Global synchronization of complex networks perturbed by the Poisson noise

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Abstract

In this paper, the problem of stochastic synchronization analysis is investigated for complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper proposes a globally exponentially synchronization criterion in mean square for complex networks perturbed by the Poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Introduction

As is known to all, complex dynamical networks (CDNs) widely exist in the real world, including food-webs, ecosystems, metabolic pathways, the Internet, the World Wide Web, social networks and global economic markets [1], [2]. Since the discoveries of the small-world feature [3] and the scale-free feature [4] of complex networks, the analysis and the control of the dynamical behaviors in complex networks have been extensively investigated in the past decades. As a significant collective behavior, the studies on the synchronization phenomena of complex dynamical networks have gained considerable research interests [5], [6], [7], [8], [9], [10], [11], [12].

On the other hand, in the real world, complex networks are often subject to environmental disturbances; especially the signal transfer within complex networks is always affected by the stochastic perturbations. Therefore, in order to reflect more realistic dynamical behaviors, many researchers have recently investigated the synchronization problems of complex networks perturbed by stochastic noises. For instance, complex networks perturbed by Brown noises have been discussed in [13], [14], [15], [16]. The synchronization problems of discrete-time stochastic complex networks with Brown noises were investigated in [13], [14]. As to the continuous case, the global exponential synchronization problem for complex dynamical networks with nonidentical nodes and Brown perturbations was studied in [15]. And the synchronization control problem for the competitive complex networks with Brown noises was investigated in [16].

However, it is well known that in the real world, beside Brown noises, there is a very common but important kind of random noises: Poisson noises. Poisson noises can be widely found in various applications such as neurophysiology systems, storage systems, queueing systems, economic systems, and so on [17], [18]. It should be pointed out that, unlike the Brown process whose almost all sample paths are continuous, the Poisson process N(t) is a jump process and has the sample paths which are right-continuous and have left limits (i.e. càdlàg). Therefore, there is a great difference between the stochastic integral with respect to the Brown process and the one with respect to the Poisson process. As a result, the dynamical behaviors of the stochastic systems driven by the Poisson process are essential different from the stochastic systems driven by the Brown process. Thus, it is very important to investigate the dynamic behaviors, such as the synchronization phenomena, for complex networks perturbed by the Poisson process. However, to the best of our knowledge, there is still no paper to discuss the synchronization problem for this kind of systems.

Motivated by above reasons, this paper investigates the synchronization problem for stochastic complex networks perturbed by the Poisson noise. By using the key tool such as the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper presents a globally exponentially synchronization criterion in mean square for complex networks perturbed by the poisson noise. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

Notation: Throughout the paper, unless otherwise specified, we will employ the following notation. Let (Ω,F,{Ft}t0,P) be a complete probability space with a natural filtration {Ft}t0 and E(·) be the expectation operator with respect to the probability measure. If A is a vector or matrix, its transpose is denoted by AT. If P is a square matrix, P>0 ( P<0) means that is a symmetric positive (negative) definite matrix of appropriate dimensions while P0 ( P0) is a symmetric positive (negative) semidefinite matrix. I stands for the identity matrix of appropriate dimensions. Let |·| denote the Euclidean norm of a vector and its induced norm of a matrix. Unless explicitly specified, matrices are assumed to have real entries and compatible dimensions. L2(Ω) denotes the space of all random variables X with E|X|2<, it is a Banach space with norm X2=(E|X|2)1/2. The symbol ‘∗’ within a matrix represents the symmetric terms of the matrix, e.g. XYZ=XYYTZ.

Section snippets

Problem formulation and preliminaries

Consider the following complex dynamical networks consisting of N nodes perturbed by the Poisson noise:dxi(t)=Axi(t)+Bf(xi(t))+j=1NgijΓxj(t)dt+σi(t,xi(t))dN(t),i=1,2,,Nwhere xi(t)=xi1(t),xi2(t),,xin(t)TRn is the state vector of the ith network at time t; A denotes a known connection matrix, B denotes the connection weight matrix; ΓRn×n is the matrix describing the inner-coupling between the subsystems at time t; G=(gij)N×N is the out-coupling configuration matrix representing the coupling

Main results

We are in the position to present our main results of the globally exponentially synchronization criterion in mean square for the complex networks perturbed by the Poisson noise.

Theorem 1

Under Assumptions Assumption 1, Assumption 2, Assumption 3, the dynamic system (1) is globally exponentially synchronized in mean square if there exist matrices P>0 and scalars >0, 1>0 such that the following LMI hold for all 1i<jNΞ=Ξ11Ξ12λP0-21I00-PP-I<0,whereΞ11=PA+ATP-NgijPΓ-NgijΓTP+λWTW-λP-1UTV-1VTU,Ξ

Numerical examples

In this section, we present two simulation examples to illustrate the effectiveness of our approach.

Example 1

Consider the following complex network consisting of three nodes.dxi(t)=Axi(t)+Bf(xi(t))+j=13gijΓxj(t)dt+σi(t,xi(t))dN(t)for all i=1,2,3, where xi(t)=xi1(t),xi2(t)TR2 is the state vector of the ith subsystem, {N(t)}t0 is a one-dimension {Ft}t0 adapted Poisson process with parameter λ=6.5. LetA=-300-3,B=-10.10.2-1.The out-coupling configuration matrices G and inner-coupling matrices Γ are

Conclusions

This paper is concerned with the problem of stochastic synchronization analysis for complex networks perturbed by the Poisson noise. Using the infinitesimal operator for stochastic differential equations driven by the Poisson process, this paper gives a globally exponentially synchronization criterion in mean square. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

On the other hand, it is worth mentioning that there are still some important

Acknowledgements

The work was supported by 2012 Yeungnam University Research Grant. Also, the work of B. Song was supported by the National Natural Science Foundation of China under Grants 61104221 and 61174029 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China under Grant 10KJB120004.

References (21)

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