Synchronization hypothesis in the Winfree model

TL;DR

This paper investigates the conditions under which synchronization occurs in the Winfree oscillator model, identifying a new hypothesis that delineates the parameter domain for synchronization regardless of oscillator count or frequency distribution.

Contribution

The authors propose a new hypothesis for synchronization in the Winfree model and demonstrate its necessity through numerical counter-examples, expanding understanding of synchronization conditions.

Findings

Synchronization domain includes both weak and strong coupling regimes.

Domain of synchronization is independent of the number of oscillators.

A numerical counter-example confirms the necessity of the hypothesis.

Abstract

We consider $N$ oscillators coupled by a mean field as in the Winfree model. The model is governed by two parameters: the coupling strength $\kappa$ and the spectrum width $\gamma$ of the frequencies of each oscillator. In the uncoupled regime, $\kappa=0$, each oscillator possesses its own natural frequency, and the difference between the phases of any two oscillators grows linearly in time. We say that $N$ oscillators are synchronized if the difference between any two phases is uniformly bounded in time. We identify a new hypothesis for the existence of synchronization. The domain in $(\gamma,\kappa)$ of synchronization contains coupling values that are both weak and strong. Moreover the domain is independent of the number of oscillators and the distribution of the frequencies. We give a numerical counter-example which shows that this hypothesis is necessary for the existence of synchronization.