Phase diagram of a generalized Winfree model

TL;DR

This paper explores a generalized Winfree model where the coupling depends on the synchronization level, revealing how this affects phase transitions and the size of incoherent regions through analytical and numerical analysis.

Contribution

It introduces a generalized Winfree model with a coupling dependent on the synchronization measure, showing how the parameter z influences phase behavior and transition types.

Findings

Incoherent phase region size is controlled by parameter z.

Original Winfree model is a special case with unique behavior.

Transition type changes from continuous to first-order as z increases.

Abstract

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, z<1 strengthens the coupling at low r (low amount of synchronization) and, at z>1, 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 one may make the incoherent behavior less typical by choosing z<1. We also find that the original Winfree model is a rather special case, indeed the partial locked behavior disappears for z>1. At fixed k and varying gamma, 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 on a similarly modified Kuramoto model.