Cluster Synchronization of Impulsive Complex Networks with Stochastic Perturbations and Time-Varying Delays

Abstract

This paper investigates the cluster synchronization of impulsive complex networks with stochastic perturbation and time-varying delays. Besides, the nodes in the complex networks are nonidentical. By utilizing the Lyapunov stability theory, stochastic analysis theory, and linear matrix inequalities (LMI), sufficient conditions are derived to guarantee the cluster synchronization. The numerical simulation is provided to show the effectiveness of the theoretical results.

1. Introduction

Over the past few years, the study of complex networks has become an important issue. The complex networks are constructed with individual units called nodes, connected by links that exhibit complex topological properties, such as coupled biological and chemical system, neural networks, social interacting species, the Internet, and the World Wide Web [1]. Thereafter, the problem of control and synchronization in complex networks has attracted increasing attention for its wide applications in real world.

From the literature, there are three common phenomena in many evolving networks: delay effects, stochastic effects, and impulsive effects. First of all, considering finite switching speed of amplifiers and finite signal propagation time, time delay is ubiquitous in the implementation of electronic networks. There are two kinds of time delays in complex dynamical networks. The inner delay time causes chaos, such as delayed neural networks and delayed Chua’s circuit system and the outer coupling delay time also exists widely, such as in communication and traffic congestion. Therefore, time delays cannot be ignored in order to simulate realistic networks; see [2–5]. Second, the nodes in dynamical networks are often subject to various types of noise and uncertainty, which can have a great influence on the behavior of dynamical networks. For instance, signals transmitted between nodes of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost; see [6, 7]. Therefore, stochastic perturbations should be included in the complex network model. Third, impulsive exchange is a common phenomenon in many evolving networks. The state of nodes is often subject to instantaneous perturbations and experience abrupt change at certain instants which may be caused by switching phenomena, frequency change, or sudden noise. For example, the states of electronic networks and biological networks are often subject to instantaneous disturbances and experience abrupt changes at certain instants, which may be caused by switching phenomenon, frequency change, or other sudden noises, which could be expressed as impulsive effects. Thus, impulsive dynamical networks, which involve sudden changes at certain discrete times, are receiving more and more attention of researchers for various fields, that is, [8–10] and the references therein. Since time delays, stochastic perturbations, and impulsive effects can heavily affect the dynamical behaviors of the networks, it is necessary to investigate the time delay, stochastic perturbations, and impulsive effects on the synchronization of dynamical networks. Actually, impulsive stochastic perturbed complex networks have attracted the interest of many researchers for their various applications in information science, economic systems, automated control systems, and so forth.

Synchronization is one of the hot topics in the investigation of complex networks. Generally speaking, synchronization is the process in which two or more dynamical systems seek to adjust a certain prescribed property of their motion to a common behavior in the limit as time tends to infinity either by virtue of coupling or by forcing [11]. Synchronization of complex network has observed wide potential applications in many different areas such as population dynamic, power system, chemical process simulation, automatic control, and many others [12–18]. Synchronization patterns are classified as complete synchronization [19], lag synchronization [20], cluster synchronization [21], phase synchronization [22], and partial synchronization [23]. Among them, the cluster synchronization has received more and more attention. 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. The clusters can be classified by different function communities contained in the networks of real world, 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 [24]. There have been already some papers focused on sufficient conditions for the global stability of cluster synchronization of some related networks; see [25, 26] and the references therein. In [27], Wang et al. investigated the exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning. Compared with the work in [27], in this paper, we would focus on the cluster synchronization of complex networks with stochastic perturbations, time-varying delays, and impulsive effects. It should also be mentioned that synchronization problems could be seen as an application of stability. There are many papers that are concerned with the stability of complex systems with time delays, such as [28–31], which inspire the studies of synchronization problems.

As everyone knows, the real-world networks normally have a large number of nodes, and it is usually impractical to control a complex network by adding the controllers to all nodes. Pinning control, in which controllers are only applied to a small fraction of nodes, is an effective way to reduce the number of controlled nodes. Chen et al. [32] proved that the complex networks with symmetric or asymmetric coupling matrix achieved synchronization by a single pinning controller. In [33], Zhao et al. proposed a pinning scheme to solve the problems how to stabilize a network onto a homogeneous state globally exponentially and how to select an optimal combination between the number of pinned nodes and the feedback control gain.

Based on the above analysis, in this paper, we study the cluster synchronization of impulsive complex networks with time-varying delays and stochastic perturbations by adding feedback controllers and impulsive controllers on a fraction of selected nodes. To obtain our main results, we first formulate a new complex network nondelayed and time-varying delayed linear coupling and vector-form stochastic perturbations. By using the Lyapunov functional method, the stochastic stability analysis theory, and linear matrix inequality technique (LMI), some novel sufficient conditions are derived to guarantee cluster synchronization of the complex networks.

Remark 1. The main tools to derive the results in this paper are Lyapunov functional method, Itô formula, and linear matrix inequality technique (LMI). It is well known that Lyapunov functional method is one of the most useful tools to handle the stability problems. Since stochastic perturbations are considered in the model of this paper, Itô formula is employed to deal with the stochastic differential equations arising in the analysis of synchronization. Linear matrix inequality technique also plays an important role in the proof of our main theorem.

The paper is organized as follows. In Section 2, a general model of impulsive complex network with both time-varying delays and stochastic perturbations is given, and some preliminaries are stated. In Section 3, some cluster synchronization criteria for such complex dynamical network are established. In Section 4, a numerical example for verifying the effectiveness of the theoretical results is provided. We conclude the paper in Section 5.

