Low-Dimensional Dynamics for Higher Order Harmonic Globally Coupled Phase Oscillator Ensemble

TL;DR

This paper extends the low-dimensional theory for globally coupled oscillators to include higher-order harmonic coupling, enabling analytical understanding of complex phenomena like asymmetrical clustering.

Contribution

It generalizes the Watanabe-Strogatz approach to arbitrary system sizes for higher-order coupling, providing new insights into complex synchronization patterns.

Findings

Analytical explanation of asymmetrical clustering.

Extension of low-dimensional theory to higher-order modes.

Discussion of phenomena beyond first-order harmonic coupling.

Abstract

The Kuramoto model, despite its popularity as a mean-field theory for many synchronization phenomenon of oscillatory systems, is limited to a first-order harmonic coupling of phases. For higher-order coupling, there only exists a low-dimensional theory in the thermodynamic limit. In this paper, we extend the formulation used by Watanabe and Strogatz to obtain a low-dimensional description of a system of arbitrary size of identical oscillators coupled all-to-all via their higher-order modes. To demonstrate an application of the formulation, we use a second harmonic globally coupled model, with a mean-field equal to the square of the Kuramoto mean-field. This model is known to exhibit asymmetrical clustering in previous numerical studies. We try to explain the phenomenon of asymmetrical clustering using the analytical theory developed here, as well as discuss certain phenomena not observed at the level of first-order harmonic coupling.