Heteroclinic Switching in Coupled Oscillator Networks: Dynamics on Odd Graphs

We review some examples of dynamics displaying sequential switching for systems of coupled phase oscillators. As an illustration we discuss a simple family 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.