Cluster Anti-Synchronization of Complex Networks with Nonidentical Dynamical Nodes

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 [1–6]; 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 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡),𝑖=1,2,…,𝑁,(2.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈𝑅^𝑛is the state vector of node 𝑖, 𝑐>0 is the coupling strength, and𝑓_𝜙𝑖∶𝑅^𝑛→𝑅^𝑛 describes the local dynamics of nodes in the 𝜙_𝑖th community. For any pair of nodes 𝑖 and 𝑗, if 𝜙_𝑖≠𝜙_𝑗, that is, nodes 𝑖 and 𝑗 belong to different communities, then𝑓_𝜙𝑖≠𝑓_𝜙𝑗, 𝜂_𝜙𝑖(𝑡) is time-varying delay. Γ₁∈𝑅^𝑛×𝑛 and Γ₂∈𝑅^𝑛×𝑛 are inner-coupling matrices, for simplicity; we assume that Γ₁, Γ₂ are diagonal matrices with positive diagonal elements, Γ₁=diag(𝜌₁,𝜌₂,…,𝜌_𝑛) with 𝜌_𝑖≥0, Γ₂=diag(𝜃₁,𝜃₂,…,𝜃_𝑛) with 𝜃_𝑖≥0.  𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁, 𝐵=(𝑏_𝑖𝑗)_𝑁×𝑁 are the weight configuration matrices. If there is a connection from node 𝑖 to node 𝑗 (𝑗≠𝑖), then 𝑎_𝑖𝑗=𝑎_𝑗𝑖>0, 𝑏_𝑖𝑗=𝑏_𝑗𝑖>0; otherwise 𝑎_𝑖𝑗=𝑎_𝑗𝑖=0, 𝑏_𝑖𝑗=𝑏_𝑗𝑖=0, and the diagonal elements of matrix 𝐴, 𝐵 are defined as 𝑎_𝑖𝑖=−^𝑁_𝑗=_{1,𝑗≠𝑖}𝑎_𝑗𝑖,𝑏_𝑖𝑖=−^𝑁_𝑗=_{1,𝑗≠𝑖}𝑏_𝑗𝑖,𝑖=1,2,…,𝑁.(2.2) Particularly, the weight configuration matrices are not assumed to be irreducible.

When the control inputs 𝑣_𝑖(𝑡), 𝑢_𝑖(𝑡)∈𝑅^𝑛 (𝑖=1,2,…,𝑁) are introduced, the controlled dynamical network with respect to network (2.1) can be written as 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡)_𝑗=1𝑥̇_𝑖𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑣_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(2.3) where 𝐽_𝜙𝑖 denotes all the nodes in the 𝜙_𝑖 th community and 𝐽⁻_𝜙𝑖 represents the nodes in the 𝜙_𝑖 th community, which have direct links with the nodes in other communities.

In this paper, let {𝐶₁,𝐶₂,…,𝐶_𝑚} denote 𝑚  (2≤𝑚≤𝑁) communities of the networks and ⋃^𝑚𝑖=1𝐶_𝑖={1,2,…𝑁}. If node 𝑖 belongs to the 𝑗th community, then we denote 𝜙_𝑖=𝑗. We employ 𝑓_𝑖(⋅) 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 𝜙_𝑖≠𝜙_𝑗, then 𝑓_𝜙𝑖≠𝑓_𝜙𝑗. Let 𝑠_𝜙𝑖(𝑡) be a solution of an isolated node in the 𝜙_𝑖 th community, that is, 𝑠_𝑖̇(𝑡)=𝑓_𝜙𝑖(𝑡,𝑠_𝑖(𝑡)), where lim_𝑡→∞‖𝑠_𝑖(𝑡)−𝑠_𝑗(𝑡)‖≠0 (𝑖≠𝑗) and the set 𝑆={𝑠₁(𝑡),𝑠₂(𝑡),…,𝑠_𝑚(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 𝑠_𝑘(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 lim_𝑡→∞‖‖𝑒_𝑖(𝑡)‖‖=lim_𝑡→∞‖‖𝑥_𝑖(𝑡)+𝑠_𝜙𝑖(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 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄𝑥+𝑦^𝑇𝑄⁻¹𝑦.(2.5)

