A Formal Setting for Network Dynamics

Abstract

This chapter is an introduction to coupled cell networks, a formal setting in which to analyse general features of dynamical systems that are coupled together in a network. Such networks are common in many areas of application. The nodes (‘cells’) of the network represent system variables, and directed edges (‘arrows’) represent how variables influence each other. Cells and arrows are assigned types, which determine the form of admissible differential equations-those compatible with the network structure. By analogy with the modern theory of dynamical systems, emphasis is placed on phenomena that are typical of entire classes of model equations with a given network structure, rather than on specific models. Such phenomena include symmetry and synchrony relations among cells, leading to a clustering effect embodied in a quotient network described by a balanced colouring. Rigid patterns of synchrony (those preserved by admissible perturbations) for equilibria and periodic states are classified by the balanced colourings. Bifurcations in which network structure can cause anomalous power-law growth rates are briefly mentioned. The formal concepts are motivated and explained in terms of typical examples.