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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.

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 [

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 [

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 [

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. [

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.

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

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 [

Consider an impulsive complex network consisting of

The initial conditions associated with (

In order to achieve the cluster synchronization objective, the feedback controllers as well as impulsive controllers are added to part of its nodes. When

Since both state coupling and impulsive coupling are considered in (

The complex network (

A continuous function

We will simply introduce the following notations:

The average impulsive interval of the impulsive sequence

The following assumptions will be used throughout this paper in establishing our synchronization condition.

Denote

Consider an

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

Suppose the assumptions (H1), (H2), and (H3) hold and

Let

Define

In Theorem

When there is no time-varying delay coupling, the network (

Suppose that the assumptions (H1), (H2), and (H3) hold and

When the time-varying delays are constant (i.e.,

Suppose the assumptions (H1), (H2), and (H3) hold and

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

Consider the following chaotic delayed neural networks:

In order to verify our results, we consider the following complex network:

The controlled complex network (

Through computation, we get

The initial conditions of the numerical simulations are as follows:

(a) The trajectories of the state variables of

(a) The time evolution of

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.

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).