2. Preliminaries

For the facility of statements, we give some definitions of notations and lemmas, which would be used in the analysis of the next sections. Similar definitions can also be found in [27]. Suppose thatℝ^𝑛andℝ^𝑛×𝑛denote the𝑛-dimensional Euclidean space and the set of all𝑛×𝑛real matrices, respectively. The superscript𝑇denotes the transpose of a matrix or vector,Tr(⋅)denotes the trace of the corresponding matrix, and𝐼_𝑛denotes the identity matrix with𝑛dimensions. For square matrices𝑀, the notation𝑀>0(<0) denotes that𝑀is positive-definite (negative-definite) matrix. Suppose that𝐴is a symmetric matrix. Denote𝜆_max(𝐴)as the largest eigenvalue of𝐴, and denote𝜆_min(𝐴)as the smallest eigenvalue of𝐴. Let(Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝒫)be a completed probability space with a filtration{ℱ_𝑡}_𝑡≥0which is right continuous andℱ₀contains all𝒫-null sets. Let𝑤(𝑡)=(𝑤₁(𝑡),𝑤₂(𝑡),…,𝑤_𝑛(𝑡))^𝑇be an𝑚-dimensional Brownian motion defined on probability space. Denote by𝐶([−𝜏,0];𝑅^𝑛)the family of continuous function𝜙from[−𝜏,0]toℝ^𝑛with the uniform norm‖𝜙‖²=sup_{−𝜏≤𝑠≤0}𝜙(𝑠)^𝑇𝜙(𝑠). Denote by𝐶²_ℱ0([−𝜏,0];ℝ^𝑛)the family of allℱ₀measurable,𝐶([−𝜏,0];ℝ^𝑛)-valued stochastic variables𝜉={𝜉(𝜃)∶−𝜏≤𝜃≤0}such that∫⁰_−𝜏𝔼‖𝜉(𝑠)‖²𝑑𝑠≤∞, where𝔼stands for the correspondent expectation operator with respect to the given probability measure𝒫.

Consider an impulsive complex network consisting of𝑁nodes and𝑚communities (𝑁>𝑚≥2) with nondelayed and time-varying delayed linear coupling and vector-form stochastic perturbations, which is described as 𝑑𝑥_𝑖(𝑡)=𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_{𝑁𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡+_{𝑁𝑗=1,𝑖≠𝑗})𝑏_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖(𝑡)𝑑𝑡+𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁,𝑡≥0,𝑡≠𝑡_𝑘𝑥,𝑘∈ℕ_𝑗𝑡⁺_𝑘−𝑥_𝑖𝑡⁺_𝑘=𝜈𝑥_𝑗𝑡⁻_𝑘−𝑥_𝑖𝑡⁻_𝑘for𝑘∈ℕ𝑖,𝑗∶𝑎_𝑖𝑗>0, (1) where𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛represents the state vector of the𝑖th node of the network;𝑓_𝜇𝑖(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))=[𝑓_𝜇𝑖1(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),𝑓_𝜇𝑖2(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))),…,𝑓_𝜇𝑖𝑛(𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡)))]^𝑇is a continuous vector-form function. If node𝑖belongs to the𝑗th community, then we let𝜇_𝑖=𝑗. We denote by𝑈_𝑖the set of all nodes in the𝑖th community and𝑈^𝐴_𝑖, which is the subset of𝑈_𝑖, is the index set of all nodes in the𝑖th community having direct connections to other communities in𝐴. And by the similar way, we can define𝑈^𝐵_𝑖. The function𝑓_𝜇𝑖(⋅)describes the local dynamics of nodes in the𝜇_𝑖th community, which is differentiable and capable of performing abundant dynamical behaviors. For any pair of indices𝑖and𝑗, if𝜇_𝑖≠𝜇_𝑗, which means node𝑖and node𝑗belong to different communities, then𝑓_𝜇𝑖≠𝑓_𝜇𝑗.Γ=diag(𝛾₁,𝛾₂,…,𝛾_𝑛)is an inner coupling of the networks, satisfying𝛾_𝑗>0,𝑗=1,2,…,𝑛.𝐴=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛and𝐵=[𝑏_𝑖𝑗]∈ℝ^𝑛×𝑛are outer coupling matrices of the networks at time𝑡and𝑡−𝜏_𝑐(𝑡), respectively, which are both symmetric, satisfying𝑎_𝑖𝑗≥0for𝑖≠𝑗,𝑎_𝑖𝑖=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑎_𝑖𝑗, and𝑏_𝑖𝑗≥0for𝑖≠𝑗,𝑏_𝑖𝑖=−∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝑏_𝑖𝑗;𝜏(𝑡)is the inner time-varying delay satisfying𝜏≥𝜏(𝑡)≥0, and𝜏_𝑐(𝑡)is the coupling time-varying delay satisfying𝜏_𝑐≥𝜏_𝑐(𝑡)≥0;𝜎_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥(𝑡−𝜏_𝑐(𝑡)))=𝜎_𝑖(𝑡,𝑥₁(𝑡),…,𝑥_𝑛(𝑡),𝑥₁(𝑡−𝜏(𝑡)),…,𝑥_𝑛(𝑡−𝜏(𝑡)),𝑥₁(𝑡−𝜏_𝑐(𝑡)),…,𝑥_𝑛(𝑡−𝜏_𝑐(𝑡)))∈ℝ^𝑛×𝑛is the noise intensity matrix, and𝑤_𝑖(𝑡)=(𝑤_𝑖1(𝑡),𝑤_𝑖2(𝑡),…,𝑤_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛is a bounded vector-form Weiner process, satisfying 𝔼𝑤_𝑖𝑗(𝑡)=0,𝔼𝑤²_𝑖𝑗(𝑡)=1,𝔼𝑤_𝑖𝑗(𝑡)𝑤_𝑖𝑗(𝑠)=0(𝑠≠𝑡). (2) In this paper, we always assume that𝐴is irreducible in the sense that there is no isolated node and|𝜈|<1, which means that corresponding impulsive effects are synchronizing.

