Improved Numerical Scheme for the Generalized Kuramoto Model

TL;DR

This paper introduces a more accurate numerical scheme for simulating the three-dimensional generalized Kuramoto model, improving long-term behavior accuracy by leveraging the model's geometric interpretation as particles on a sphere undergoing rigid rotations.

Contribution

The paper develops a novel numerical scheme based on the geometric interpretation of the model, outperforming conventional methods in accuracy and stability for long-term simulations.

Findings

The new scheme accurately reproduces known analytic results.

It captures the expected behavior of 3D oscillators in various scenarios.

It reveals discrepancies in traditional methods leading to potential misinterpretations.

Abstract

We present an improved and more accurate numerical scheme for a generalization of the Kuramoto model of coupled phase oscillators to the three-dimensional space. The present numerical scheme relies crucially on our observation that the generalized Kuramoto model corresponds to particles on the unit sphere undergoing rigid body rotations with position-dependent angular velocities. We demonstrate that our improved scheme is able to reproduce known analytic results and capture the expected behavior of the three-dimensional oscillators in various cases. On the other hand, we find that the conventional numerical method, which amounts to a direct numerical integration with the constraint that forces the particles to be on the unit sphere at each time step, may result in inaccurate and misleading behavior especially in the long time limit. We analyze in detail the origin of the discrepancy between the two methods and present the effectiveness of our method in studying the limit cycle of the Kuramoto oscillators.