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

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 [1–13]; many control approaches have been developed to synchronize complex networks such as feedback control, adaptive control, pinning control, impulsive control, and intermittent control [14–21].

Cluster synchronization means that nodes in the same group synchronize with each other, but there is no synchronization between nodes in different groups [22–25]; 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 by𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗𝐻₂𝑥_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^𝑛 is the state vector of node 𝑖;𝑓_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, then 𝑓_𝜙𝑖≠𝑓_𝜙𝑗⋅𝜂_𝜙𝑖(𝑡),𝜏_𝜙𝑖(𝑡), is a time-varying delay. 𝐻₁(⋅) and 𝐻₂(⋅) are nonlinear functions. 𝑐 is coupling strength. 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 the 𝑎_𝑖𝑗>0,𝑏_𝑖𝑗>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 matrix is not assumed to be symmetric, irreducible.

When the control inputs 𝑢_𝑖(𝑡)∈𝑅^𝑛 and 𝑣_𝑖(𝑡)∈𝑅^𝑛(𝑖=1,2,…,𝑁) are introduced, the controlled dynamical network with respect to network (2.1) can be written as𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)_𝑗=1𝑥̇_𝑖𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝑣_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽_𝜙𝑖,(2.3) where 𝐽_𝜙𝑖 denotes all the nodes in the 𝜙_𝑖th community and 𝐽_𝜙𝑖 represents the nodes in the 𝜙_𝑖th community which have direct links with the nodes in other communities.

The study presents the mathematical definition of the cluster synchronization.

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. Let 𝑠_𝑖(𝑡) be the solution of the system 𝑠_𝑖̇(𝑡)=𝑓_𝜙𝑖(𝑡,𝑠_𝑖(𝑡),𝑠_𝑖(𝑡−𝜏_𝜙𝑖(𝑡))),(𝑖=1,2,…,𝑚) where lim_𝑡→∞‖𝑠_𝑖(𝑡)−𝑠_𝑗(𝑡)‖≠0(𝑖≠𝑗); the set 𝑆={𝑠₁(𝑡),𝑠₂(𝑡),…,𝑠_𝑚(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 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠_𝜙𝑖(𝑡) for 𝑖=1,2,…,𝑁, where 𝑠_𝜙𝑖(𝑡) satisfies 𝑠̇_𝜙𝑖(𝑡)=𝑓_𝜙𝑖(𝑡,𝑠_𝜙𝑖(𝑡),𝑠_𝜙𝑖(𝑡−𝜏_𝜙𝑖(𝑡))).

Definition 2.2 (see [19]). Let {1,2,…,𝑁} be the 𝑁 nodes of the network and {𝐶₁,𝐶₂,…,𝐶_𝑚} be the 𝑚 communities, respectively. A network with 𝑚 communities is said to realize cluster synchronization if lim_𝑡→∞𝑒_𝑖(𝑡)=0 and lim_𝑡→∞‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≠0 for 𝜙_𝑖≠𝜙_𝑗.

Lemma 2.3. For any two vectors 𝑥 and 𝑦, a matrix Q>0 with compatible dimensions, one has 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄𝑥+𝑦^𝑇𝑄⁻¹𝑦.

Assumption 2.4. For the vector valued function 𝑓_𝜙𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏_𝜙𝑖)), assuming that there exist positive constants 𝛼_𝜙𝑖>0,𝛾_𝜙𝑖>0 such that f satisfies the semi-Lipschitz condition 𝑥_𝑖(𝑡≤)−𝛼𝑦_𝑖𝜙𝑖(𝑡)^𝑇𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖−𝑓_𝜙𝑖𝑡,𝑦_𝑖(𝑡),𝑦_𝑖𝑡−𝜏_𝜙𝑖+𝛾_𝜙𝑖𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)^𝑇𝑥_𝑖(𝑡)−𝑦_𝑖(𝑡)𝑥_𝑖𝑡−𝜏_𝜙𝑖−𝑦_𝑖𝑡−𝜏_𝜙𝑖^𝑇𝑥_𝑖𝑡−𝜏_𝜙𝑖−𝑦_𝑖𝑡−𝜏_𝜙𝑖,(2.4) for all 𝑥,𝑦∈𝑅^𝑛 and 𝜏_𝜙𝑖(𝑡)≥0.𝑖=1,2,…,𝑁.

