Coherence–incoherence patterns in a ring of non-locally coupled phase oscillators

We consider a paradigmatic spatially extended model of non-locally coupled phase oscillators which are uniformly distributed within a one-dimensional interval and interact depending on the distance between their sites' modulo periodic boundary conditions. This model can display peculiar spatio-temporal patterns consisting of alternating patches with synchronized (coherent) or irregular (incoherent) oscillator dynamics, hence the name coherence–incoherence pattern, or chimera state. For such patterns we formulate a general bifurcation analysis scheme based on a hierarchy of continuum limit equations. This provides the possibility of classifying known coherence–incoherence patterns and of suggesting directions for the search for new ones.