.5).
If (3.4) is satisfied, from Lemma 2.4, it can be easily seen that 𝑑𝑉/𝑑𝑡<0, and thus (3.2) could achieve cluster synchronization.

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.

Theorem 3.2. Suppose that the coupling matrix 𝐴 and 𝐵 in (3.2) satisfy 𝐴𝐌𝟏(𝐤) and 𝐵𝐌𝟏(𝐤). The control strength 𝜉𝑖 in (3.2) is defined as ̇𝜉𝑖(𝑡)=𝛾𝑖𝑒𝑇𝑖(𝑡)𝑃𝑒𝑖(𝑡)for𝑖𝐽,(3.10) where 𝛾𝑖 is a positive constant. Let 𝑃=diag{𝑝1,𝑝2,,𝑝𝑛} be a positive definite diagonal matrix and Δ=diag{𝛿1,𝛿2,,𝛿𝑛} be a diagonal matrix such that 𝑓(𝑥,𝑡)QUAD(𝑃,Δ,𝜂). For 𝑘=1,2,,𝑛, if Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝑝𝑘Ξ+𝑐2𝑝4Λ2𝑘𝐵𝐵𝑇<0,(3.11) where Ξ=diag{𝜉1,𝜉2,,𝜉𝑁}. 𝜉𝑖 is a constant and satisfies 𝜉𝑖=0 if 𝑖𝐽. Then, for any initial values, the solution 𝑥1(𝑡), 𝑥2(𝑡), , 𝑥𝑁(𝑡) of the system (3.1) under the control (3.2) can achieve cluster synchronization and satisfies (3.3).

Proof. Choose a Lyapunov function as 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+2Λ𝑡𝑡𝜏𝑒𝑖(𝜃)𝑇𝑒𝑖+(𝜃)𝑑𝜃𝑖𝐽𝜉𝑖𝜉𝑖22𝛾𝑖.(3.12) Note that 𝑓(𝑥,𝑡)QUAD(𝑃,Δ,𝜂). We differentiate (3.12) along (3.7) and have 𝑑𝑉=𝑑𝑡𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃̇𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏)𝑒𝑖+(𝑡𝜏)𝑖𝐽𝜉𝑖𝜉𝑖𝑒𝑇𝑖(𝑡)𝑃𝑒𝑖=(𝑡)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝑥𝑖𝑠̂𝑖(𝑡),𝑡𝑓(𝑡),𝑡+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝑖𝐽𝑒𝑖(𝑡)𝑇𝑃𝜉𝑖𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑇𝑖(𝑡𝜏)𝑒𝑖(𝑡𝜏).(3.13) By using similar calculations in (3.9), we have 𝑑𝑉𝑑𝑡𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑒𝑘(𝑡)𝑇,𝑒𝑘(𝑡𝜏)𝑇×Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝑝𝑘Ξ𝑐2𝑝𝑘𝐵𝑐2𝑝𝑘𝐵𝑇𝑒Λ𝐼𝑘(𝑡),𝑒𝑘.(𝑡𝜏)(3.14) Noticing the inequalities in (3.11), we could obtain 𝑑𝑉(𝑡)/𝑑𝑡<0. Thus, we get 𝑥𝑖̂𝑖(𝑡)𝑠(𝑡), and ̇𝜉𝑖(𝑡)0. By Cauchy convergence principle, 𝜉𝑖(𝑡) converges.

4. Cluster Synchronization of LCODEs with Nonidentical Nodes

The complex network considered in this section is ̇𝑥𝑖(𝑡)=𝑓𝜇𝑖𝑥𝑖(𝑡),𝑡+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗(𝑡𝜏),𝑖=1,2,,𝑁,(4.1) where 𝑥𝑖=(𝑥1𝑖,𝑥2𝑖,,𝑥𝑛𝑖)𝑇𝑅𝑛 are the state variables of node 𝑖. The complex network (4.1) has 𝑁 nodes and 𝑚 communities with 𝑁>𝑚2. If node 𝑖 belongs to the 𝑗th community, then we let 𝜇𝑖=𝑗. We denote by 𝑈𝑖 the set of all nodes in the 𝑖th community and let 𝑈𝐴𝑖, which is the subset of 𝑈𝑖, be the index set of all nodes in the 𝑖th community having direct connections to other communities in 𝐴. And by the similar way, we can define 𝑈𝐵𝑖. The function 𝑓𝜇𝑖() describes the local dynamics of nodes in the 𝜇𝑖th community, which is differentiable and capable of performing abundant dynamical behaviors. For any pair of indices 𝑖 and j, if 𝜇𝑖𝜇𝑗, which means that node 𝑖 and node 𝑗 belong to different communities, then 𝑓𝜇𝑖𝑓𝜇𝑗. The constant 𝑐>0 denotes the coupling strength. 𝜏 is the time delay in couplings. For 𝐴=(𝑎𝑖𝑗)𝑅𝑛×𝑛 and 𝐵=(𝑏𝑖𝑗)𝑅𝑛×𝑛 are the coupling configuration matrices with zero-sum rows, which represent the topological structure of the network. Take 𝐴 for an example and 𝐵 has the same properties. 𝑎𝑖𝑗>0 if there is a connection and is zero otherwise. Also, 𝐴 and 𝐵 need not be symmetric.

