On oscillator death in the Winfree model

TL;DR

This paper rigorously proves that in the Winfree oscillator model, sufficiently strong coupling leads to oscillator death for almost all initial conditions, confirming and extending numerical observations with mathematical guarantees.

Contribution

It provides the first rigorous proof of oscillator death in the Winfree model under strong coupling, using novel analytical techniques and volumetric arguments.

Findings

Coupling strength exceeding twice the maximal intrinsic frequency guarantees convergence.

Oscillator death occurs in finite time for almost all initial conditions.

Results are robust across various interaction functions.

Abstract

We show that for the standard sinusoidal Winfree model, a coupling strength exceeding twice the maximal magnitude of the intrinsic frequencies guarantees the convergence of the system for Lebesgue almost every initial data. This is proven by first showing, via an order parameter bootstrapping argument, that the pathwise critical coupling strength is upper bounded by a function of the order parameter, and then showing by a volumetric argument that for Lebesgue almost every data the order parameter cannot stay below and be bounded away from 1 for all time; this is a Winfree model counterpart of the analysis of Ha and the author (2020) performed for the Kuramoto model. Using concentration of measure and the aforementioned volumetric argument, we show that, except possibly on a set of very small measure, oscillator death is observed in finite time; this rigorously demonstrates the existence of the oscillator death regime numerically observed by Ariaratnam and Strogatz (2001). These results are robust under many other choices of interaction functions often considered for the Winfree model. We demonstrate that the asymptotic dynamics described in this paper are sharp by analyzing the equilibria of the Winfree model, and we bound the total number of equilibria using a polynomial description.