Assumption 2.5. 𝜂_𝜙𝑖(𝑡) and 𝜏_𝜙𝑖(𝑡) is a differential function with 0≤𝜂̇_𝜙𝑖(𝑡)≤𝜀≤1 and 0≤𝜏._𝜙𝑖(𝑡)≤𝜀≤1. Clearly, this assumption is certainly ensured if the delay 𝜂_𝜙𝑖(𝑡) and 𝜏_𝜙𝑖(𝑡) is constant.

Assumption 2.6 (34) (Global Lipschitz Condition). Suppose that there exist nonnegative constants 𝜗,𝛽, for all 𝑡∈𝑅₊, such that for any time-varying vectors 𝑥(𝑡),𝑦(𝑡)∈𝑅^𝑛‖‖𝐻₁(𝑥)−𝐻₁(𝑦)‖‖≤𝜗‖𝑥−𝑦‖,‖‖𝐻₂(𝑥)−𝐻₂(𝑦)‖‖≤𝛽‖𝑥−𝑦‖,(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 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 as𝑒_𝑖̇(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)−𝐻₁𝑠_𝜙𝑖(𝑡)_𝑗=1𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)+^𝑁_𝑖=1+𝑎_𝑖𝑗𝐻₁𝑠_𝜙𝑖(𝑡)_𝑁_𝑖=1𝑒_𝑖̇𝑏_𝑖𝑗𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)−𝐻₁𝑠_𝜙𝑖(𝑡)_𝑗=1−𝑣_𝑖𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽_𝜙𝑖,(3.1) where 𝑓_𝜙𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏_𝜙𝑖(𝑡)))=𝑓_𝜙𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏_𝜙𝑖(𝑡)))−𝑓_𝜙𝑖(𝑡,𝑠_𝜙𝑖(𝑡),𝑠_𝜙𝑖(𝑡−𝜏_𝜙𝑖(𝑡))) for 𝑖=1,2,…,𝑁.

According to the diffusive coupling condition (2.2) of the matrix 𝐴,𝐵 we have𝑐^𝑁_𝑖=1𝑎_𝑖𝑗𝐻₁𝑠_𝜙𝑖(𝑡)+𝑐^𝑁_𝑖=1𝑏_𝑖𝑗𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)=0,𝑖∈𝐽_𝜙𝑖−𝐽_𝜙𝑖.(3.2) On the basis of this property, for achieving cluster synchronization, we design controllers as follows:𝑢_𝑖(𝑡)=⎧⎪⎨⎪⎩−𝑐^𝑁_𝑖=1𝑎_𝑖𝑗𝐻₁𝑠_𝜙𝑖(𝑡)−𝑐^𝑁_𝑖=1𝑣_𝑖𝑏_𝑖𝑗𝐻₂𝑠_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).

Theorem 3.1. Suppose assumptions 2.4–2.5 hold. Consider the network (2.1) via control law (3.3). If the following conditions hold: 𝛼+𝜗𝑐𝜆_max(𝑄)+12𝛽²𝑐²𝜆_max𝑃𝑃^𝑇+11−𝜀𝛾+21<𝑑,(3.4) where 𝛼=max(𝛼_𝜙1,𝛼_𝜙2,…,𝛼_𝜙𝑚),𝛾=max(𝛾_𝜙1,𝛾_𝜙2,…,𝛾_𝜙𝑚). Then, the systems (2.3) is cluster synchronization.

