Efficient moment-based approach to the simulation of infinitely many heterogeneous phase oscillators

TL;DR

This paper introduces a moment-based method for efficiently simulating the dynamics of infinitely many heterogeneous phase oscillators, extending applicability beyond the Ott-Antonsen theory and enabling new insights into complex oscillator systems.

Contribution

The authors develop a Fourier-Hermite moment approach for Gaussian heterogeneities, providing a more efficient alternative to direct simulation and allowing analysis of complex oscillator models.

Findings

The moment-based scheme accurately reproduces the dynamics of the Kuramoto model.

It outperforms direct simulation in computational efficiency.

New results are obtained for bimodal and nonpairwise interaction models.

Abstract

The dynamics of ensembles of phase oscillators are usually described considering their infinite-size limit. In practice, however, this limit is fully accessible only if the Ott-Antonsen theory can be applied, and the heterogeneity is distributed following a rational function. In this work we demonstrate the usefulness of a moment-based scheme to reproduce the dynamics of infinitely many oscillators. Our analysis is particularized for Gaussian heterogeneities, leading to a FourierHermite decomposition of the oscillator density. The Fourier-Hermite moments obey a set of hierarchical ordinary differential equations. As a preliminary experiment, the effects of truncating the moment system and implementing different closures are tested in the analytically solvable Kuramoto model. The moment-based approach proves to be much more efficient than the direct simulation of a large oscillator ensemble. The convenience of the moment-based approach is exploited in two illustrative examples: (i) the Kuramoto model with bimodal frequency distribution, and (ii) the 'enlarged Kuramoto model' (endowed with nonpairwise interactions). In both systems we obtain new results inaccessible through direct numerical integration of populations.