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Journal of Applied Mathematics
Volume 2012, Article ID 347570, 12 pages
http://dx.doi.org/10.1155/2012/347570
Research Article

Cluster Anti-Synchronization of 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 5 June 2012; Accepted 19 June 2012

Academic Editor: Chong Lin

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 antisynchronization scheme in the time-varying delays coupled complex dynamical networks with nonidentical nodes. Based on the community structure of the networks, the controllers are designed differently between the nodes in one community that have direct connections to the nodes in other communities and the nodes without direct connections with the nodes in other communities strategy; some sufficient criteria are derived to ensure cluster anti-synchronization of the network model. Particularly, the weight configuration matrix is not assumed to be irreducible. The numerical simulations are performed to verify the effectiveness of the theoretical results.

1. Introduction

During the past decade, the research on synchronization and dynamical behavior analysis of complex network systems has become a new and important direction in this field [16]; many control approaches and many different synchronization phenomena have been developed, such as impulsive control, pinning control and complete synchronization, phase synchronization, cluster synchronization, mixed synchronization, and generalized synchronization, which have been investigated since ten years ago in [7, 8] and references therein.

Cluster synchronization means that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups [9, 10]. Wang et al. [11] investigated the cluster synchronization of the dynamical networks with community structure and nonidentical nodes and with identical local dynamics for all individual nodes in each community, which were considered by using some feedback control schemes. Wu and Lu [12] 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. Qin and Chen [13] investigated the stability of selected cluster synchronization in coupled Josephson equations. Ma et al. [14] showed that the arbitrarily selected cluster synchronization manifolds could be stabilized by constructing a special coupled matrix for connected chaotic networks. Wu et al. [15] investigated the anti-synchronization (AS) problem of two general complex dynamical networks with nondelayed and delayed coupling using pinning adaptive control method. Based on Lyapunov stability theory and Barbalat lemma, a sufficient condition is derived to guarantee the AS between two networks with nondelayed and delayed coupling. However, there is few theoretical result on the cluster anti-synchronization of linearly coupled complex networks with time-varying delays coupling.

Motivated by the aforementioned discussions, this paper aims to analyze the cluster anti-synchronization problem for the time-varying delays coupled complex dynamical networks. The main contributions of this paper are threefold: (1) the local dynamics in each community are identical, but those of different communities are nonidentical. (2) For achieving the synchronization, based on the community structure of the networks, the controllers are designed differently between the nodes in one community which have direct connections to the nodes in other communities and the nodes without direct connections with the nodes in other communities strategy; some sufficient criteria are derived to ensure cluster anti-synchronization of the network model. (3) According to Lyapunov stability theory, the sufficient conditions for achieving cluster anti-synchronization are obtained analytically. Compared with some similar designs, our controllers are very simple.

The paper is organized as follows: the network model is introduced followed by some definitions, lemmas, and hypotheses in Section 2. The cluster anti-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 time-varying delays coupling can be described by ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡),𝑖=1,2,,𝑁,(2.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),,𝑥𝑖𝑛(𝑡))𝑇𝑅𝑛is the state vector of node 𝑖, 𝑐>0 is the coupling strength, and𝑓𝜙𝑖𝑅𝑛𝑅𝑛 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, then𝑓𝜙𝑖𝑓𝜙𝑗, 𝜂𝜙𝑖(𝑡) is time-varying delay. Γ1𝑅𝑛×𝑛 and Γ2𝑅𝑛×𝑛 are inner-coupling matrices, for simplicity; we assume that Γ1, Γ2 are diagonal matrices with positive diagonal elements, Γ1=diag(𝜌1,𝜌2,,𝜌𝑛) with 𝜌𝑖0, Γ2=diag(𝜃1,𝜃2,,𝜃𝑛) with 𝜃𝑖0.  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 𝑎𝑖𝑗=𝑎𝑗𝑖>0, 𝑏𝑖𝑗=𝑏𝑗𝑖>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 matrices are not assumed to be irreducible.

When the control inputs 𝑣𝑖(𝑡), 𝑢𝑖(𝑡)𝑅𝑛 (𝑖=1,2,,𝑁) are introduced, the controlled dynamical network with respect to network (2.1) can be written as ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,𝑖=1,2,,𝑁,̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,𝑖=1,2,,𝑁,(2.3) where 𝐽𝜙𝑖 denotes all the nodes in the 𝜙𝑖 th community and 𝐽𝜙𝑖 represents the nodes in the 𝜙𝑖 th community, which have direct links with the nodes in other communities.