Let 𝑢𝑖(𝑡)𝑅𝑛, 𝑖=1,,𝑁, be the control inputs, then the controlled dynamical network with respect to (4.1) can be described by ̇𝑥𝑖(𝑡)=𝑓𝜇𝑖𝑥𝑖(𝑡),𝑡+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑥𝑗(𝑡𝜏)+𝑢𝑖(𝑡),𝑖=1,2,,𝑁.(4.2) We define the error variables by 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)𝑠𝜇𝑖(𝑡) for 𝑖=1,2,,𝑁, where 𝑠𝜇𝑖=(𝑠1𝜇𝑖,𝑠2𝜇𝑖,,𝑠𝑛𝜇𝑖)𝑇𝑅𝑛 satisfies ̇𝑠𝜇𝑖(𝑡)=𝑓𝜇𝑖(𝑠𝜇𝑖(𝑡)), which describes the identical local dynamics for the nodes in the 𝜇𝑖th community. The 𝑁 nodes are said to achieve cluster synchronization if lim𝑡||||𝑒𝑖||||(𝑡)=0,𝑖=1,,𝑁,(4.3) which means that the nodes within 𝜇𝑖th community are fully synchronized to dynamic state 𝑠𝜇𝑖, while nodes in separate communities behave independently. Define a set 𝑀=(𝑠𝜇1,𝑠𝜇2,,𝑠𝜇𝑁)𝑅𝑛×𝑁 as the cluster synchronization manifold for network (4.2). In fact, condition (4.3) implies that the manifold 𝑀 is stable.

According to the above definition of error variables, we can write the corresponding error system with respect to (4.2) as ̇𝑒𝑖(𝑡)=𝑓𝜇𝑖𝑥𝑖(𝑡)𝑓𝜇𝑖𝑠𝜇𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑠𝜇𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)+𝑐𝑁𝑗=1𝑏𝑖𝑗𝑠𝜇𝑗(𝑡𝜏)+𝑢𝑖(𝑡),𝑖=1,,𝑁.(4.4)

Since 𝐴 and 𝐵 are zero-row-sum matrices, we have 𝑁𝑗=1𝑎𝑖𝑗𝑠𝜇𝑗(𝑡)=0,𝑖𝑈𝑖𝑈𝐴𝜇𝑖,𝑁𝑗=1𝑏𝑖𝑗𝑠𝜇𝑗(𝑡)=0,𝑖𝑈𝑖𝑈𝐵𝜇𝑖.(4.5) Let 𝑑𝑖 stands for the feedback control strength. We design a local feedback control as 𝑢𝑖(𝑡)=𝑙𝑐𝑑𝐴𝑖𝑒𝑖(𝑡)𝑐𝑁𝑗=1𝑎𝑖𝑗𝑠𝜇𝑗(𝑡)𝑐𝑑𝐵𝑖𝑒𝑖(𝑡𝜏)𝑐𝑁𝑗=1𝑏𝑖𝑗𝑠𝜇𝑗𝑈(𝑡𝜏),𝑖𝐴𝜇𝑖𝑈𝐵𝜇𝑖0,otherwise,(4.6) where 𝑑𝐴𝑖=𝑑𝑖>0 for 𝑈𝑖𝐴𝜇𝑖 and 𝑑𝐵𝑖=𝑑𝑖>0 for 𝑈𝑖𝐵𝜇𝑖, which means we put control on the nodes that have communications with nodes in other different clusters. By intuition, the terms 𝑐𝑑𝐴𝑖𝑒𝑖(𝑡) and 𝑐𝑑𝐵𝑖𝑒𝑖(𝑡𝜏) in (4.6) is used to synchronize all nodes in the same cluster, while the remainder terms 𝑐𝑁𝑗=1𝑎𝑖𝑗𝑠𝜇𝑗(𝑡) and 𝑐𝑁𝑗=1𝑏𝑖𝑗𝑠𝜇𝑗(𝑡𝜏) in the controller is to weaken the mutual influences among clusters at the intersection nodes. It is easy to verify that the manifold 𝑀 is an invariant manifold for the network (4.2).