Assumption 2.3 (see [15]). For any 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛)^𝑇∈𝑅^𝑛, 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)^𝑇∈𝑅^𝑛, there exists a positive constant 𝐿 such that, (𝑦−𝑥)^𝑇(𝑓(𝑡,𝑦)−𝑓(𝑡,𝑥))≤𝐿(𝑦−𝑥)^𝑇Γ(𝑦−𝑥), 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 𝑥∈𝑅^𝑛.

Assumption 2.5. 𝜂_𝜙𝑖(𝑡) is a differential function with 0≤𝜂̇_𝜙𝑖(𝑡)≤𝜀≤1. Clearly, this assumption is certainly ensured if the delay 𝜂_𝜙𝑖(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 𝑠_𝜙𝑖(𝑡). Let synchronization errors 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)+𝑠_𝜙𝑖(𝑡) for 𝑖=1,2,…,𝑁, according to system (2.1); the error dynamical system can be derived as 𝑒_𝑖̇(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ₁𝑒_𝑗(𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ₂𝑒_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝑐^𝑁_𝑖=1−𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑠_𝜙𝑖(𝑡)_𝑖=1𝑒_𝑖̇𝑏_𝑖𝑗Γ₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑒_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗Γ₂𝑒_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑣_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(3.1) where 𝑓_𝜙𝑖(𝑡,𝑒_𝑖(𝑡))=𝑓_𝜙𝑖(𝑡,𝑥_𝑖(𝑡))+𝑓_𝜙𝑖(𝑡,𝑠_𝜙𝑖(𝑡)), for 𝑖=1,2,…,𝑁.

According to the diffusive coupling condition (2.2) of the matrix A, B, we have 𝑐^𝑁_𝑖=1𝑎_𝑖𝑗Γ₁𝑠_𝜙𝑖(𝑡)+𝑐^𝑁_𝑖=1𝑏_𝑖𝑗Γ₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)=0,𝑖∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖.(3.2) On the basis of this property, for achieving cluster anti-synchronization, we design the controllers as follows: 𝑢_𝑖(𝑡)=𝑐^𝑁_𝑖=1𝑎_𝑖𝑗Γ₁𝑠_𝜙𝑖(𝑡)+𝑐^𝑁_𝑖=1𝑣_𝑖𝑏_𝑖𝑗Γ₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽⁻_𝜙𝑖,(𝑡)=−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖,(3.3) where 𝑑̇_𝑖=𝑘_𝑖𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡), 𝑑_𝑖 are the feedback strength and 𝑘_𝑖 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 𝑒_𝑗̃(𝑡)=(𝑒_1𝑗(𝑡),𝑒_2𝑗(𝑡),…,𝑒_𝑁𝑗(𝑡))^𝑇, 𝐷=diag(𝑑₁,…,𝑑_𝑁)^𝑇; then we have the following result.