In this paper, 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. The local dynamics of individual nodes in different communities are assumed to be nonidentical, that is, if 𝜙𝑖𝜙𝑗, then 𝑓𝜙𝑖𝑓𝜙𝑗. Let 𝑠𝜙𝑖(𝑡) be a solution of an isolated node in the 𝜙𝑖 th community, that is, ̇𝑠𝑖(𝑡)=𝑓𝜙𝑖(𝑡,𝑠𝑖(𝑡)), where lim𝑡𝑠𝑖(𝑡)𝑠𝑗(𝑡)0 (𝑖𝑗) and the set 𝑆={𝑠1(𝑡),𝑠2(𝑡),,𝑠𝑚(𝑡)} is used as the cluster anti-synchronization manifold for network (2.3). Cluster anti-synchronization can be realized if and only if the manifold S is stable, where 𝑠𝑘(𝑡) may be an equilibrium point, a periodic orbit, or even a chaotic orbit.

Definition 2.1 (see [15]). The dynamical network (2.1) is said to achieve cluster anti-synchronization (CAS) if lim𝑡𝑒𝑖(𝑡)=lim𝑡𝑥𝑖(𝑡)+𝑠𝜙𝑖(𝑡)=0,𝑖=1,2,,𝑁.(2.4)

Lemma 2.2. For any two vectors x and y, a matrix Q>0 with compatible dimensions, one has 2𝑥𝑇𝑦𝑥𝑇𝑄𝑥+𝑦𝑇𝑄1𝑦.(2.5)

Assumption 2.3 (see [15]). For any 𝑥=(𝑥1,𝑥2,,𝑥𝑛)𝑇𝑅𝑛, 𝑦=(𝑦1,𝑦2,,𝑦𝑛)𝑇𝑅𝑛, there exists a positive constant 𝐿 such that, (𝑦𝑥)𝑇(𝑓(𝑡,𝑦)𝑓(𝑡,𝑥))𝐿(𝑦𝑥)𝑇Γ(𝑦𝑥), where Γ is a positive definite matrix. Here 𝑥 and 𝑦 are time-varying vectors.

Assumption 2.4. 𝑓(𝑥,𝑡) is an odd function of 𝑥, that is, 𝑓(𝑥,𝑡)=𝑓(𝑥,𝑡) for arbitrary 𝑥𝑅𝑛.

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

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 as ̇𝑒𝑖𝑓(𝑡)=𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑒𝑗𝑡𝜂𝜙𝑖(𝑡)𝑐𝑁𝑖=1𝑎𝑖𝑗Γ1𝑠𝜙𝑖(𝑡)𝑐𝑁𝑖=1𝑏𝑖𝑗Γ2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,𝑖=1,2,,𝑁,̇𝑒𝑖𝑓(𝑡)=𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑒𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,𝑖=1,2,,𝑁,(3.1) where 𝑓𝜙𝑖(𝑡,𝑒𝑖(𝑡))=𝑓𝜙𝑖(𝑡,𝑥𝑖(𝑡))+𝑓𝜙𝑖(𝑡,𝑠𝜙𝑖(𝑡)), for 𝑖=1,2,,𝑁.

According to the diffusive coupling condition (2.2) of the matrix A, B, we have 𝑐𝑁𝑖=1𝑎𝑖𝑗Γ1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑖=1𝑏𝑖𝑗Γ2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)=0,𝑖𝐽𝜙𝑖𝐽𝜙𝑖.(3.2) On the basis of this property, for achieving cluster anti-synchronization, we design the controllers as follows: 𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑎𝑖𝑗Γ1𝑠𝜙𝑖(𝑡)+𝑐𝑁𝑖=1𝑏𝑖𝑗Γ2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖,(3.3) where ̇𝑑𝑖=𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡), 𝑑𝑖 are the feedback strength and 𝑘𝑖 are arbitrary positive constants.

It is easy to see that the synchronization of the controlled complex network (2.1) is achieved if the zero solution of the error system (3.1) is globally asymptotically stable, which is ensured by the following theorem.

Let ̃𝑒𝑗(𝑡)=(𝑒1𝑗(𝑡),𝑒2𝑗(𝑡),,𝑒𝑁𝑗(𝑡))𝑇, 𝐷=diag(𝑑1,,𝑑𝑁)𝑇; then we have the following result.

Theorem 3.1. Suppose Assumptions 2.32.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold: 𝜆max(Γ)𝐿+𝑐𝜆max1(𝑄)+2𝑐𝜆max𝑃𝑃𝑇+𝑐2(1𝜀)<𝑑,(3.4) then the system (2.3) is cluster anti-synchronization, where 𝑑 is sufficiently large positive constant to be determined.

