Exact solutions of the Kuramoto model with asymmetric higher order interactions of arbitrary order

TL;DR

This paper derives exact solutions for the Kuramoto model with asymmetric higher order interactions, revealing complex bifurcations, stability phenomena, and phase transitions in coupled oscillator systems.

Contribution

It introduces a method to derive reduced equations for arbitrary higher order interactions in the Kuramoto model, highlighting the role of an effective order in the dynamics.

Findings

Existence of bi- and tri-stability in oscillator systems.

Synchronization via a third order phase transition.

Explicit solutions for specific interaction forms.

Abstract

Higher order interactions can lead to new equilibrium states and bifurcations in systems of coupled oscillators described by the Kuramoto model. However, even in the simplest case of 3-body interactions there are more than one possible functional forms, depending on how exactly the bodies are coupled. Which of these forms is better suited to describe the dynamics of the oscillators depends on the specific system under consideration. Here we show that, for a particular class of interactions, reduced equations for the Kuramoto order parameter can be derived for arbitrarily many bodies. Moreover, the contribution of a given term to the reduced equation does not depend on its order, but on a certain effective order, that we define. We give explicit examples where bi and tri-stability is found and discuss a few exotic cases where synchronization happens via a third order phase transition.