Adaptive synchronization of two nonlinearly coupled complex dynamical networks with delayed coupling

Abstract

This paper investigates the adaptive synchronization between two nonlinearly delay-coupled complex networks with the bidirectional actions and nonidentical topological structures. Based on LaSalle’s invariance principle, some criteria for the synchronization between two coupled complex networks are achieved via adaptive control. To validate the proposed methods, the unified chaotic system as the nodes of the networks are analyzed in detail, and numerical simulations are given to illustrate the theoretical results.

Introduction

A complex network is a large set of interconnected nodes, in which a node is a fundamental unit having specific contents and exhibiting dynamical behavior, typicality. Complex network models have been shown to exist in many different areas in real world including the Internet, the World Wide Web, biological neural networks, social connection networks, etc., and thus become an important part of our daily lives. In past decades, some typical complex networks such that the random networks, small-world networks and scale-free networks are most noticable [1], [2], [3]. Recently, the synchronization of complex networks, which is a significant and interesting phenomenon among various complex dynamical behaviours, has been a focus for many scientists from various fields, for example, mathematics, sociology and biology, etc., [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Wang and Chen [10], [11] studied local synchronization of the small-world and scale-free networks by transverse stability to the synchronization manified. Lu and Chen [13], [14] proposed a new approach to analyze the stability of synchronization manifold of linearly coupled dynamical systems with or without the coupling delay. The synchronization in nonlinearly coupled dynamical network is studied in [16]. By using the invariance principle of differential equations, some simple linear feedback controllers with dynamical updated strengths were constructed to make the dynamical network synchronize an isolate node. Wu and Chen [17] studied global synchronization criteria of linearly coupled neural network systems with time-varying coupling. Arenas et al. [18] proposed the advances in the comprehension of synchronization phenomenon when oscillating elements are constrained to interact in a complex network topology and several applications of synchronization in complex networks to different disciplines. Liu et al. [19] investigated synchronization of complex delayed dynamical networks with nonlinearly coupled nodes. In Ref. [20], Guo et al. studied the problem of global synchronization for nonlinearly coupled complex networks with either non-delayed or delayed coupling.

However, most of the above references focused on synchronization in a network that was called inner synchronization because it was a collective behavior within a network. In reality, synchronization between two or more complex networks regardless of synchronization of the inner network, which is called outer synchronization, always does exist in our lives, which indicates that it is necessary and significant to investigate the dynamics between two coupled networks. Li et al. [21] firstly investigated the synchronization between two unidirectionally coupled complex networks with identical topological structures. Later, the outer synchronization was investigated by several other researchers in Refs. [22], [23], [24], [25], [26], etc. Tang et al. [22] discussed the synchronization between two complex dynamical networks with nonidentical topological structures via using adaptive control method. Li et al. [25] discussed the outer synchronization of two coupled discrete-time network. Wu et al. [26] studied the problem of generalized outer synchronization between two complex dynamical networks with different topologies and diverse node dynamics. Actually, there are lots of active forms between two or more complex networks, for example communicated by signals, bidirectional actions and special nodes, etc. In Refs. [21], [22], [23], [24], [25], [26], the authors only considered the coupling of network itself. However, in this paper, we not only consider their own network of coupling, but also take into account the time delay and the bidirectional actions, that is, the interaction from one network to another. Therefore, motivated by the above discussions, the aim of this paper is to discuss the outer synchronization of two nonlinearly delay-coupled dynamical networks with inter-network actions and nonidentical topological structures.

The rest of this paper is organized as follows. The drive-response nonlinearly coupled complex dynamical network models and some preliminaries are presented in Section 2. In Section 3, some sufficient conditions for the synchronization are derived by the adaptive method. In Section 4, two illustrative examples are given for supporting the theory results. Finally, conclusions are given in Section 5.