Proof. From Assumptions 2.3 and 2.4, we get (𝑦+𝑥)𝑇(𝑓(𝑡,𝑦)+𝑓(𝑡,𝑥))=(𝑦(𝑥))𝑇(𝑓(𝑡,𝑦)𝑓(𝑡,𝑥))𝐿(𝑦+𝑥)𝑇Γ(𝑦+𝑥)(3.5) for 𝑥,𝑦𝑅𝑛.
Construct the following Lyapunov functional: 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝑐(𝑡)+2(1𝜀)𝑁𝑖=1𝑡𝑡𝜂𝜙𝑖(𝑡)𝑒𝑇𝑖(𝜉)𝑒𝑖1(𝜉)𝑑𝜉+2𝑁𝑖=1(𝑑𝑖𝑑)2𝑘𝑖.(3.6) Calculating the derivative of 𝑉(𝑡), we have ̇𝑉(𝑡)=𝑁𝑖=1𝑒𝑇𝑖𝑓(𝑡)𝜙𝑖𝑡,𝑒𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑒𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑒𝑗𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖+𝑐(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)𝑐1̇𝜂𝜙𝑖(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖+(𝑡)𝑁𝑖=1𝑑𝑖𝑒𝑑𝑇𝑖(𝑡)𝑒𝑖(𝑡).(3.7)
By the Assumptions 2.32.5, we have ̇𝑉(𝑡)𝑁𝑖=1𝐿𝑒𝑇𝑖(𝑡)Γ𝑒𝑖(𝑡)+𝑐𝑁𝑁𝑖=1𝑗=1𝑎𝑖𝑗𝑒𝑇𝑖(𝑡)Γ1𝑒𝑗(𝑡)+𝑐𝑁𝑁𝑖=1𝑗=1𝑏𝑖𝑗𝑒𝑇𝑖(𝑡)Γ2𝑒𝑗𝑡𝜂𝜙𝑖+𝑐(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝑐(𝑡)2𝑁𝑖=1𝑒𝑇𝑖𝑡𝜂𝜙𝑖𝑒(𝑡)𝑖𝑡𝜂𝜙𝑖(𝑡)𝑁𝑖=1𝑑𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)𝑁𝑖=1𝐿𝑒𝑇𝑖(𝑡)Γ𝑒𝑖(𝑡)+𝑐𝑒𝑇𝐴Γ1𝑒+𝑐𝑒𝑇𝐵Γ2𝑒𝑡𝜂𝜙𝑖+𝑐(𝑡)2(1𝜀)𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑒𝑖𝑐(𝑡)2𝑒𝑇𝑡𝜂𝜙𝑖𝑒(𝑡)𝑡𝜂𝜙𝑖(𝑡)𝑑𝑒𝑇(𝑡)𝑒(𝑡).(3.8)
Let 𝑒(𝑡)=(𝑒𝑇1(𝑡),𝑒𝑇2(𝑡),,𝑒𝑇𝑁(𝑡))𝑇𝑅𝑛𝑁, 𝑄=(𝐴Γ1), 𝑃=(𝐵Γ2), where represents the Kronecker product. Then ̇𝑉(𝑡)𝜆max(Γ)𝐿𝑒𝑇(𝑡)𝑒(𝑡)+𝑐𝑒𝑇(𝑡)𝑄𝑒(𝑡)+𝑐𝑒𝑇(𝑡)𝑃𝑒𝑡𝜂𝜙𝑖+𝑐(𝑡)𝑒2(1𝜀)𝑇𝑐(𝑡)𝑒(𝑡)2𝑒𝑇𝑡𝜂𝜙𝑖𝑒(𝑡)𝑡𝜂𝜙𝑖(𝑡)𝑑𝑒𝑇(𝑡)𝑒(𝑡).(3.9) By Lemma 2.2, we have ̇𝑉(𝑡)𝜆max(Γ)𝐿𝑒𝑇(𝑡)𝑒(𝑡)+𝑐𝑒𝑇𝑐(𝑡)𝑄𝑒(𝑡)+2𝑒𝑇(𝑡)𝑃𝑃𝑇𝑒𝑐(𝑡)+𝑒2(1𝜀)𝑇(𝑡)𝑒(𝑡)𝑑𝑒𝑇𝜆(𝑡)𝑒(𝑡)max(Γ)𝐿+𝑐𝜆max1(𝑄)+2𝑐𝜆max𝑃𝑃𝑇+𝑐𝑒2(1𝜀)𝑑𝑇(𝑡)𝑒(𝑡).(3.10) Therefore, if we have 𝜆max(Γ)𝐿+𝑐𝜆max(𝑄)+(1/2)𝑐𝜆max(𝑃𝑃𝑇)+𝑐/2(1𝜀)<𝑑, then ̇𝑉(𝑡)𝑒𝑇(𝑡)𝑒(𝑡).(3.11)
Choose 𝜆max(Γ)𝐿+𝑐𝜆max(𝑄)+(1/2)𝑐𝜆max(𝑃𝑃𝑇)+𝑐/2(1𝜀)+1<𝑑. Thus, one obtains ̇𝑉(𝑡)𝑒𝑇(𝑡)𝑒(𝑡)0. It is obvious that the largest invariant set contained in set ̇𝐸={𝑉(𝑡)=0}={𝑒𝑖(𝑡)=0,𝑖=1,2,,𝑁} is 𝑄={𝑒𝑖(𝑡)=0,𝑑𝑖(𝑡)=𝑑,𝑖=1,2,,𝑁}. Based on LaSalle invariance principle, starting with any initial values of the error dynamical system, the trajectory asymptotically converges to the largest invariant 𝑄 which implies that lim𝑡𝑒𝑖(𝑡)=0 for 𝑖=1,2,,𝑁. Therefore, cluster anti-synchronization in the network (2.3) is achieved under the controllers (3.3). This completes the proof.

