This paper investigates the exponential synchronization problem of stochastic complex dynamical networks with impulsive perturbation and Markovian switching. The complex dynamical networks consist of

Since Watts and Strogatz wrote their pioneering work [

Exponential synchronization is a more favorite property since it gives a fast convergence rate to the synchronous solution. In [

It has been widely reported that networks have finite modes which switch from one mode to another at different times, and such a switching signal can be governed by a Markovian chain. Markovian jump networks are of great significance in modeling a class of complex networks with finite network modes, and many relevant results have been reported in the literature (see, e.g., [

In the real world, there exist a number of networks in which the state of nodes is usually subject to instantaneous perturbations and experiences abrupt change at certain instants which may be caused by switching phenomena, frequent change, and other sudden noise. Such networks are described by impulsive differential networks [

In this paper, we are concerned with the analysis issue for exponential synchronization of stochastic complex dynamical networks with impulsive perturbations and Markovian switching. According to two different cases of complex dynamical networks, synchronous networks and asynchronous networks, some sufficient conditions are presented to ensure the exponential synchronization of stochastic complex dynamical network with impulsive perturbations and Markovian switching, and the upper bound of impulsive gain is evaluated. Two numerical examples are included to show the effectiveness of our results. The main contributions of this paper can be highlighted as follows.

Impulsive effects, noise perturbations, and switchings are considered for modeling the coupled complex networks simultaneously, which has been rarely investigated.

By using the average dwell time approach, M-matrix approach, Lyapunov theory, and stochastic analysis, some sufficient conditions are presented to ensure the exponential synchronization of stochastic complex dynamical networks with impulsive perturbations and Markovian switching.

The notations are quite standard. Throughout this paper,

Let

In this paper, we consider a class of stochastic complex dynamical networks, which is described as follows:

One important consideration in practical networks is the existence of impulsive perturbations and the impulse of each node which does not emerge at the same time. Hence, the impulsive perturbations network is described by

For all

In [

In [

The primary object here is to deal with the exponential synchronization problem of the stochastic complex dynamical network (

Define

For the purpose of the exponential synchronization of the stochastic complex dynamical network (

The function

The noise intensity matrix

In order to derive the main results, the following definitions and lemmas are necessary in this paper.

Consider a stochastic differential equation with Markovian switching of the form

System (

The dynamical network (

The average impulsive interval of the impulsive sequence

Let

In this section, we propose some criteria of exponential synchronization in mean square for stochastic complex dynamical networks with impulsive perturbations and Markovian switching.

Let Assumptions

Choose a nonnegative Lyapunov function as follows:

For

Without any loss of generality, we assume that there are

Fix

For each

According to Assumptions

Consider that the coupling matrix

According to condition (

Let

Based on Lemma

It is easy to get

On the other hand, from the construction of

For

It is easy to see that

The proof is completed.

We assume that there exist infinite time points

Let Assumptions

In Theorem

Assume that

Let Assumptions

The Lyapunov function is the same as that in Theorem

Based on Lemma

Because of (

It follows from the Gronwall's inequality that

On the other hand, from the construction of

The proof can be completed by following the same steps as that in Theorem

For each

Let Assumptions

In this section, we present two numerical simulations to illustrate the feasibility and effectiveness of our results.

Consider that a stochastic complex network model consists of five nodes and two modes, which is described as follows:

The synchronization state

The nonlinear function

Markov chain generated by the probability transition matrix

Desynchronizing impulsive sequence.

The trajectories of the state variables of

The trajectories of the state variables of

Let

The synchronization state

The initial conditions for this simulation are

Markov chain generated by the probability transition matrix

Synchronizing impulsive sequence.

The trajectories of the error variables of

The trajectories of the error variables of

The trajectories of the error variables of

In this paper, we have dealt with the exponential synchronization problem of complex dynamical networks with impulsive perturbations and Markovian switching. An

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by the National Natural Science Foundation of China (61075060), the Key Foundation Project of Shanghai (12JC1400400), and the Innovation Program of Shanghai Municipal Education Commission (12zz064, 13zz050).