Global analysis of a continuum model for pulse-coupled oscillators

We consider phase oscillators on the circle interacting through an impulsive instantaneous coupling. In contrast with previous studies on related pulse-coupled models, global stability results are obtained in the continuum limit. For the nonlinear transport equation governing the evolution of the oscillators, we propose (under technical assumptions) a global Lyapunov function which is induced by a total variation distance between quantile densities. The monotone time evolution of the Lyapunov function completely characterizes the dichotomic behavior of the oscillators: either the oscillators converge in finite time to a synchronous state or they asymptotically converge to an asynchronous state uniformly spread on the circle. The results of the present paper apply to various popular phase oscillators models (e.g. the well-known leaky integrate-and-fire model) and provide a novel approach for the (global) analysis of pulse-coupled oscillators.