Synchronization of identical oscillators on a sphere: exact results with external forces and higher-order interactions

TL;DR

This paper analyzes the synchronization behavior of identical oscillators on a sphere with higher-order interactions and external forces, providing exact results and dimensional reduction techniques.

Contribution

It introduces a novel approach to incorporate three- and four-body interactions into the Kuramoto model on a sphere, enabling full dimensional reduction.

Findings

Dynamics on the equator resemble the standard Kuramoto model with renormalized coupling.

Outside the equator, the system exhibits a set of periodic orbits.

Bifurcation curves are mapped as functions of system parameters.

Abstract

We study the dynamics of the Kuramoto model on the sphere under higher-order interactions and an external periodic force. For identical oscillators, we introduce a novel way to incorporate three- and four-body interactions into the dynamics of the order parameter, allowing for a full dimensional reduction of this system. We discuss how such reduction can be implemented in two different ways and how they are related. When restricted to the equator, the dynamics is similar to that of the usual Kuramoto model, up to an interesting renormalization of the coupling constants. Outside this plane, the motion reduces to a two-parameter set of periodic orbits. We also locate the bifurcation curves of the system as functions of different parameters.