The initial conditions associated with (1) are 𝑥_𝑖(𝑠)=𝜉_𝑖(𝑠),−𝜏̌≤𝑠≤0,𝑖=1,2,…,𝑁, (3) where𝜏̌=max{𝜏,𝜏_𝑐},𝜉_𝑖∈𝐶^𝑏_ℱ0([−𝜏̌,0],ℝ^𝑛)with the norm‖𝜉_𝑖‖²=sup_{−̌𝜏≤𝑠≤0}𝜉_𝑖(𝑠)^𝑇𝜉_𝑖(𝑠). We define𝑠_𝜇𝑖=(𝑠_𝜇𝑖1,𝑠_𝜇𝑖2,…,𝑠_𝜇𝑖𝑛)^𝑇∈𝑅^𝑛that satisfies 𝑑𝑠_𝜇𝑖(𝑡)=𝑓_𝜇𝑖𝑡,𝑠_𝜇𝑖(𝑡),𝑠_𝜇𝑖(𝑡−𝜏(𝑡))−𝑏_𝑖𝑖Γ𝑠_𝜇𝑖(𝑡−𝜏(𝑡))−𝑠_𝜇𝑖(𝑡)𝑑𝑡, (4) which describes the identical local dynamics for the nodes in the𝜇_𝑖th community. Obviously, the synchronization state in the same community𝑠_𝜇𝑖(𝑡)is uniform. Therefore, we assume that𝑏₁₁=𝑏₂₂=⋯=𝑏_𝑁𝑁.

In order to achieve the cluster synchronization objective, the feedback controllers as well as impulsive controllers are added to part of its nodes. When𝑡≠𝑡_𝑘for𝑘∈ℕ𝑢_𝑖(𝑡)=−𝜀_𝑖𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)−∑^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑗(𝑡)𝑖∈𝑈^𝐴_𝜇𝑖⧵𝑈^𝐵_𝜇𝑖,−𝜀_𝑖𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)−∑^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑗(𝑡)𝑖∈𝑈^𝐵_𝜇𝑖⧵𝑈^𝐴_𝜇𝑖,−𝜀_𝑖𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)−∑^𝑁_𝑗=1𝑎_𝑖𝑗Γ𝑠_𝜇𝑗(𝑡)−∑^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑠_𝜇𝑗(𝑡)𝑖∈𝑈^𝐴_𝜇𝑖∩𝑈^𝐵_𝜇𝑖,0,𝑖otherwise, (5) and when𝑡=𝑡_𝑘for𝑘∈ℕ𝑒_𝑖𝑡⁺_𝑘=𝜈𝑒_𝑖𝑡⁻_𝑘for𝑖∈𝑈^𝐴_𝜇𝑖∪𝑈^𝐵_𝜇𝑖, (6) where𝜀_𝑖>0(𝑖∈𝑈^𝐴_𝜇𝑖∪𝑈^𝐵_𝜇𝑖)represents control gain and by defining𝜀_𝑖=0for𝑖otherwise, we can denoteΞ=diag{𝜀₁,𝜀₂,…,𝜀_𝑁}. Define the synchronization error as𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠_𝜇𝑖(𝑡)(𝑖=1,2,…,𝑁), according to the controllers (5) and (6), the error system is derived as follows: 𝑑𝑒_𝑖(𝑡)=𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝜇𝑖𝑡,𝑠_𝜇𝑖(𝑡),𝑠_𝜇𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑎_𝑖𝑗Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)−𝑏_𝑖𝑖Γ𝑒_𝑖𝑡−𝜏_𝑐(𝑡)−𝑒_𝑖(𝑡)−𝜀_𝑖𝑒_𝑖(𝑡)𝑑𝑡+𝜎_𝑖(𝑡)((𝑡,𝑥𝑡,𝑥,𝑥𝑡−𝜏))𝑡−𝜏_𝑐(𝑡)𝑑𝑤_𝑖(𝑡),for𝑖=1,2,…,𝑁,𝑡≥0,𝑡≠𝑡_𝑘𝑒,𝑘∈ℕ_𝑖𝑡⁺_𝑘−𝑒_𝑗𝑡⁺_𝑘=𝜈𝑒_𝑖𝑡⁻_𝑘−𝑒_𝑗𝑡⁻_𝑘for𝑖,𝑗∶𝑎_𝑖𝑗>0,𝑡=𝑡_𝑘𝑒,𝑘∈ℕ_𝑖𝑡⁺_𝑘=𝜈𝑒_𝑖𝑡⁻_𝑘for𝑖∈𝑈^𝐴_𝜇𝑖∪𝑈^𝐵_𝜇𝑖,𝑡=𝑡_𝑘,𝑘∈ℕ. (7)

Remark 2. Since both state coupling and impulsive coupling are considered in (1), it is reasonable to expect that both the state-feedback controller (5) and the impulsive controller (6) will play important roles in the stabilizing process of the complex network (1). Thus, the controller designed in this paper consists of both the state-feedback controller (5) and the impulsive controller (6).

