On synchronization in Kuramoto models on spheres

TL;DR

This paper investigates synchronization dynamics in Kuramoto models on spheres, revealing that higher-dimensional spheres facilitate faster synchronization in real models, while complex models' order parameters are dimension-independent, supported by theoretical derivations and simulations.

Contribution

It derives closed-form equations for order parameters in Kuramoto models on spheres and compares synchronization rates across dimensions, highlighting differences between real and complex models.

Findings

Synchronization is faster on higher-dimensional spheres.

Real order parameters satisfy the same ODE regardless of dimension.

Theoretical predictions match simulations with hundreds of oscillators.

Abstract

We analyze two classes of Kuramoto models on spheres that have been introduced in previous studies. Our analysis is restricted to ensembles of identical oscillators with the global coupling. In such a setup, with an additional assumption that the initial distribution of oscillators is uniform on the sphere, one can derive equations for order parameters in closed form. The rate of synchronization in a real Kuramoto model depends on the dimension of the sphere. Specifically, synchronization is faster on higher-dimensional spheres. On the other side, real order parameter in complex Kuramoto models always satisfies the same ODE, regardless of the dimension. The derivation of equations for real order parameters in Kuramoto models on spheres is based on recently unveiled connections of these models with geometries of unit balls. Simulations of the system with several hundreds of oscillators yield perfect fits with the theoretical predictions, that are obtained by solving equations for the order parameter.