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.
Highlights
The emergence of synchronization in large populations of interacting units is one of the most well-known cooperative phenomena across a number of disciplines [1,2]
We have systematically studied the dynamical properties of steady states in the Kuramoto model with higherorder interactions in the thermodynamic limit
The phase model serves as a representative nonlinear higher-order coupling that has been shown to exhibit novel collective dynamics for phase transitions to synchronization
Summary
The emergence of synchronization in large populations of interacting units is one of the most well-known cooperative phenomena across a number of disciplines [1,2]. Extensions of the Kuramoto model have been traditionally limited to pairwise interaction between oscillators, in which only the first-order harmonic of the phase difference in the coupling function is considered. The dynamical origin of such novel collective behavior was further addressed from a microscopic perspective [37] Despite these advances, a fundamental problem still lies in understanding how higherorder interactions give rise to the observed multicluster states and the structure of their corresponding spectral properties in the thermodynamic limit. The aim of this paper is to provide a detailed analysis of the dynamical properties of multicluster states occurring in oscillator ensembles with higher-order coupling.
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