Definition 3. The complex network (4) is said to be exponentially synchronized if the trivial solution of system (9) satisfies ^𝑁_𝑖=1𝔼𝑒_𝑖𝑡,𝑡₀,𝜉_𝑖²≤𝐾𝑒^−𝜅𝑡, (8) where𝐾>0and𝜅>0, for any initial data𝜉_𝑖∈𝒞^𝑏_ℱ0([−𝜏,0];ℝ^𝑛).

Definition 4 (see [6]). A continuous function𝑓(𝑡,𝑥,𝑦)∶[0,+∞]×ℝ^𝑛×ℝ^𝑛→ℝ^𝑛is said to be in theQUADfunction class, denoted as𝑓∈QUAD(𝑃,Δ,𝜂,𝜁), for given matrixΓ=diag{𝛾₁,𝛾₂,…,𝛾_𝑛}, if there exist positive definite diagonal matrices𝑃=diag{𝑝₁,𝑝₂,…,𝑝_𝑛}, diagonal matricesΔ=diag{𝛿₁,𝛿₂,…,𝛿_𝑛},and constants𝜂>0,𝜁>0, such that𝑓satisfies the following condition: 𝑥−𝑦^𝑇𝑃𝑓(𝑡,𝑥,𝑧)−𝑓𝑡,𝑦,𝑤−ΔΓ𝑥−𝑦≤−𝜂𝑥−𝑦^𝑇𝑥−𝑦+𝜁(𝑧−𝑤)^𝑇(𝑧−𝑤) (9) for all𝑥,𝑦,𝑧,𝑤∈ℝ^𝑛.

We will simply introduce the following notations: 𝑝̌=max𝑝₁,𝑝₂,…,𝑝_𝑛,𝑝=min𝑝₁,𝑝₂,…,𝑝_𝑛,𝛿̌=max𝛿₁,𝛿₂,…,𝛿_𝑛. (10)

Definition 5 (see [34]). The average impulsive interval of the impulsive sequence𝜁={𝑡₁,𝑡₂,…}is less than𝑇_𝑎, if there exist a positive integer𝑁₀and a positive number𝑇_𝑎, such that 𝑁_𝜁(𝑇,𝑡)≥𝑇−𝑡𝑇_𝑎−𝑁₀,∀𝑇≥𝑡≥0, (11) where𝑁_𝜁(𝑇,𝑡)denotes the number of impulsive times of the impulsive sequence𝜁in the time interval(𝑡,𝑇).

The following assumptions will be used throughout this paper in establishing our synchronization condition.(H1)𝑏₁₁=𝑏₂₂=⋯=𝑏_𝑁𝑁=𝑏.(H2)𝜏(𝑡)and𝜏_𝑐(𝑡)are abounded and continuously differentiable functions, satisfying0<𝜏(𝑡)≤𝜏,𝜏̇(𝑡)<𝜏<1,0<𝜏_𝑐(𝑡)≤𝜏_𝑐, and𝜏̇_𝑐(𝑡)<𝜏_𝑐<1. Denote𝜏̌‌=max{𝜏,𝜏_𝑐}.(H3)Denote𝜎(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒(𝑡−𝜏_𝑐(𝑡)))=𝜎(𝑡,𝑒₁(𝑡),…,𝑒_𝑁(𝑡),𝑒₁(𝑡−𝜏(𝑡)),…,𝑒_𝑁(𝑡−𝜏(𝑡)),𝑒₁(𝑡−𝜏_𝑐(𝑡)),…,𝑒_𝑁(𝑡−𝜏_𝑐(𝑡))). There exist appropriate dimensional positive definite constant matricesΥ_𝑖1,Υ_𝑖2, andΥ_𝑖3for𝑖=1,2,…,𝑁such that Tr𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)^𝑇×𝜎_𝑖𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)),𝑒𝑡−𝜏_𝑐(𝑡)≤^𝑁_𝑗=1𝑒_𝑗(𝑡)^𝑇Υ_𝑖1𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑒_𝑗(𝑡−𝜏(𝑡))^𝑇Υ_𝑖2𝑒_𝑗((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑒_𝑗𝑡−𝜏_𝑐(𝑡)^𝑇Υ_𝑖3𝑒_𝑗𝑡−𝜏_𝑐(𝑡). (12)

Lemma 6 (see [35] (Itô formula)). Consider an𝑛-dimensional stochastic differential equation (𝑡)((𝑡)(((𝑡)((𝑡).𝑑𝑥=𝑓𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝑡+𝜎𝑡,𝑥,𝑥𝑡−𝜏))𝑑𝜔 (13) Let𝐶^2,1(ℝ₊×ℝ^𝑛;ℝ₊)denote the family of all nonnegative functions𝑉(𝑡,𝑥)onℝ₊×ℝ^𝑛, which are twice continuously differentiable in𝑥and once differentiable in𝑡. If   𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛;ℝ₊), define an operatorℒ𝑉fromℝ₊×ℝ^𝑛toℝby ℒ𝑉(𝑡,𝑥)=𝑉_𝑡(𝑡,𝑥)+𝑉_𝑥()𝑓+1𝑡,𝑥𝑡,𝑥,𝑦2Tr𝜎𝑡,𝑥,𝑦^𝑇𝑉_𝑥𝑥𝜎𝑡,𝑥,𝑦, (14) where𝑉_𝑡(𝑡,𝑥)=𝜕𝑉(𝑡,𝑥)/𝜕𝑡,𝑉_𝑥(𝑡,𝑥)=(𝜕𝑉(𝑡,𝑥)/𝜕𝑥₁,…,𝜕𝑉(𝑡,𝑥)/𝜕𝑥_𝑛),𝑉_𝑥𝑥(𝑡,𝑥)=(𝜕²𝑉(𝑡,𝑥)/𝜕𝑥_𝑖𝑥_𝑗)_𝑛×𝑛.If𝑉∈𝐶^2,1(ℝ₊×ℝ^𝑛;ℝ₊); then for any∞>𝑡>𝑡₀≥0, 𝔼𝑉(𝑡,𝑥(𝑡))=𝔼𝑉𝑡₀,𝑥𝑡₀+𝔼^𝑡_𝑡0ℒ𝑉(𝑠,𝑥(𝑠))𝑑𝑠 (15) as long as the expectations of the integrals exist.

