^{1,2}

^{1,3}

^{3}

^{1}

^{2}

^{3}

This paper investigates synchronization problem of nonlinearly coupled dynamical networks, and an effectively impulsive control scheme is proposed to synchronize the network onto the objective state. Based on the stability analysis of impulsive differential equations, a low-dimensional sufficient condition is derived to guarantee the exponential synchronization in virtual of average impulsive interval. A numerical example is given to illustrate the effectiveness and feasibility of the proposed methods and results.

Synchronization of complex networks is an important topic that has drawn a great deal of attention from diverse fields including physics, biology, neuroscience, and mathematics [

Due to its potential applications in many different areas, the synchronization of complex dynamical networks has been widely discussed in the last decade. For example, in [

Most recently, another synchronization technique, based on impulsive control, has been reported and developed in [

Motivated by the above discussions, the aim of this paper is to discuss the impulsive synchronization of nonlinearly-coupled complex networks. Based on the stability analysis of impulsive functional differential equations, some sufficient synchronization criteria are derived in virtual of average impulsive interval.

The main contributions of this paper are as follows. First, this paper uses the concept of “average impulsive interval” to obtain the synchronization criterion. It makes the result much less conservative than previous results since the strict requirement on the upper bound and lower bound of the impulsive interval, which always appear in the previous results, is not necessary any more. Second, the model considered in this paper is nonlinearly coupled network, which includes linearly coupled network and array of linearly coupled systems as special cases.

The outline of this paper is given as follows. In Section

Throughout this paper, some mathematical notations are used.

Consider a complex dynamical network consisting of

In order to achieve the synchronization of the complex dynamical network (

There are some definitions and denotations that are necessary for presenting the main results as follows.

The nonlinear-coupled dynamical network is said to be exponentially synchronized to the original point if there exist some constants

The function

The average impulsive interval of the impulsive sequence

Suppose that we are mainly interested in achieving synchronization of the network (

Consider the nonlinearly-coupled complex network (

Construct a Lyapunov function in the form of

Let

Let

Due to the introduction of the concept “average impulsive interval”, the requirement on the lower bound and upper bound of impulsive interval is removed in Theorem

In this section, based on the results obtained in the previous section, we consider the impulsive control of four nonlinearly-coupled canonical Lorenz systems to show the effectiveness of our results. The network is described as follows:

The double-scroll attractor of the Lorenz system.

In this case, we can prove that the coupled Lorenz system satisfies the

Evolution of the state variable

In this paper, the synchronization of nonlinearly-coupled networks has been investigated. By using the impulsive controllers, the nonlinearly-coupled dynamical networks can be synchronized to the original point. A criterion for the synchronization is derived by using the stability analysis of impulsive differential equations and the concept of average impulsive interval. A numerical example is finally given to illustrate the effectiveness and feasibility of the proposed method and result. One of the future research topics would be extending the present results to the synchronization of nonlinearly coupled networks by impulsively controlling a small fraction of nodes.

The work of J. D. Cao was supported by the National Natural Science Foundation of China under Grant 11072059, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 20070286003, and the Natural Science Foundation of Jiangsu Province of China under Grant BK2009271. The work of J. Q. Lu was supported by the National Natural Science Foundation of China (NSFC) under Grants 11026182 and 61175119, the Natural Science Foundation of Jiangsu Province of China under Grant BK2010408, Program for New Century Excellent Talents in University (NCET-10-0329), and the Alexander von Humboldt Foundation of Germany.