Topological phase transition in the periodically forced Kuramoto model

TL;DR

This paper uncovers a previously unrecognized abrupt phase transition in the periodically forced Kuramoto model, driven by topological changes in the order-parameter space, beyond traditional bifurcation analysis.

Contribution

It reveals a new type of phase transition in the model that is topologically driven and not detectable by standard bifurcation analysis.

Findings

Discovered an abrupt transition from oscillations to wobbling rotations.

Identified the transition as topologically caused by a singular point.

Showed the transition does not fit classical bifurcation categories.

Abstract

A complete bifurcation analysis of explicit dynamical equations for the periodically forced Kuramoto model was performed in [L. M. Childs and S. H. Strogatz. Chaos 18 , 043128 (2008)], identifying all bifurcations within the model. We show that the phase diagram predicted by this analysis is incomplete. Our numerical analysis of the equations reveals that the model can also undergo an abrupt phase transition from oscillations to wobbly rotations of the order parameter under increasing field frequency or decreasing field strength. This transition was not revealed by bifurcation analysis because it is not caused by a bifurcation, and can neither be classified as first nor second-order since it does not display critical phenomena characteristic of either transition. We discuss the topological origin of this transition and show that it is determined by a singular point in the order-parameter space.