Existence of hysteresis in the Kuramoto model with bimodal frequency distributions

TL;DR

This paper examines the conditions under which hysteresis occurs in the Kuramoto model with bimodal frequency distributions, revealing that hysteresis depends on the width of the central dip rather than its depth.

Contribution

The study demonstrates that hysteresis in the Kuramoto model is not solely determined by the proximity to unimodal distributions but also by the width of the central dip in bimodal distributions.

Findings

Hysteresis depends on the width of the central dip in bimodal distributions.

Proximity to the unimodal-bimodal border does not guarantee hysteresis.

Hysteresis can occur even when the central dip is very shallow, if its width is sufficiently large.

Abstract

We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one, due to the shallowness the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model with a suitable family of bimodal distributions.