Proof. Construct the following Lyapunov functional: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛾1−𝜀^{𝑡𝑡−𝜏}_{𝜙𝑖𝑁(𝑡)}_𝑖=1+1𝑒^𝑇𝑖(𝜃)𝑒_𝑖(𝜃)𝑑𝜃2(1−𝜀)^{𝑡𝑡−𝜂}_{𝜙𝑖𝑁(𝑡)}_𝑖=1𝑒^𝑇𝑖(𝜃)𝑒_𝑖(𝜃)𝑑𝜃+12^𝑁_𝑖=1𝑑_𝑖−𝑑²𝑘_𝑖.(3.5) Calculating the derivative of 𝑉(𝑡), we have 𝑉̇(𝑡)=^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖̇(𝑡)+11−𝜀𝛾+12^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝛾1−𝜏.^𝜙𝑖(𝑡)1−𝜀^𝑁_𝑖=1−1−𝜂̇_𝜙𝑖𝑒^𝑇𝑖𝑡−𝜏_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜏_𝜙𝑖(𝑡)(𝑡)2(1−𝜀)^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)+^𝑁_𝑖=1𝑑_𝑖=−𝑑𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)_𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)−𝐻₁𝑠_𝜙𝑖(𝑡)_𝑗=1+1𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡)1−𝜀𝛾+21^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝛾1−𝜏.^𝜙𝑖(𝑡)1−𝜀^𝑁_𝑖=1−1−𝜂̇_𝜙𝑖𝑒^𝑇𝑖𝑡−𝜏_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜏_𝜙𝑖(𝑡)(𝑡)2(1−𝜀)^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)+^𝑁_𝑖=1𝑑_𝑖−𝑑𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡).(3.6) By assumptions 2.4–2.6, we obtain ≤𝛼^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝛾^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜏_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝜗𝑐^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)^𝑁_𝑗=1+𝛽𝑐_𝑁𝑎_𝑖𝑗𝑒_𝑗(𝑡)_𝑖=1𝑒^𝑇𝑖(𝑡)^𝑁_𝑗=1𝑏_𝑖𝑗𝑒_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑁_𝑖=1+1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)1−𝜀𝛾+21^𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)−𝛾^𝑁_𝑖=1−1𝑒^𝑇𝑖𝑡−𝜏_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜏_𝜙𝑖(𝑡)2^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)+^𝑁_𝑖=1𝑑_𝑖≤𝛼_𝑁−𝑑𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)_𝑖=1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)+𝜗𝑐𝑒^𝑇(𝐴⊗𝐼)𝑒+𝛽𝑐𝑒^𝑇(𝐵⊗𝐼)𝑒𝑡−𝜂_𝜙𝑖(𝑡)+11−𝜀𝛾+21^𝑁_𝑖=1−1𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)2^𝑁_𝑖=1𝑒^𝑇𝑖𝑡−𝜂_𝜙𝑖(𝑡)𝑒_𝑖𝑡−𝜂_𝜙𝑖(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.7) Let 𝑒(𝑡)=(𝑒^𝑇1(𝑡),𝑒^𝑇2(𝑡),…,𝑒^𝑇𝑁(𝑡))^𝑇∈𝑅^𝑛𝑁,𝑄=(𝐴⊗𝐼),𝑃=(𝐵⊗𝐼), where ⊗ represents the Kronecker product. Then 𝑉̇(𝑡)≤𝛼𝑒^𝑇(𝑡)𝑒(𝑡)+𝜗𝑐𝑒^𝑇(𝑡)𝑄𝑒(𝑡)+𝛽𝑐𝑒^𝑇(𝑡)𝑃𝑒𝑡−𝜂_𝜙𝑖(𝑡)+11−𝜀𝛾+21𝑒^𝑇−1(𝑡)𝑒(𝑡)2𝑒^𝑇𝑡−𝜂_𝜙𝑖(𝑡)𝑒𝑡−𝜂_𝜙𝑖(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.8) By the Lemma 2.3, we have ≤𝛼𝑒^𝑇(𝑡)𝑒(𝑡)+𝜗𝑐𝑒^𝑇(𝑡)𝑄𝑒(𝑡)+12(𝛽𝑐)²𝑒^𝑇(𝑡)𝑃𝑃^𝑇𝑒(𝑡)+11−𝜀𝛾+21𝑒^𝑇≤𝛼+𝜗𝑐𝜆_max(𝑡)𝑒(𝑡)−𝑑𝑒^𝑇(𝑡)𝑒(𝑡)(𝑄)+12𝛽²𝑐²𝜆_max𝑃𝑃^𝑇+11−𝜀𝛾+21−𝑑𝑒^𝑇(𝑡)𝑒(𝑡).(3.9) Therefore, if we have 𝛼+𝜗𝑐𝜆_max(𝑄)+(1/2)𝛽²𝑐²𝜆_max(𝑃𝑃^𝑇)+(1/(1−𝜀))(𝛾+(1/2))<𝑑 then 𝑉̇(𝑡)≤0.(3.10) Theorem 3.1 is proved completely.
We can conclude that, for any initial values, the solutions 𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡) of the system (2.3) satisfy lim_𝑡→∞∑^𝑚𝑘=1∑_{𝑖∈𝐶𝑘}‖𝑥_𝑖(𝑡)−𝑠_𝑘(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 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡),𝑖=1,2,…,𝑁.(3.11)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢_𝑖(𝑡)=⎧⎪⎨⎪⎩−𝑐^𝑁_𝑖=1𝑣_𝑖𝑏_𝑖𝑗𝐻₂𝑠_𝜙𝑖𝑡−𝜂_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖,(𝑡)=𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖−𝐽_𝜙𝑖.(3.12)

