Spectrum of extensive multiclusters in the Kuramoto model with higher-order interactions

TL;DR

This paper explores a Kuramoto model extension with three-way higher-order interactions, revealing complex phenomena like multiclusters, multistability, and abrupt desynchronization, and analyzes their stability in large populations.

Contribution

It provides a rigorous spectral stability analysis of multicluster states in a higher-order Kuramoto model with novel dynamical behaviors.

Findings

Higher-order interactions induce multiclusters and multistability.

Spectral analysis distinguishes stability based on frequency distribution support.

Finite support distributions can lead to fully phase-locked, potentially stable states.

Abstract

Globally coupled ensembles of phase oscillators serve as useful tools for modeling synchronization and collective behavior in a variety of applications. As interest in the effects of simplicial interactions (i.e., non-additive, higher-order interactions between three or more units) continues to grow we study an extension of the Kuramoto model where oscillators are coupled via three-way interactions that exhibits novel dynamical properties including clustering, multistability, and abrupt desynchronization transitions. Here we provide a rigorous description of the stability of various multicluster states by studying their spectral properties in the thermodynamic limit. Not unlike the classical Kuramoto model, a natural frequency distribution with infinite support yields a population of drifting oscillators, which in turn guarantees that a portion of the spectrum is located on the imaginary axes, resulting in neutrally stable or unstable solutions. On the other hand, a natural frequency distribution with finite support allows for a fully phase-locked state, whose spectrum is real and may be linearly stable or unstable.