3. Main Results

We would give some sufficient conditions for the cluster synchronization of impulsive complex networks with time-varying delays and stochastic perturbations in this section.

Theorem 7. Suppose the assumptions (H1), (H2), and (H3) hold and𝑓_𝜇𝑖∈𝑄𝑈𝐴𝐷(𝑃,Δ,𝜂,𝜁). If there exist positive constants𝛼and𝛽, such that 𝐴+𝛿̌+𝑏𝐼_𝑁−Ξ+𝛼𝐼_𝑁𝐵−𝑏𝐼_𝑁2𝐵^𝑇−𝑏𝐼_𝑁2−𝛽𝐼_𝑁̌≤0,0≤𝜏≤1−𝑏+𝑐𝑎, (16)(17) where𝛾>0is the smallest root of the equation 𝑏𝛾−𝑎+1−𝜏𝑒^𝛾𝜏+𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐𝜆=0,𝑎=^min2𝜂𝐼_𝑛−𝑝̌∑^𝑁_𝑖=1Υ_𝑖1+2𝛼𝑃Γ𝑝̌,𝜆𝑏=^max∑^𝑁_𝑖=1𝑃Υ_𝑖2+2𝜁𝐼_𝑁𝑝,𝜆𝑐=^max∑^𝑁_𝑖=1𝑃Υ_𝑖3+2𝛽𝑃Γ𝑝, (18)(19) then the solution𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)of system (4) could reach cluster synchronization under controller (5) when the average impulsive interval is less than𝑇_𝑎, and0<𝜈²+(𝑏𝜏/(1−𝜏))𝑒^𝛾𝜏+(𝑐𝜏_𝑐/(1−𝜏_𝑐))𝑒^𝛾𝜏𝑐<1.

