Rotating Waves of Nonlocally Coupled Oscillators on the Annulus

Abstract

Rotating waves are a common occurrence in large scale recordings of brain oscillations. We examine the existence, stability, and form of rigid rotating waves in a nonlocally coupled phase model on the annulus. For odd interaction functions, it is possible to write down an explicit solution. When interactions are more general, we prove existence of the waves and use perturbation theory to estimate the shape, frequency, and stability of the waves. For large holes, we show that the phase satisfies a Burgers-type equation. We show that as the hole in the annulus decreases, the waves lose stability. Through numerical simulations, we suggest that the bifurcation that occurs with the shrinking hole is a saddle-node infinite cycle and gives rise to so-called spiral chimeras.