 
Summary: Heteroclinic Switching in Coupled Oscillator
Networks: Dynamics on Odd Graphs
Peter Ashwin, Gábor Orosz, and Jon Borresen
Abstract We review some examples of dynamics displaying sequential switching
for systems of coupled phase oscillators. As an illustration we discuss a simple fam
ily of coupled phase oscillators for which one can find robust heteroclinic networks
between unstable cluster states. For N = 2k + 1 oscillators we show that there
can be open regions in parameter space where the heteroclinic networks have the
structure of an odd graph of order k; a class of graphs known from permutation
theory. These networks lead to slow sequential switching between cluster states that
is driven by noise and/or imperfections in the system. The dynamics observed is
of relevance to modelling the emergent complex dynamical behaviour of coupled
oscillator systems, e.g. for coupled chemical oscillators and neural networks.
1 Introduction
Coupled phase oscillator networks provide a set of models that are very useful in
a variety of applications ranging from theoretical and computational neuroscience
[10, 11, 17, 18] to coupled chemical reactors [21]. They provide models that are
amenable to variety of approaches aimed at understanding the emergent phenom
ena of such nonlinear dynamical systems. These approaches include "continuum
approximations" as well as detailed studies of the dynamics and bifurcations of