Proof. Let𝑒^𝑘̃(𝑡)=(𝑒_1𝑘(𝑡),𝑒_2𝑘(𝑡),…,𝑒_𝑁𝑘(𝑡))^𝑇,𝑘=1,2,…,𝑛. Define a Lyapunov function 𝑉((𝑡(𝑡)=1𝑡,𝑒))≡𝑉2^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡). (20) When𝑡∈[𝑡_𝑘,𝑡_𝑘+1), according to Lemma 6, we have ((𝑡=ℒ𝑉𝑡,𝑒))_𝑁𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝜇𝑖𝑡,𝑠_𝜇𝑖(𝑡),𝑠_𝜇𝑖((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑎_𝑖𝑗Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑏_𝑖𝑗Γ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)−𝑏Γ𝑒_𝑖𝑡−𝜏_𝑐(𝑡)−𝑒_𝑖(𝑡)+𝜀_𝑖𝑒_𝑖(𝑡)+12Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)^𝑇×𝑃𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝜇𝑖(𝑡,𝑠(𝑡),𝑠(𝑡−𝜏(𝑡)))−ΔΓ𝑒_𝑖(𝑡+_𝑁𝑖=1)𝑒_𝑖(𝑡)^𝑇𝑃ΔΓ𝑒_𝑖(𝑡)+^𝑁𝑁_𝑖=1𝑗=1𝑎_𝑖𝑗𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑒_𝑗(𝑡)+^𝑁𝑁_𝑖=1𝑗=1𝑏_𝑖𝑗𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)−^𝑁_𝑖=1𝑏𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑒_𝑖𝑡−𝜏_𝑐(𝑡)+^𝑁_𝑖=1𝑏𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑒_𝑖(𝑡)−^𝑙_𝑖=1𝜀_𝑖𝑒_𝑖(𝑡)^𝑇𝑃Γ𝑒_𝑖(𝑡)+12^𝑁_𝑖=1Tr𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)^𝑇×𝑃𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)≤−𝜂^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑒_𝑖(𝑡)+𝜁^𝑁_𝑖=1𝑒_𝑖(𝑡−𝜏(𝑡))^𝑇𝑒_𝑖((𝑡+𝑡−𝜏))_𝑛𝑘=1𝑝_𝑘𝛾_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴+𝛿̌+𝑏𝐼_𝑁−Ξ𝑒^𝑘̃(𝑡)+𝑒^𝑘̃(𝑡)^𝑇𝐵−𝑏𝐼_𝑁𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)+𝑝̌2^𝑁𝑁_𝑗=1𝑖=1𝑒_𝑗(𝑡)^𝑇Υ_𝑖1𝑒_𝑗(𝑡)+𝑒_𝑗(𝑡−𝜏(𝑡))^𝑇Υ_𝑖2𝑒_𝑗(𝑡−𝜏(𝑡))+𝑒_𝑗𝑡−𝜏_𝑐(𝑡)^𝑇Υ_𝑖3𝑒_𝑗𝑡−𝜏_𝑐(𝑡)≤^𝑁_𝑗=1𝑒_𝑗(𝑡)^𝑇−𝜂𝐼_𝑛+𝑝̌2^𝑁_𝑖=1Υ_𝑖1−𝛼₁𝑃Γ𝑒_𝑗(𝑡)+^𝑁_𝑗=1𝑒_𝑗(𝑡−𝜏(𝑡))^𝑇𝜁𝐼_𝑁+𝑝̌2^𝑁_𝑖=1Υ_𝑖2𝑒_𝑗((𝑡+𝑡−𝜏))_𝑁𝑗=1𝑒_𝑗𝑡−𝜏_𝑐(𝑡)^𝑇𝑝̌2^𝑁_𝑖=1Υ_𝑖3+𝛽₁𝑃Γ𝑒_𝑗𝑡−𝜏_𝑐(𝑡)+^𝑛_𝑘=1𝑝_𝑘𝛾_𝑘𝑒^𝑘̃(𝑡)^𝑇𝐴+𝛿̌+𝑏𝐼_𝑁−Ξ+𝛼₁𝐼_𝑁𝑒^𝑘̃(𝑡)+𝑒^𝑘̃(𝑡)^𝑇𝐵−𝑏𝐼_𝑁𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)−𝛽₁𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡)^𝑇𝑒^𝑘̃𝑡−𝜏_𝑐(𝑡). (21) From condition (16), we obtain ℒ𝑉(𝑡,𝑒(𝑡))≤−𝑎𝑉(𝑡)+𝑏𝑉(𝑡−𝜏(𝑡))+𝑐𝑉𝑡−𝜏_𝑐(𝑡). (22)
Define 𝑊(𝑡)≡𝑊(𝑡,𝑒(𝑡))=𝑒^𝛾𝑡𝑉(𝑡,𝑒(𝑡)), (23) where𝛾is the largest root of (18). For𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we calculate the operator ℒ𝑊(𝑡,𝑒(𝑡))=𝑒^𝛾𝑡𝛾𝑉(𝑡)+ℒ𝑉(𝑡)≤𝑒^𝛾𝑡𝛾𝑉(𝑡)−𝑎₁𝑉(𝑡)+𝑏𝑉(𝑡−𝜏(𝑡))+𝑐𝑉𝑡−𝜏_𝑐(𝑡). (24) Therefore, by the generalized Itô formula, we have that, for any𝑡>𝑡₀≥0, 𝑊(𝑡)=𝑊𝑡₀+𝔼^𝑡_𝑡0ℒ𝑊(𝑠)𝑑𝑠. (25) For𝑡∈[0,𝑡₁), we have 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝔼𝑉(0)+𝔼^𝑡₀𝑒^𝛾𝑠𝛾𝑉(𝑠)−𝑎𝑉(𝑠)+𝑏𝑉(𝑠−𝜏(𝑠))+𝑐𝑉𝑠−𝜏_𝑐(𝑠)≤𝑝̌𝑑𝑠2^𝑁_𝑖=1𝔼𝜉_𝑖²+𝛾−𝑎^𝑡₀𝑒^𝛾𝑠𝔼𝑉(𝑠)𝑑𝑠+𝑏𝑒^𝛾𝜏𝑡₀𝑒^{𝛾(𝑠−𝜏(𝑠))}𝔼𝑉(𝑠−𝜏(𝑠))𝑑𝑠+𝑐𝑒^{𝛾𝜏𝑐𝑡}₀𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}𝔼𝑉𝑠−𝜏_𝑐(𝑠)𝑑𝑠. (26) By changing variable𝑠−𝜏(𝑠)=𝑢, we have ^𝑡₀𝑒^{𝛾(𝑠−𝜏(𝑠))}𝔼𝑉(𝑠−𝜏(𝑠))𝑑𝑠=₍₀₎^{𝑡−𝜏(𝑡)}_−𝜏𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢𝜏̇(𝑠)≤1−_𝑡−𝜏𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢1−𝜏≤𝑝̌2𝜏1−𝜏^𝑁_𝑖=1𝔼𝜉_𝑖²+11−𝜏^𝑡₀𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢. (27) Substituting variable𝑠−𝜏_𝑐(𝑠)=𝑢gives ^𝑡₀𝑒^{𝛾(𝑠−𝜏𝑐(𝑠))}𝔼𝑉𝑠−𝜏_𝑐(𝑠)𝑑𝑠=^{𝑡−𝜏𝑐(𝑡)}_{−𝜏𝑐(0)}𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢1−𝜏̇_𝑐(𝑡)≤𝑝̌2𝜏_𝑐1−𝜏_𝑐^𝑁_𝑖=1𝔼𝜉_𝑖²+11−𝜏_𝑐^𝑡₀𝑒^𝛾𝑢𝔼𝑉(𝑢)𝑑𝑢. (28) Taking (27) and (28) into (26) and by condition (18), we obtain (𝑡)≤𝑝̌𝔼𝑉21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑁}_𝑖=1𝔼𝜉_𝑖²𝑒^−𝛾𝑡. (29) When𝑡∈[𝑡₁,𝑡₂), we have𝔼𝑉(𝑡₁)=𝜈²𝔼𝑉(𝑡⁻₁)and 𝑒^𝛾𝑡𝔼𝑉(𝑡)≤𝑒^𝛾𝑡1𝔼𝑉𝑡₁+𝔼^𝑡_𝑡1𝑒^𝛾𝑠𝛾𝑉(𝑠)−𝑎𝑉(𝑠)+𝑏𝑉(𝑠−𝜏(𝑠))+𝑐𝑉𝑠−𝜏_𝑐(𝑠)𝑑𝑠≤𝑒^𝛾𝑡1𝔼𝑉𝑡₁+𝑏1−𝜏𝑒^{𝛾𝜏𝑡1}_𝑡1−𝜏𝑒^𝛾𝑠(𝑠)+𝑐𝔼𝑉𝑑𝑠1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑡1}_{𝑡1−𝜏𝑐}𝑒^𝛾𝑠(𝑠)≤𝑝̌𝔼𝑉𝑑𝑠21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑁}_𝑖=1𝔼𝜉_𝑖². (30) Similarly, we can prove that, when𝑡∈[𝑡_𝑘,𝑡_𝑘+1), we have 𝑒^𝛾𝑡(𝑡)≤𝑝̌𝔼𝑉21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑘𝑁}_𝑖=1𝔼𝜉_𝑖². (31) If0<𝜈²+(𝑏𝜏/(1−𝜏))𝑒^𝛾𝜏+(𝑐𝜏_𝑐/(1−𝜏_𝑐))𝑒^𝛾𝜏𝑐<1, we get (𝑡)≤𝑝̌𝔼𝑉21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑘𝑁}_𝑖=1𝔼𝜉_𝑖²𝑒^−𝛾𝑡≤𝑝̌21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐𝑡/𝑇𝑏−𝑁0}×^𝑁_𝑖=1𝔼𝜉_𝑖²𝑒^−𝛾𝑡=𝑝̌21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐−𝑁0}×𝑒^{((ln(𝜈2+(𝑏𝜏/(1−𝜏))𝑒𝛾𝜏+(𝑐𝜏𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏))/𝑇𝑏𝑁)𝑡}_𝑖=1𝔼𝜉_𝑖²𝑒^−𝛾𝑡≤𝐾𝑒^{(((ln(𝜈2+(𝑏𝜏/(1−𝜏))𝑒𝛾𝜏+(𝑐𝜏𝑐/(1−𝜏𝑐))𝑒𝑐𝛾𝜏))/𝑇𝑎)−𝛾)𝑡}, (32) where 𝑝̌𝐾=21+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^𝛾𝜏𝑐×𝜈²+𝑏𝜏1−𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐1−𝜏_𝑐𝑒^{𝛾𝜏𝑐−𝑁0𝑁}_𝑖=1𝔼𝜉_𝑖²𝑒^−𝛾𝑡. (33) The proof is then completed.