Corollary 3.3. When 𝐵=0, network (2.1) is translated into 𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁.(3.13)
We design the controllers, as follows, then the complex networks can also achieve synchronization, where 𝑢_𝑖(𝑡)=⎧⎪⎨⎪⎩−𝑐^𝑁_𝑖=1𝑣_𝑖𝑎_𝑖𝑗𝐻₁𝑠_𝜙𝑖(𝑡)−𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖,(𝑡)=𝑑_𝑖𝑒_𝑖(𝑡),𝑖∈𝐽_𝜙𝑖−𝐽_𝜙𝑖,(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:𝑥̇_𝑖(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)_𝑗=1𝑥̇_𝑖𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)+𝑢_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖,(𝑡)=𝑓_𝜙𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖𝑡−𝜏_𝜙𝑖(𝑡)+𝑐^𝑁_𝑗=1+𝑐_𝑁𝑎_𝑖𝑗𝐻₁𝑥_𝑗(𝑡)_𝑗=1𝑏_𝑖𝑗𝐻₂𝑥_𝑗𝑡−𝜂_𝜙𝑖(𝑡)−𝑣_𝑖(𝑡),𝜙_𝑖(𝑡)∈𝐽_𝜙𝑖−𝐽_𝜙𝑖,𝑖=1,2,…,𝑁,(4.1) where 𝑥𝜏_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),𝑥_𝑖3(𝑡))^𝑇,𝑓₁(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏₁(𝑡)))=𝐷₁𝑥_𝑖(𝑡)+ℎ₁₁(𝑥_𝑖(𝑡))+ℎ₁₂(𝑥_𝑖(𝑡−𝜏₁(𝑡))),𝑓₂(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−₂𝜏(𝑡)))=𝐷₂𝑥_𝑖(𝑡)+ℎ₂₁(𝑥_𝑖(𝑡))+ℎ₂₂(𝑥_𝑖(𝑡−𝜏₂(𝑡)))+𝑉,𝑓₃(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏₃(𝑡)))=𝐷₃𝑥_𝑖(𝑡)+ℎ₃₁(𝑥_𝑖(𝑡))+ℎ₃₂(𝑥_𝑖(𝑡−₃(𝑡))).𝑘₁=𝑘₂=⋯=𝑘_𝑁=10,𝑐=1,𝐻₁(𝑥)=sin𝑥,𝐻₂(𝑥)=cos𝑥.

In simulation, we choose [ℎ0,₁₁0(𝑥,0_𝑖).2=]_𝑇(0,−𝑥_𝑖1𝑥_𝑖3,𝑥_𝑖1𝑥_𝑖2)^𝑇,ℎ₁₂(𝑥_𝑖)=(0,5𝑥_𝑖2,0)^𝑇,ℎ₂₁(𝑥_𝑖)=(0,0,𝑥_𝑖1𝑥_𝑖3)^𝑇,ℎ₂₂(𝑥_𝑖)=(𝑥_𝑖1,0,0)^𝑇,𝑉=𝑒,ℎ₃₁(𝑥_𝑖)=(3.247(|𝑥_𝑖1+1|−|𝑥_𝑖1−1|),0,0)^𝑇,ℎ₃₂(𝑥_𝑖)=(0,0,−3.906sin(0.5𝑥_𝑖1))^𝑇,𝜏₁(𝑡)=𝑒^𝑡/(1+_𝑡),𝜏₂(𝑡)=2𝑒^𝑡/(1+𝑒^𝑡),𝜏₃(𝑡)=0.5𝑒^𝑡/(1+𝑒^𝑡),𝐷₁=⎡⎢⎢⎢⎢⎣−201801400−0083⎤⎥⎥⎥⎥⎦,𝐷²=⎡⎢⎢⎢⎢⎣0100−0.12−−101.2⎤⎥⎥⎥⎥⎦,𝐷₃=⎡⎢⎢⎢⎢⎣−2.10169−1−1901.53−0.101636⎤⎥⎥⎥⎥⎦.(4.2) Taking the weight configuration coupling matrices𝐴=𝐵=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣−−000121−110002−011002−001102−011002−001102⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.3) The following quantities are utilized to measure the process of cluster synchronization𝐸(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,(4.4) 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 1–4, we see the time evolution of the synchronization errors. The numerical results show that Theorem 3.1 is effective.

Figure 1 

Time evolution of the synchronization errors 𝐸(𝑡).

Figure 2 

Time evolution of the synchronization errors 𝐸12(𝑡).

Figure 3 

Time evolution of the synchronization errors 𝐸13(𝑡).

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.