Corollary 3.2. When A = 0, network (2.1) is translated into ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡),𝑖=1,2,,𝑁.(3.12)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑏𝑖𝑗Γ2𝑠𝜙𝑖𝑡𝜂𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖.(3.13)

Corollary 3.3. When 𝐵=0, network (2.1) is translated into ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡),𝑖=1,2,,𝑁.(3.14)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢𝑖(𝑡)=𝑐𝑁𝑖=1𝑎𝑖𝑗Γ1𝑠𝜙𝑖(𝑡)𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖,𝑣𝑖(𝑡)=𝑑𝑖𝑒𝑖(𝑡),𝑖𝐽𝜙𝑖𝐽𝜙𝑖.(3.15)

4. Illustrative Examples

In this section, several numerical examples are provided to illustrate the proposed synchronization methods. The nodes dynamics are the following well-known modified Chua’s circuit [16] with different system parameters. Considering the following network: ̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑢𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖,𝑖=1,2,,𝑁,̇𝑥𝑖(𝑡)=𝑓𝜙𝑖𝑡,𝑥𝑖(𝑡)+𝑐𝑁𝑗=1𝑎𝑖𝑗Γ1𝑥𝑗(𝑡)+𝑐𝑁𝑗=1𝑏𝑖𝑗Γ2𝑥𝑗𝑡𝜂𝜙𝑖(𝑡)+𝑣𝑖(𝑡),𝜙𝑖(𝑡)𝐽𝜙𝑖𝐽𝜙𝑖,𝑖=1,2,,𝑁,(4.1) where 𝑥𝑖(𝑡)=(𝑥1𝑖(𝑡),𝑥2𝑖(𝑡),𝑥3𝑖(𝑡))𝑇, 𝑓1(𝑡,𝑥𝑖(𝑡))=𝐷1𝑥𝑖(𝑡)+1(𝑥𝑖(𝑡)), 𝑓2(𝑡,𝑥𝑖(𝑡))=𝐷2𝑥𝑖(𝑡)+2(𝑥𝑖(𝑡)), 𝑓3(𝑡,𝑥𝑖(𝑡))=𝐷3𝑥𝑖(𝑡)+3(𝑥𝑖(𝑡)), 1(𝑥𝑖)=2(𝑥𝑖)=3(𝑥𝑖)=((20/7)𝑥3𝑖1,0,0)𝑇, Γ1=Γ2=Diag{1,1,1}, 𝜂1(𝑡)=0.1𝑒𝑡1+𝑒𝑡,𝜂2(𝑡)=2𝑒𝑡1+𝑒𝑡,𝜂3(𝑡)=0.8𝑒𝑡1+𝑒𝑡,𝐷1=1071001110120,𝐷2=1071001110130,𝐷3=107,.1001110140𝑐=0.01,𝑑=10,𝐴=𝐵=211211130121103110111410111041110002(4.2) Simulation results are given in Figures 1, 2, and 3. Cluster anti-synchronization is achieved by the controller (3.3).

fig1
Figure 1: The trajectories of nodes in the first community.
fig2
Figure 2: The trajectories of nodes in the second community.
fig3
Figure 3: The trajectories of nodes in the third community.

The following quantities are utilized to measure the process of cluster anti-synchronization 𝐸(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,(4.3) 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 anti-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 4, 5, 6, and 7.

347570.fig.004
Figure 4: Time evolution of the synchronization error 𝐸(𝑡).
347570.fig.005
Figure 5: Time evolution of the synchronization error 𝐸12(𝑡).
347570.fig.006
Figure 6: Time evolution of the synchronization error 𝐸23(𝑡).
347570.fig.007
Figure 7: Time evolution of the synchronization error 𝐸13(𝑡).

5. Conclusions

The cluster anti-synchronization in community networks has been studied in this paper, based on the community structure of the networks. Particularly, weight configuration matrix is not assumed to be irreducible. Some simple and useful criteria are derived by constructing an effective control scheme. The synchronization criteria are independent of time delay. Finally, the developed techniques are applied in three complex community networks. The numerical simulations are performed to verify the effectiveness of the theoretical results.

Acknowledgments

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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