Remark 8. In Theorem 7, we can see that the speed of stabilizing process depends on the dynamical structure and stochastic perturbations of the complex network, as well as the impulsive effects|𝜈|and the average impulsive interval𝑇_𝑎. The smaller|𝜈|and𝑇_𝑎are, the higher the speed of the stabilizing process will be.

When there is no time-varying delay coupling, the network (1) can be changed as 𝑑𝑥_𝑖(𝑡)=𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_{𝑁𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)𝑑𝑡+𝜎_𝑖(𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)))𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,𝑁,𝑡≥0,𝑡≠𝑡_𝑘𝑥,𝑘∈ℕ,_𝑗𝑡⁺_𝑘−𝑥_𝑖𝑡⁺_𝑘=𝜈𝑥_𝑗𝑡⁻_𝑘−𝑥_𝑖𝑡⁻_𝑘for𝑘∈ℕ𝑖,𝑗∶𝑎_𝑖𝑗>0. (34)

Corollary 9. Suppose that the assumptions (H1), (H2), and (H3) hold and𝑓_𝜇𝑖∈𝑄𝑈𝐴𝐷(𝑃,Δ,𝜂,𝜁). If there exist positive constants𝛼such that 𝐴+𝛿̌𝐼_𝑁+𝛼𝐼_𝑁−Ξ≤0,0≤𝜏≤1−𝑏𝑎, (35) where𝛾>0is the smallest root of the equation 𝑏𝛾−𝑎+1−𝜏𝑒^𝛾𝜏𝜆=0,𝑎=^min2𝜂𝐼_𝑛−𝑝̌∑^𝑁_𝑖=1Υ_𝑖1+2𝛼𝑃Γ𝑝̌,𝜆𝑏=^max∑^𝑁_𝑖=1𝑃Υ_𝑖2+2𝜁𝐼_𝑁𝑝, (36) then the solution𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)of system (34) could reach cluster synchronization under the controller (5) when the average impulsive interval is less than𝑇_𝑎and0<𝜈²+(𝑏𝜏/(1−𝜏))𝑒^𝛾𝜏<1.

When the time-varying delays are constant (i.e.,𝜏(𝑡)=𝜏,𝜏_𝑐(𝑡)=𝜏_𝑐), we get the following corollary.

Corollary 10. Suppose the assumptions (H1), (H2), and (H3) hold and𝑓_𝜇𝑖∈𝑄𝑈𝐴𝐷(𝑃,Δ,𝜂,𝜁). If there exist positive constants𝛼and𝛽, such that 𝐴+𝛿̌+𝑏𝐼_𝑁−Ξ+𝛼𝐼_𝑁𝐵−𝑏𝐼_𝑁2𝐵^𝑇−𝑏𝐼_𝑁2−𝛽𝐼_𝑁≤0, (37) where𝛾>0is the smallest root of the equation 𝛾−𝑎+𝑏𝑒^𝛾𝜏+𝑐𝑒^𝛾𝜏𝑐𝜆=0,𝑎=^min2𝜂𝐼_𝑛−𝑝̌∑^𝑁_𝑖=1Υ_𝑖1+2𝛼𝑃Γ𝑝̌,𝜆𝑏=^max∑^𝑁_𝑖=1𝑃Υ_𝑖2+2𝜁𝐼_𝑁𝑝,𝜆𝑐=^max∑^𝑁_𝑖=1𝑃Υ_𝑖3+2𝛽𝑃Γ𝑝, (38) then the solution𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑁(𝑡)of system (4) could reach cluster synchronization under controller (5) when the average impulsive interval is less than𝑇_𝑎and0<𝜈²+𝑏𝜏𝑒^𝛾𝜏+𝑐𝜏_𝑐𝑒^𝛾𝜏𝑐<1.