Theorem 4.1. Let 𝑃=diag{𝑝1,𝑝2,,𝑝𝑛} be a positive definite diagonal matrix and Δ=diag{𝛿1,𝛿2,,𝛿𝑛} be a diagonal matrix such that 𝑓𝜇𝑖(𝑥,𝑡)QUAD(𝑃,Δ,𝜂) for 1𝑖𝑁. For 𝑘=1,2,,𝑛, if Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝐷𝐴+𝑐2𝑝4Λ2𝑘𝐵𝐷𝐵𝐵𝐷𝐵𝑇<0,(4.7) where 𝐷𝐴=diag{𝑑𝐴1,𝑑𝐴2,,𝑑𝐴𝑁} and 𝐷𝐵=diag{𝑑𝐵1,𝑑𝐵2,,𝑑𝐵𝑁}. Λ is a given positive constant. Then, for any initial values, the solution 𝑥1(𝑡), 𝑥1(𝑡), , 𝑥𝑁(𝑡) of the system (4.1) under the control (4.6) can achieve cluster synchronization and satisfies (4.3).

Proof. Denote 𝑒𝑘(𝑡)=(𝑒𝑘1(𝑡),𝑒𝑘2(𝑡),,𝑒𝑘𝑁(𝑡))𝑇, 𝑘=1,2,,𝑛. Define a Lyapunov-Krasovskii function as 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+2Λ𝑡𝑡𝜏𝑒𝑖(𝜃)𝑇𝑒𝑖.(𝜃)𝑑𝜃(4.8)
Differentiating (4.8) along the solution of error system (4.4) under the control (4.6) gives 𝑑𝑉(𝑡)=𝑑𝑡𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃̇𝑒𝑖(𝑡)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑖(𝑡𝜏)𝑇𝑒𝑖=(𝑡𝜏)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝜇𝑖𝑥𝑖(𝑡)𝑓𝜇𝑖𝑠𝜇𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)𝑁𝑖=1𝑐𝑑𝐴𝑖𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝑁𝑖=1𝑐𝑑𝐵𝑖𝑒𝑖(𝑡)𝑇𝑃𝑒𝑗(𝑡𝜏)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑖(𝑡𝜏)𝑇𝑒𝑖=(𝑡𝜏)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑓𝜇𝑖𝑥𝑖(𝑡)𝑓𝜇𝑖𝑠𝜇𝑖(𝑡)Δ𝑒𝑖+(𝑡)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃Δ𝑒𝑖+(𝑡)𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑐𝑁𝑗=1𝑎𝑖𝑗𝑒𝑗(𝑡)𝑁𝑖=1𝑐𝑑𝐴𝑖𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖(𝑡)+𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑐𝑁𝑗=1𝑏𝑖𝑗𝑒𝑗(𝑡𝜏)𝑁𝑖=1𝑐𝑑𝐵𝑖𝑒𝑖(𝑡)𝑇𝑃𝑒𝑗(𝑡𝜏)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑖(𝑡𝜏)𝑇𝑒𝑖(𝑡𝜏)𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑝𝑘𝛿𝑘𝑒𝑘(𝑡)𝑇𝑒𝑘(𝑡)+𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐴𝑆𝑒𝑘(𝑡)𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐷𝐴𝑒𝑘(𝑡)+𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐵𝐷𝐵𝑒𝑘(𝑡𝜏)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑖(𝑡𝜏)𝑇𝑒𝑖(𝑡𝜏)=𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝛿𝑘𝐼𝑁𝐴+𝑐𝑆𝐷𝐴𝑒𝑘(𝑡)+𝑐𝑛𝑘=1𝑝𝑘𝑒𝑘(𝑡)𝑇𝐵𝐷𝐵𝑒𝑘(𝑡𝜏)+Λ𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)Λ𝑁𝑖=1𝑒𝑖(𝑡𝜏)𝑇𝑒𝑖(𝑡𝜏)=𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)+𝑛𝑘=1𝑒𝑘(𝑡)𝑇,𝑒𝑘(𝑡𝜏)𝑇×Λ+𝑝𝑘𝛿𝑘𝐼+𝑐𝑝𝑘𝐴𝑆𝐷𝐴𝑐2𝑝𝑘𝐵𝐷𝐵𝑐2𝑝𝑘𝐵𝐷𝐵𝑇𝑒Λ𝐼𝑘(𝑡),𝑒𝑘.(𝑡𝜏)(4.9)
Noticing the inequalities of (4.7), from Lemma 2.4, we obtain 𝑑𝑉(𝑡)𝑑𝑡𝜂𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑒𝑖(𝑡)<0.(4.10)
Thus, we get 𝑥𝑖(𝑡)𝑠𝜇𝑖(𝑡).

