Classical solutions of a mean field system for pulse-coupled oscillators: long time asymptotics versus blowup

TL;DR

This paper introduces a new reformulation of the mean-field system for pulse-coupled oscillators, providing insights into long-term behavior, steady states, convergence rates, and conditions for blow-up, enhancing understanding of oscillator dynamics.

Contribution

It presents a novel inverse distribution function approach that clarifies long-time dynamics and blow-up phenomena in pulse-coupled oscillator models.

Findings

Established conditions for existence of steady states.

Derived rates of convergence to equilibrium.

Identified mechanisms leading to finite-time blow-up.

Abstract

We introduce a novel reformulation of the mean-field system for pulse-coupled oscillators. It is based on writing a closed equation for the inverse distribution function associated to the probability density of oscillators with a given phase in a suitable time scale. This new framework allows to show a hidden contraction/expansion of certain distances leading to a full clarification of the long-time behavior, existence of steady states, rates of convergence, and finite time blow-up of classical solutions for a large class of monotone phase response functions. In the process, we get insights about the origin of obstructions to global-in-time existence and uniform in time estimates on the firing rate of the oscillators.