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This paper investigates the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method. The switching parameters are modeled by a continuous-time, finite-state Markov chain, and the complex network is subject to noise perturbations, Markovian switching, and internal and outer time-varying delays. Sufficient conditions for mean-square exponential synchronization are obtained by using the Lyapunov-Krasovskii functional, Itö’s formula, and the linear matrix inequality (LMI), and numerical examples are given to demonstrate the validity of the theoretical results.

A complex network is a structure that is made up of a large set of nodes (also called vertices) that are interconnected to varying extents by a set of links (also called edges). Coupled biological systems such as neural networks and socially interacting animal species are simple examples of complex networks and so too is the Internet and the World Wide Web [

Chaos synchronization is a phenomenon that has been widely investigated since it was first discovered by Pecora and Carroll in 1990, and it is a 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 [

Stochastic perturbations and time delays are important considerations when simulating realistic complex networks because signals traveling along real physical systems are usually randomly perturbed by the environmental elements [

Pinning control is a technique that applies controllers to only a small fraction of the nodes in a network, and the technique is important because it greatly reduces the number of controlled nodes for real-world complex networks (which, in most cases, is huge). In fact, pinning control can be so effective for some networks that a single pinning controller is required for synchronization, namely, for complex networks that have either a symmetric or an asymmetric coupling matrix (Chen et al. [

In this paper, we study the mean-square exponential synchronization of stochastic time-varying delayed complex networks with Markovian switching by using the pinning control method. We consider a stochastic complex network with internal time-varying delayed couplings, Markovian switching, and Wiener processes. We prove some sufficient conditions for mean-square exponential synchronization of these networks by applying the Lyapunov-Krasovskii functional method and the linear matrix inequality (LMI).

This paper is organized as follows. In Section

Throughout this paper,

Let

Consider a complex network consisting of

The initial conditions associated with (

Since the Markov chains

Let

The complex network (

A continuous function

The function class QUAD includes almost all the well-known chaotic systems with or without delays such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua's circuit, the logistic delayed differential system, the delayed Hopfield neural network, and the delayed CNNs. We shall simply write

The following assumptions will be used throughout this paper for establishing the synchronization conditions.

Let

Consider a stochastic delayed differential equation with Markovian switching of the form

Let assumptions (H1) and (H2) be true and let

By (

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

Let assumptions (H1) and (H2) be true and let

When

Let assumptions (H1) and (H2) be true, and let

In this section, we present some numerical simulation results that validate the theorem of the previous section.

Consider the chaotic delayed neural network

The initial conditions for this simulation are

The trajectories of the state variables of

The time evolution of

Next, let

The trajectories of the state variables of

The time evolution of

In this paper, we investigated the synchronization problem for stochastic complex networks with Markovian switching and nondelayed and time-varying delayed couplings. Specifically, we achieved global exponential synchronization by applying a pinning control scheme to a small fraction of the nodes and derived sufficient conditions for global exponential stability of synchronization in mean square. Finally, we considered some numerical examples that illustrate the theoretical analysis.

This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Project (2009B090300355), and the Shenzhen Basic Research Project (JC201006010743A, JC200903120040A). The research of the authors is partially supported by the Hong Kong Polytechnic University Grant G-U996 and Hong Kong Government GRF Grant B-Q21F.