Theorem 3.1. Suppose Assumptions 2.3–2.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold: 𝜆_max(Γ)𝐿+𝑐𝜆_max(𝑄)+12𝑐𝜆^max𝑃𝑃^𝑇+2(1𝑐−𝜀)<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,𝑦∈𝑅^𝑛.
Construct the following Lyapunov functional: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝑐2(1−𝜀)^𝑁_𝑖=1^{𝑡𝑡−𝜂}_{𝜙𝑖(𝑡)}𝑒^𝑇𝑖(𝜉)𝑒_𝑖(𝜉)𝑑𝜉+12^𝑁_𝑖=1(𝑑_𝑖−𝑑)²𝑘_𝑖.(3.6) Calculating the derivative of 𝑉(𝑡), we have 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑓_𝜙𝑖𝑡,𝑒_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ₁𝑒_𝑗(𝑡)+𝑐^𝑁_𝑗=1+𝑐𝑏_𝑖𝑗Γ₂𝑒_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡)2(1−𝜀)^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑐1−𝜂̇^𝜙𝑖(𝑡)2(1−𝜀)^𝑁_𝑖=1+𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)_𝑁_𝑖=1𝑑_𝑖−𝑑𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.7)
By the Assumptions 2.3–2.5, we have 𝑉̇(𝑡)≤^𝑁_𝑖=1𝐿𝑒^𝑇𝑖(𝑡)Γ𝑒_𝑖(𝑡)+𝑐^𝑁_𝑁𝑖=1_𝑗=1𝑎_𝑖𝑗𝑒^𝑇𝑖(𝑡)Γ₁𝑒_𝑗(𝑡)+𝑐^𝑁_𝑁𝑖=1_𝑗=1+𝑐𝑏_𝑖𝑗𝑒^𝑇𝑖(𝑡)Γ₂𝑒_𝑗𝑡−𝜂_𝜙𝑖(𝑡)2(1−𝜀)^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝑐2^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)−^𝑁_𝑖=1≤𝑑𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)_𝑁_𝑖=1𝐿𝑒^𝑇𝑖(𝑡)Γ𝑒_𝑖(𝑡)+𝑐𝑒^𝑇𝐴⊗Γ₁𝑒+𝑐𝑒^𝑇𝐵⊗Γ₂𝑒𝑡−𝜂_𝜙𝑖(𝑡)+𝑐2(1−𝜀)^𝑁_𝑖=1−𝑐𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)2𝑒^𝑇𝑡−𝜂_𝜙𝑖(𝑡)𝑒𝑡−𝜂_𝜙𝑖(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.8)
Let 𝑒(𝑡)=(𝑒^𝑇1(𝑡),𝑒^𝑇2(𝑡),…,𝑒^𝑇𝑁(𝑡))^𝑇∈𝑅^𝑛𝑁, 𝑄=(𝐴⊗Γ₁), 𝑃=(𝐵⊗Γ₂), where ⊗ represents the Kronecker product. Then 𝑉̇(𝑡)≤𝜆_max(Γ)𝐿𝑒^𝑇(𝑡)𝑒(𝑡)+𝑐𝑒^𝑇(𝑡)𝑄𝑒(𝑡)+𝑐𝑒^𝑇(𝑡)𝑃𝑒𝑡−𝜂_𝜙𝑖(𝑡)+𝑐2(1−𝜀)𝑒^𝑇−𝑐(𝑡)𝑒(𝑡)2𝑒^𝑇𝑡−𝜂_𝜙𝑖(𝑡)𝑒𝑡−𝜂_𝜙𝑖(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.9) By Lemma 2.2, we have 𝑉̇(𝑡)≤𝜆_max(Γ)𝐿𝑒^𝑇(𝑡)𝑒(𝑡)+𝑐𝑒^𝑇(𝑡)𝑄𝑒(𝑡)+𝑐2𝑒^𝑇(𝑡)𝑃𝑃^𝑇𝑒(𝑡)+𝑐2(1−𝜀)𝑒^𝑇≤𝜆(𝑡)𝑒(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡)_max(Γ)𝐿+𝑐𝜆_max(𝑄)+12𝑐𝜆^max𝑃𝑃^𝑇+2(1𝑐−𝜀)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.10) Therefore, if we have 𝜆_max(Γ)𝐿+𝑐𝜆_max(𝑄)+(1/2)𝑐𝜆_max(𝑃𝑃^𝑇)+𝑐/2(1−𝜀)<𝑑, then 𝑉̇(𝑡)≤−𝑒^𝑇(𝑡)𝑒(𝑡).(3.11)
Choose 𝜆_max(Γ)𝐿+𝑐𝜆_max(𝑄)+(1/2)𝑐𝜆_max(𝑃𝑃^𝑇)+𝑐/2(1−𝜀)+1<𝑑. Thus, one obtains 𝑉̇(𝑡)≤−𝑒^𝑇(𝑡)𝑒(𝑡)≤0. It is obvious that the largest invariant set contained in set 𝐸={𝑉̇(𝑡)=0}={𝑒_𝑖(𝑡)=0,𝑖=1,2,…,𝑁} is 𝑄={𝑒_𝑖(𝑡)=0,𝑑_𝑖(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 lim_𝑡→∞‖𝑒_𝑖(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 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡),𝑖=1,2,…,𝑁.(3.12)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢_𝑖(𝑡)=𝑐^𝑁_𝑖=1𝑣_𝑖𝑏_𝑖𝑗Γ₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽⁻_𝜙𝑖,(𝑡)=−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖.(3.13)