5. Numerical Simulation

In this section, we give numerical simulations to verify the theorems obtained in Section 3 and Section 4. Consider the unified system, 𝑥𝑓(𝑥,𝑡,𝛼)=(25𝛼+10)2𝑥1,(2835𝛼)𝑥1𝑥1𝑥3+(29𝛼1)𝑥2,𝑥1𝑥28+𝛼3𝑥3.(5.1) The system is chaotic for all 𝛼[0,1]. If 𝛼=0, the system (5.1) is Lorenz’s attractor. The ultimate bound and positively invariant set for system (5.1) is given in [22]. For 0𝛼<1/29, 𝑥22+(𝑥328+35𝛼)2𝐶2 and 𝑥21𝐶2, where 𝐶=(2835𝛼)(8+𝛼)23.(5+88𝛼)(129𝛼)(5.2) Thus, we have |𝑥1|𝐶, |𝑥2|𝐶 and |𝑥3|𝐶+2835𝛼. Let 𝑃=diag{𝑝1,𝑝2,𝑝3} and 𝑥𝑠=(𝑒1,𝑒2,𝑒3)𝑇, we obtain (𝑥𝑠)𝑇𝑃(𝑓(𝑥,𝑡,𝛼)𝑓(𝑠,𝑡,𝛼))(25𝛼+10)𝑝1𝑒21+(29𝛼1)𝑝2𝑒228+𝛼3𝑝3𝑒23+(25𝛼+10)𝑝1+(2835𝛼)𝑝2+𝑝2||𝑒(𝐶+2835𝛼)1𝑒2||+𝑝3𝐶||𝑒1𝑒3||+𝑝3𝑝2𝑥1𝑒2𝑒3+𝑝2𝑝3𝑒1𝑒2𝑒3.(5.3) Let 𝑝2=𝑝3, we have (𝑥𝑠)𝑇𝑃(𝑓(𝑥,𝑡,𝛼)𝑓(𝑠,𝑡,𝛼))(25𝛼+10)𝑝1+𝑝3𝐶2𝜗+(25𝛼+10)𝑝1+(2835𝛼)𝑝2+𝑝2(𝐶+2835𝛼)2𝜈𝑒21+(29𝛼1)𝑝2+(25𝛼+10)𝑝1+(2835𝛼)𝑝2+𝑝2(𝐶+2835𝛼)𝑒2𝜈22+8+𝛼3𝑝3+𝑝3𝐶𝑒2𝜗23𝑝1𝛿1𝑒𝜂21+𝑝2𝛿2𝑒𝜂22+𝑝3𝛿3𝑒𝜂23.(5.4) Set 𝑃=diag{5,1,1}, Δ=diag{23,36,17},𝜀=2 and 𝜗=1. Let 𝛼=0, 𝛼=0.01 and 𝛼=0.03 respectively in (5.1) since the nodes are nonidentical in different communities. If 𝛼=0, we have 𝜂1=3.2074 and 𝑓(𝑥,0)QUAD(𝑃,Δ,𝜂1). If 𝛼=0.01, we have 𝜂2=2.0243 and 𝑓(𝑥,0.01)QUAD(𝑃,Δ,𝜂2). If 𝛼=0.03, we have 𝜂3=0.1547 and 𝑓(𝑥,0.02)QUAD(𝑃,Δ,𝜂3).

5.1. Simulation to Cluster Synchronization with Identical Nodes

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 1.

fig1
Figure 1: The controlled complex network (3.2). (a) the topological structure of matrix A; (b) The sketch of matrix A: the point (𝑖,𝑗) is denoted as “" if 𝑎𝑖𝑗>0 and the point (𝑖,𝑗) is denoted as “×" if 𝑎𝑖𝑗<0; (c) the topological structure of matrix B; (d) the sketch of matrix B: the point (𝑖,𝑗) is denoted as “" if 𝑏𝑖𝑗>0 and the point (𝑖,𝑗) is denoted as “×" if 𝑏𝑖𝑗<0.

