Bifurcations in the Kuramoto model with external forcing and higher-order interactions

TL;DR

This paper explores how external forcing and higher-order interactions in the Kuramoto model lead to complex bifurcation scenarios with multiple synchronization states, revealing rich dynamics and bifurcation structures.

Contribution

It introduces the combined effects of external forcing and higher-order interactions in the Kuramoto model, uncovering new bifurcation phenomena and multiple asymptotic states.

Findings

11 different asymptotic states identified

Bifurcation scenarios include saddle-node, Hopf, and homoclinic bifurcations

Regions of bi-stability show duplicated bifurcation manifolds

Abstract

Synchronization is an important phenomenon in a wide variety of systems comprising interacting oscillatory units, whether natural (like neurons, biochemical reactions, cardiac cells) or artificial (like metronomes, power grids, Josephson junctions). The Kuramoto model provides a simple description of these systems and has been useful in their mathematical exploration. Here we investigate this model combining two common features that have been observed in many systems: external periodic forcing and higher-order interactions among the elements. We show that the combination of these ingredients leads to a very rich bifurcation scenario that produces 11 different asymptotic states of the system, with competition between forced and spontaneous synchronization. We found, in particular, that saddle-node, Hopf and homoclinic manifolds are duplicated in regions of parameter space where the unforced system displays bi-stability.