Third order interactions shift the critical coupling in multidimensional Kuramoto models

TL;DR

This paper investigates how third order (three-body) interactions influence the synchronization transition in multidimensional Kuramoto models, revealing a shift in critical coupling and complex bifurcation phenomena.

Contribution

It demonstrates that three-body interactions raise the critical coupling for synchronization in higher dimensions, except in 2D where cancellation occurs, and explores resulting bi-stability and hysteresis.

Findings

Three-body interactions increase the critical coupling in D>2.

In 2D, three-body interactions cancel out, not affecting the critical point.

Simulations in 3D and 4D show complex synchronization dynamics.

Abstract

The study of higher order interactions in the dynamics of Kuramoto oscillators has been a topic of intense recent research. Arguments based on dimensional reduction using the Ott-Antonsen ansatz show that such interactions usually facilitate synchronization, giving rise to bi-stability and hysteresis. Here we show that three body interactions shift the critical coupling for synchronization towards higher values in all dimensions, except $D=2$, where a cancellation occurs. After the transition, three and four body interactions combine to facilitate synchronization. Similar to the 2-dimensional case, bi-stability and hysteresis develop for large enough higher order interactions. We show simulations in $D=3$ and $4$ to illustrate the dynamics.