Generalized synchronization between two complex dynamical networks with time-varying delay and nonlinear coupling.(Research Article)(Report)

1. Introduction

Over the past decade, complex networks have gained a lot of attention in various fields, such as sociology, biology, physical sciences, mathematics, and engineering [1- 5]. A complex network is a large number of interconnected nodes, in which each node represents a unit (or element) with certain dynamical system and edge represents the relationship or connection between two units (or elements). Synchronization is one of the most important dynamical properties of dynamical systems, there are different kinds of methods to realize synchronization such as active control [6], feedback control [7], adaptive control [8], impulsive control [9], passive method [10], and so forth. Synchronization of complex networks includes complete synchronization (CS)[11, 12], projective synchronization (PS) [13,14], phase synchronization [15,16], generalized synchronization (GS)[17,18],and so on.

As a sort of synchronous behavior, GS is an extension of CS and PS, so GS is more widespread than CS and PS in nature and in technical applications. GS of chaos system has been widely researched. However, most of theoretical results about synchronization of complex networks focus on CS and PS. Especially, due to the complexity of GS, the theoretical results for GS are lacking, but GS of complex networks is attracting special attention; in [17], the author gives a novel definition of GS on networks and a numerical simulation example. Reference [18] applies the auxiliary-system approach to study paths to globally generalized synchronization in scale-free networks of identical chaotic oscillators.

Recently, GS of drive-response chaos systems is investigated by the nonlinear control theory in [19]. In this letter, we extend this method to investigate GS between two complex networks, and some criterions for GS are deduced.

This letter is organized as follows. In Section 2, the definition of GS between the drive-response complex networks is given and some preliminary knowledge, including three assumptions and one lemma is also introduced. By employing the Lyapunov theory and Barbalat lemma, some schemes are designed to construct response networks to realize GS with respect to the given nonlinear smooth mapping. In Section 3, two numerical examples are given to demonstrate the effectiveness of the proposed method in Section 2. Finally, conclusions are given in Section 4.

2. GS Theorems between Two Complex Networks with Nonlinear Coupling

2.1. Definition and Assumptions

Definition 2.1. Suppose [x.sub.i] = [([x.sub.i1], [x.sub.i2],...,[x.sub.in]).sup.T] [member of] [R.sup.n], [y.sub.i] = [([y.sub.i1],[y.sub.i2], ..., [y.sub.in]).sup.T] [member of] [R.sup.n], i = 1,2, ..., N are the state variables of the drive network and the response network, respectively. Given the smooth vector function [PHI]: [R.sup.n] [right arrow] [R.sup.n] the drive network and response network are said to achieve GS with respect to [PHI]. If

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.1)

where [e.sub.i](t)= [x.sub.i](t) - [PHI]([y.sub.i](t)),i = 1,2, ..., N, the norm [parallel x [parallel] of a vector x is defined as [parallel]x[parallel] = [([x.sup.T]x).sup.1/2].

Remark 2.2. If [PHI]([y.sub.i]) = [y.sub.i], then GS is CS in [20]. If [PHI]([y.sub.i]) = [lambda][y.sub.i], then GS is PS in [13,14].

In this paper, we consider a general complex dynamical network with time-varying nonlinear coupling and consisting of N nonidentical nodes:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (2.2)

where [x.sub.i] = [([x.sub.i1],[x.sub.i2], ..., [x.sub.in]).sup.T] [member of] [R.sup.n], i = 1,2, ..., N are the state variables of the drive network, [f.sup.i] : [R.sup.n] …