M. Golubitsky, K. Josic and E. Shea-Brown

Winding numbers and averaged frequencies in phase oscillator networks

J. Nonlinear Science. 16 (2006) 201-231.

We study networks of coupled phase oscillators and show that network architecture can force relations between average frequencies of the oscillators. The main tool of our analysis is coupled cell theory developed by Stewart, Golubitsky, Pivato, and Torok, which provides precise relations between network architecture and the corresponding class of ODEs in R^M, and gives conditions for the flow-invariance of certain polydiagonal subspaces for all coupled systems with a given network architecture. The theory generalizes the notion of fixed-point subspaces for subgroups of network symmetries and directly extends to networks of coupled phase oscillators.

For systems of coupled phase oscillators (but not generally for ODEs in R^M), invariant polydiagonal subsets of codimension one arise naturally and strongly restrict the network dynamics. We say that two oscillators i and j coevolve if the polydiagonal theta_i = theta_j is flow-invariant and show that the average frequencies of these oscillators must be equal. Given a network architecture, it is shown that coupled cell theory provides a direct way of testing when two oscillators coevolve. We give generalization of these results to pairs of synchronous sets of phase oscillators using quotient networks, and discuss implications for networks of spiking cells and those connected through buffers that implement coupling dynamics. We conclude by showing how oscillators in a network can be partitioned into collections with closely related dynamics.