M. Golubitsky and I. Stewart

Nonlinear dynamics of networks: the groupoid formalism

Bull. Amer. Math. Soc. 43 (2006) 305-364.

Coupled systems of differential equations can be viewed as a network: a directed graph, whose nodes represent state variables, and whose directed edges represent coupling among these variables. The architecture of a network is an annotated graph with sets of node and arrow symbols; two nodes with the same symbol have the same type of state variables and two arrows with the same arrow symbol have the same coupling. Each network has an associated class of admissible coupled systems. We ask: what aspects of the dynamics of admissible vector fields are forced by network architecture?

A network of dynamical systems is not just a dynamical system with a high-dimensional phase space. It is also equipped with a canonical set of observables --- the states of the individual nodes of the network. Comparison between the dynamics of different nodes is therefore possible and notions of synchrony can be explored. Moreover, the form of the admissible vector fields are constrained by network topology. The result is that admissible vector fields exhibit a rich and new range of typical phenomena, only a few of which are yet properly understood.

In this survey we discuss how a strong form of synchrony (two or more nodes have exactly the same time series) is a consequence of network architecture; and how synchrony-breaking bifurcations from synchronous equilibria are changed by network architecture. The correspondence between graph and synchrony is based on local symmetries, the set of which forms a groupoid. Comparisons with previous results based on network symmetries and symmetry-breaking bifurcations are made.