Phase diagram of a generalized Winfree model

We study the phase diagram of a generalized Winfree model. The modification is such that the coupling depends on the fraction of synchronized oscillators, a situation which has been noted in some experiments on coupled Josephson junctions and mechanical systems. We let the global coupling k be a function of the Kuramoto order parameter r through an exponent z such that z=1 corresponds to the standard Winfree model, z1 strengthens the coupling at low r low amount of synchronization, and at z1, the coupling is weakened for low r. Using both analytical and numerical approaches, we find that z controls the size of the incoherent phase region and that one may make the incoherent behavior less typical by choosing z1. We also find that the original Winfree model is a rather special case; indeed, the partial locked behavior disappears for z1. At fixed k and varying , the stability boundary of the locked phase corresponds to a transition that is continuous for z<1 and first order for z>1. This change in the nature of the transition is in accordance with a previous study of a similarly modified Kuramoto model.