Kuramoto model on the $D$-dimensional torus

TL;DR

This paper generalizes the Kuramoto model to particles on a D-dimensional torus, revealing a first order phase transition and analyzing the synchronization dynamics through mean field theory and numerical validation.

Contribution

It introduces a multidimensional Ott-Antonsen ansatz for the model and characterizes the synchronization transition as a saddle-node bifurcation.

Findings

The model exhibits a first order phase transition unlike the traditional version.

Synchronization arises from a saddle-node bifurcation.

Numerical simulations validate the theoretical analysis.

Abstract

We propose a generalization of the Kuramoto model of interacting oscillators in which the particles move on the surface of a $D$-dimensional torus. In contrast with the traditional one-dimensional version, this model has a first order phase transition. We establish its mean field dynamics by means of a multidimensional Ott-Antonsen ansatz, and show that synchronization arises from a saddle-node bifurcation, while the incoherent state is always stable. Our theoretical calculations are validated by numerical simulations.