4. Numerical Simulation

In this section, we give numerical simulation to verify the theorem given in the previous section.

Consider the following chaotic delayed neural networks: 𝑓_𝑖((𝑡)((𝑡(𝑡)𝑡,𝑥,𝑥𝑡−𝜏)))=−𝐶𝑥+𝐴_𝑖𝑓(𝑥(𝑡))+𝐵_𝑖𝑔(𝑥(𝑡−𝜏(𝑡))), (39) where𝑓(𝑥)=𝑔(𝑥)=tanh(𝑥),𝜏(𝑡)=1, 𝐶=1001,𝐴₁=,𝐵2−0.1−54.5₁=−1.5−0.1−0.2−4,𝐴₂=,𝐵2−0.2−54.5₂=.−1.2−0.1−0.1−4 (40) Take𝑃=diag{1,2}andΔ=diag{5,11,5}and we have𝜂=0.15,𝜁=3.25. Hence the condition (9) can be satisfied for𝑖=1,2(see [6]).

In order to verify our results, we consider the following complex network: 𝑑𝑥_𝑖(𝑡)=𝑓_𝜇𝑖𝑡,𝑥_𝑖(𝑡),𝑥_𝑖((𝑡+𝑡−𝜏))_{10𝑗=1,𝑖≠𝑗}𝑎_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+¹⁰_{𝑗=1,𝑖≠𝑗}𝑏_𝑖𝑗Γ𝑥_𝑗𝑡−𝜏_𝑐(𝑡)−𝑥_𝑖(𝑡)𝑑𝑡+𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)𝑑𝑤_𝑖(𝑡),𝑖=1,2,…,20, (41) where Γ=1001,𝜏_𝑐(𝑡)𝑒=0.1^𝑡1+𝑒^𝑡,𝜎_𝑖𝑡,𝑥(𝑡),𝑥(𝑡−𝜏(𝑡)),𝑥𝑡−𝜏_𝑐(𝑡)=0.1×diag𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡). (42) In this simulation, we consider a network with the topological structure of Figure 1 for 20 nodes with 2 communities. The nodes numbered𝑖=1,2,…,10belong to the first community, while the rest of the nodes belong to the second community. By the pinning scheme above, we pin the nodes1,7,8,9,10,11,12,13,14, and20.

(a)

(b)

Figure 1 

The controlled complex network (41). (a) The topological structure of matrix A. (b) The topological structure of matrix B.

Through computation, we get𝜏=1,𝜏=0,𝜏_𝑐=0.1,𝜏_𝑐=0.1, andΥ_𝑖𝑗=0.01𝐼₂for𝑖=1,2,…,𝑁and𝑗=1,2. Let the control strength𝜀_𝑖=90. By using the Matlab LMI toolbox, we can obtain the following solution for the conditions of Theorem 7:𝑎=67.5,𝑏=5.3,𝑐=0.7,𝛼=45.5,𝛽=0.2,𝑇=0.8, and𝜈=0.7.

The initial conditions of the numerical simulations are as follows:𝑥_𝑖𝑗(𝑡)=𝜉_𝑖𝑗,𝑖=1,2,…,10,𝑗=1,2, for all𝑡∈[−1,0], where𝜉_𝑖𝑗are chosen randomly in[−10,10]. By using the Euler-Maruyama method, the simulation is carried out withΔ=0.01. The trajectories of the state variables of𝑥_𝑖1and𝑥_𝑖2(𝑖=1,2,…,10)in system (41) under pinning control are shown in Figure 2. And Figure 3 exhibits the time evolutions of synchronization errors under pinning control.

(a)

(b)

Figure 2 

(a) The trajectories of the state variables of𝑥𝑖1(𝑖=1,2,…,10)in system (41) under pinning control; (b) The trajectories of the state variables of𝑥𝑖2(𝑖=1,2,…,10)in system (41) under pinning control.

(a)

(b)

Figure 3 

(a) The time evolution of𝑒𝑖1(𝑖=1,2,…,10) in system (41) under pinning control. (b) The time evolution of𝑒𝑖2(𝑖=1,2,…,10) in system (41) under pinning control.

5. Conclusion

In this paper, we investigated the cluster synchronization of impulsive complex networks with time-varying delay coupling and stochastic perturbations. Specifically, we achieved global exponential synchronization by applying pinning control scheme to a small fraction of nodes and derived sufficient conditions for the global exponential stability of synchronization. Finally, for clarity of exposition, a numerical example was considered to illustrate the theoretical analysis by using Matlab.

Acknowledgments

The authors thank the referees and the editor for their valuable comments on this paper. This work is supported by the Natural Science Foundation of China (Grant no. 61273220), Guangdong Education University Industry Cooperation Projects (Grant no. 2009B090300355), the Shenzhen Basic Research Project (JC201006010743A, JCYJ20120613105730482), and 2011 Foundation for Distinguished Young Talents in Higher Education of Guangdong (LYM11115).