Effects of network topology on the synchronized behaviour of coupled nonlinear oscillators: a case study

Abstract

The concept of network is ubiquitous to various disciplines and includes different phenomena that emerge in a collection of interacting systems. These may be technology-based, as in energy-distribution and transportation systems as well as in telecommunications, or they may appear naturally, as in the case of ecosystems, social or neuronal networks and biological systems at a cellular level. Even though the nature of each constituting system and the interconnections among them differ drastically from one discipline to another, at the level of mathematical abstraction, they all possess common features and may be analysed via common approaches. In particular, on the level of modelling, networks can be viewed as a set of nodes and links that represent individual dynamical systems and the interactions among them, respectively and the behaviour of isolated systems is described via nonlinear models. Even taken separately, both, the complexity of network topology and the nonlinear nature of individual dynamics can lead to the appearence of non-trivial network behaviour. However, the interplay of these two characteristics entails large scale collective behaviour , in which some form of global coordination arises out of the local interactions among initially disordered systems. From such a perspective, network behaviour is often regarded as the dichotomy of two related processes: the emergence of collective behaivour and the reorganisation of the individual systems relative to the latter; these two processes can be described as a two-levels system in which a macroscopic level corresponds to the large-scale network behaviour and the microscopic level considers the network from the point of view of the individual systems that compose it. Indirectly, through the notion of synchronisation, behaviour at the microscopic level is well-studied in dynamic control theory-see eg, Blekhman et al. (1997); Boccaletti et al. (2006); Brown and Kocarev (2000), using a variety of tools that stem both from the dynamical systems and automatic control domains for the synchronisation analysis of complex (networked) systems. Nonetheless, the analysis This article is supported by Government of Russian Federation (grant 074-U01). of emergent behaviour is hardly explored in the control-theory community.