Let 𝑠1(𝑡), 𝑠2(𝑡), 𝑠3(𝑡) be the solution of the uncoupling system ̇𝑠𝑖(𝑡)=𝑓(𝑠𝑖(𝑡),𝑡) with initial values 𝑠1(0)=[1,2,3]𝑇, 𝑠2(0)=[4,5,6]𝑇, and 𝑠3(0)=[7,8,9]𝑇. Define 𝐸(𝑡)=(1/30)30𝑖=1||𝑥𝑖(𝑡)𝑠̃𝑖(𝑡)||2. If lim𝑡𝐸(𝑡)=0, the complex network achieves cluster synchronization.

For the controlled dynamic network (4.6). Let 𝜏=0.1, 𝜉𝑖=275, 𝑖𝐽={7,8,9,10,11,12,13,28,29,30}, 𝑐=5, Λ=0.0934 in the Theorem 3.1. It can be easily verified that such parameters fits (4.7).

Figures 2(a), 2(b), and 2(c) give the behavior of 𝑥(𝑡), as well as Figure 2(d) which shows how 𝐸(𝑡) evolve in pinning united chaotic attractor complex network with initial values chosen randomly in the interval [10,10].

fig2
Figure 2: The controlled complex network (3.2). (a, b, c) The behavior of 𝑥(𝑡) in 3 communities. (d) The error 𝐸(𝑡).

For the controlled dynamic network (4.6) with adaptive control strength 𝑑𝑖(𝑡), let Ξ=Ξ and keep other parameters the same as selected above, which makes (3.11) correct.

Figure 3 shows the behavior of 𝑥(𝑡) and how 𝐸(𝑡) evolve in pinning Lorenz chaotic attractor complex network with adaptive control strength and the initial values are chosen randomly in the interval [10,10].

fig3
Figure 3: The controlled complex network (3.2). (a,b,c) The behavior of 𝑥(𝑡) in 3 communities. (d) The error 𝐸(𝑡). (e) The adaptive control strength 𝜉(t).
5.2. Simulation to Cluster Synchronization with Nonidentical Nodes

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 4

fig4
Figure 4: The controlled complex network (3.2). (a) The topological structure of matrix A; (b) the sketch of matrix A: the point (𝑖,𝑗) is denoted as “" if 𝑎𝑖𝑗>0; (c) the topological structure of matrix B; (d) the sketch of matrix B: the point (𝑖,𝑗) is denoted as “" if 𝑏𝑖𝑗>0.

Let 𝑠1(𝑡), 𝑠2(𝑡), 𝑠3(𝑡) be the solution of the uncoupling system ̇𝑠𝑖(𝑡)=𝑓(𝑠𝑖(𝑡),𝑡) with initial values 𝑠1(0)=[1,2,3]𝑇, 𝑠2(0)=[4,5,6]𝑇, and 𝑠3(0)=[7,8,9]𝑇. Define 𝐸(𝑡)=(1/30)30𝑖=1||𝑥𝑖(𝑡)𝑠̃𝑖(𝑡)||2. If lim𝑡𝐸(𝑡)=0, the complex network achieves cluster synchronization.

For the controlled dynamic network (4.6). Let 𝜏=0.1, 𝐷𝐴=10.2173𝐼, 𝐷𝐴=0.0171𝐼, 𝑐=6, and Λ=0.0071 in the Theorem 4.1. It can be easily verified that such parameters fits (4.7).

Figures 5(a), 5(b), and 5(c) give the behavior of 𝑥(𝑡), as well as Figure 5(d) which shows how 𝐸(𝑡) evolve in pinning united chaotic attractor complex network with initial values chosen randomly in the interval [10,10].

fig5
Figure 5: The controlled complex network (3.2). (a, b, c) The behavior of 𝑥(𝑡) in 3 communities. (d) The error 𝐸(𝑡).

6. Conclusion

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.

Acknowledgments

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).