Corollary 3.3. When 𝐵=0, network (2.1) is translated into 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁.(3.14)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢_𝑖(𝑡)=𝑐^𝑁_𝑖=1𝑣_𝑖𝑎_𝑖𝑗Γ₁𝑠_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽⁻_𝜙𝑖,(𝑡)=−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖.(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: 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡)_𝑗=1𝑥̇_𝑖𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗Γ₁𝑥_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗Γ₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑣_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽⁻_𝜙𝑖,𝑖=1,2,…,𝑁,(4.1) where 𝑥_𝑖(𝑡)=(𝑥_1𝑖(𝑡),𝑥_2𝑖(𝑡),𝑥_3𝑖(𝑡))^𝑇, 𝑓₁(𝑡,𝑥_𝑖(𝑡))=𝐷₁𝑥_𝑖(𝑡)+ℎ₁(𝑥_𝑖(𝑡)), 𝑓₂(𝑡,𝑥_𝑖(𝑡))=𝐷₂𝑥_𝑖(𝑡)+ℎ₂(𝑥_𝑖(𝑡)), 𝑓₃(𝑡,𝑥_𝑖(𝑡))=𝐷₃𝑥_𝑖(𝑡)+ℎ₃(𝑥_𝑖(𝑡)), ℎ₁(𝑥_𝑖)=ℎ₂(𝑥_𝑖)=ℎ₃(𝑥_𝑖)=((−20/7)𝑥^3𝑖1,0,0)^𝑇, Γ₁=Γ₂=Diag{1,1,1}, 𝜂₁(𝑡)=0.1𝑒^𝑡1+𝑒^𝑡,𝜂₂(𝑡)=2𝑒^𝑡1+𝑒^𝑡,𝜂₃(𝑡)=0.8𝑒^𝑡1+𝑒^𝑡𝐷₁,=⎡⎢⎢⎢⎢⎣−10107−−11012100⎤⎥⎥⎥⎥⎦,𝐷₂=⎡⎢⎢⎢⎢⎣−10107−−11013100⎤⎥⎥⎥⎥⎦,𝐷₃=⎡⎢⎢⎢⎢⎣−10𝑐=0.01,𝑑=10,𝐴=𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−−111121−101113−101103−−110024−110124107−001112−⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦−110.14010⎤⎥⎥⎥⎥⎦,(4.2) Simulation results are given in Figures 1, 2, and 3. Cluster anti-synchronization is achieved by the controller (3.3).

(a)

(b)

(c)

Figure 1 

The trajectories of nodes in the first community.

(a)

(b)

(c)

Figure 2 

The trajectories of nodes in the second community.

(a)

(b)

(c)

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)=^𝑁_𝑖=1𝐸‖‖𝑥_𝑖(𝑡)+𝑠_𝜙𝑖(𝑡)₁₂‖‖,𝐸₁₃(𝑡)=‖‖𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡)‖‖,𝑢∈𝐶₁,𝑣∈𝐶₂,𝐸₂₃(𝑡)=‖‖𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡)‖‖,𝑢∈𝐶₁,𝑣∈𝐶₃,(𝑡)=‖‖𝑥_𝑢(𝑡)−𝑥_𝑣(𝑡)‖‖,𝑢∈𝐶₂,𝑣∈𝐶₃,(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.

Figure 4 

Time evolution of the synchronization error 𝐸(𝑡).

Figure 5 

Time evolution of the synchronization error 𝐸12(𝑡).

Figure 6 

Time evolution of the synchronization error 𝐸23(𝑡).

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.