Adaptive synchronization in complex dynamical networks with coupling delays for general graphs

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Abstract

A sufficient condition of adaptive synchronization in dynamical networks with coupling delays is gained, where the coupling configuration corresponds to a weighted graph. One can reduce edges to original graph such that the network more quickly achieves synchronization. Numerical simulations are given to illustrate the efficiency of theoretical results.

Introduction

In general, complex networks consist of a large number of nodes and link among them, in which a node is a fundamental cell with specific activity. So complex networks and graphs are closely contacted each other. The dynamics on complex networks is one on graphs, though the graphs may have different characteristics, e.g. classical random graph model [1], small-world model [2], [3], scale-free model [4], or others are related closely to natural structure.

Synchronization in complex dynamical networks is a universal phenomenon in various fields of science and society. There are many works on the synchronization in complex dynamical networks [5], [6], [7], [8], [9]. Due to the finite speeds of transmission and spreading as well as traffic congestion, a signal or influence traveling through a network often is associated with time delays. Real-world complex systems, particularly in biological and physical systems, are time-delay systems. Thus in recent years, a lot of efforts have been made to study the synchronization of dynamical coupled systems with delays [10], [11], [12], [13], [14], [15], [16], [17], [18].

Underlying these researches imply that the structural properties of a network must have some bearing on the synchronization [19], [20]. In addition, as pointed in [20], Lu and Cao introduced an adaptive synchronization method by enhancing the coupling strength automatically under a simple updated law. However, their work is limited to tree-like networks (In fact, a tree is a graph without cycles.), which cannot be applied to general networks, and delay effect on synchronization is also unconsidered. In this paper, we will propose an adaptive synchronization method for general networks or graphs with coupling time-delays. Based on the invariant principle of functional differential equations, the global synchronization will be realized by designing adaptive controllers. Finally, the numerical simulations are given to illustrate our theoretical results.

Section snippets

Preliminaries

In this section, we now introduce some notations and preliminaries. Consider the delay complex dynamical network consisting of N linearly and diffusively coupled identical nodes, with full diagonal coupling, and each node is an n-dimensional dynamical oscillator which can be chaotic. The state equations of the network areẋi=f(xi)+∑j=1NaijΓxj(t-τ)+ui(t),i=1,2,…,N,where xi = (xi1, xi2, … , xin)T ∈ Rn is a state vector of node i, f(xi) = (f1(xi), f2(xi), … , fn(xi))T : Rn → Rn is a given nonlinear vector valued

Main results

In this section, we will use state feedback control method and invariant principle to investigate adaptive synchronization of complex network (2). To achieve synchronization, we design the adaptive controllers as:u=-k(t)(IN⊗Γ)x,where k(t) is the time-varying gain. To guarantee negative feedback, the adaptive gain is designed as:k̇(t)=βxT(L⊗Γ)x,where β is a positive constant to be determined.

The main result of this paper is stated as follows:

Theorem 1

Suppose that F is Lipschitz continuous and the

Numerical simulations

In this section, we present a numerical example to illustrate the theoretical results. In particular, we consider complex dynamical networks with five nodes as shown in Fig. 1, where each node is a Lorenz system:ẋi1ẋi2ẋi3=a(xi2-xi1)cxi1-xi1xi3-xi2xi1xi2-bxi3,i=1,2,3,4,5,which is chaotic when a=10,b=83,c=28. Choose the inner coupling link matrix to be Γ = diag{1, 2, 3}.

In the numerical simulations, we use the Runge–Kutta method to solve the delayed differential equations by τ = 0.5 and randomly

Conclusions

In this paper, we proposed an adaptive synchronization method for general networks or graphs with coupling time-delays. The global synchronization for general networks is realized by designing adaptive controllers. Finally, numerical simulations show that synchronization in complex dynamical networks with time-delays for general connected graphs can be easily achieved. Based on spectral graph theory, we can reduce edges to original graph such that the network more quickly achieves

Acknowledgements

This research is supported by National Natural Science Foundation of China (11071001, 11071002), Doctoral Fund of Ministry of Education of China (20093401110001), Major Program of Educational Commission of Anhui Province of China (KJ2010ZD02), 211 Project of Anhui University (KJTD002B), Program of Natural Science of Collages of Anhui Province (KJ2011A020), Fund of Outstanding Young Scientists in Higher Education Institutions of Anhui Province (2011SQRL126).

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