References

  1. S. H. Strogatz and I. Stewart, “Coupled oscillators and biological synchronization,” Scientific American, vol. 269, no. 6, pp. 102–109, 1993. View at Scopus
  2. C. M. Gray, “Synchronous oscillations in neuronal systems: mechanisms and functions,” Journal of Computational Neuroscience, vol. 1, no. 1-2, pp. 11–38, 1994. View at Publisher · View at Google Scholar · View at Scopus
  3. L. Glass, “Synchronization and rhythmic processes in physiology,” Nature, vol. 410, no. 6825, pp. 277–284, 2001. View at Publisher · View at Google Scholar · View at Scopus
  4. M. S. De Vieira, “Chaos and synchronized chaos in an earthquake model,” Physical Review Letters, vol. 82, no. 1, pp. 201–204, 1999. View at Scopus
  5. L. Kuhnert, K. I. Agladze, and V. I. Krinsky, “Image processing using light-sensitive chemical waves,” Nature, vol. 337, no. 6204, pp. 244–247, 1989. View at Scopus
  6. S. Wang, J. Kuang, J. Li, Y. Luo, H. Lu, and G. Hu, “Chaos-based secure communications in a large community,” Physical Review E, vol. 66, no. 6, Article ID 065202, 4 pages, 2002. View at Publisher · View at Google Scholar · View at Scopus
  7. S. Boccaletti, J. Kurths, G. Osipov, D. L. Valladares, and C. S. Zhou, “The synchronization of chaotic systems,” Physics Report, vol. 366, no. 1-2, pp. 1–101, 2002. View at Publisher · View at Google Scholar · View at Scopus
  8. Z. Zheng and G. Hu, “Generalized synchronization versus phase synchronization,” Physical Review E, vol. 62, no. 6, pp. 7882–7885, 2000. View at Scopus
  9. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “From phase to lag synchronization in coupled chaotic oscillators,” Physical Review Letters, vol. 78, no. 22, pp. 4193–4196, 1997. View at Scopus
  10. V. N. Belykh, I. V. Belykh, and E. Mosekilde, “Cluster synchronization modes in an ensemble of coupled chaotic oscillators,” Physical Review E, vol. 63, no. 3, Article ID 036216, 4 pages, 2001. View at Scopus
  11. M. G. Rosenblum, A. S. Pikovsky, and J. Kurths, “Phase synchronization of chaotic oscillators,” Physical Review Letters, vol. 76, no. 11, pp. 1804–1807, 1996. View at Scopus
  12. C. van Vreeswijk, “Partial synchronization in populations of pulse-coupled oscillators,” Physical Review E, vol. 54, no. 5, pp. 5522–5537, 1996. View at Scopus
  13. Z. J. Ma, Z. R. Liu, and G. Zhang, “A new method to realize cluster synchronization in connected chaotic networks,” Chaos, vol. 16, no. 2, Article ID 023103, 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. W. Wu, W. J. Zhou, and T. P. Chen, “Cluster synchronization of linearly coupled complex networks under pinning control,” IEEE Transactions on Circuits and Systems I, vol. 56, no. 4, pp. 829–839, 2009. View at Publisher · View at Google Scholar · View at Scopus
  15. J. Zhou, X. Wu, W. Yu, M. Small, and J. A. Lu, “Pinning synchronization of delayed neural networks,” Chaos, vol. 18, no. 4, Article ID 043111, 2008. View at Publisher · View at Google Scholar · View at Scopus
  16. W. L. Lu, B. Liu, and T. P. Chen, “Cluster synchronization in networks of coupled nonidentical dynamical systems.,” Chaos, vol. 20, no. 1, Article ID 013120, p. 12, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. W. L. Lu, B. Liu, and T. P. Chen, “Cluster synchronization in networks of distinct groups of maps,” European Physical Journal B, vol. 77, no. 2, pp. 257–264, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. K. H. Wang, X. C. Fu, and K. Z. Li, “Cluster synchronization in community networks with nonidentical nodes,” Chaos, vol. 19, no. 2, Article ID 023106, 10 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  19. J. Cao, Z. Wang, and Y. Sun, “Synchronization in an array of linearly stochastically coupled networks with time delays,” Physica A, vol. 385, no. 2, pp. 718–728, 2007. View at Publisher · View at Google Scholar · View at Scopus
  20. T. P. Chen, X. W. Liu, and W. L. Lu, “Pinning complex networks by a single controller,” IEEE Transactions on Circuits and Systems I, vol. 54, no. 6, pp. 1317–1326, 2007. View at Publisher · View at Google Scholar · View at Scopus
  21. W. Guo, F. Austin, and S. Chen, “Global synchronization of nonlinearly coupled complex networks with non-delayed and delayed coupling,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 6, pp. 1631–1639, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. D. M. Li, J. A. Lu, X. Q. Wu, and G. R. Chen, “Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 844–853, 2006. View at Publisher · View at Google Scholar · View at Scopus