The mathematics of asymptotic stability in the Kuramoto model

TL;DR

This paper reviews the mathematical understanding of asymptotic stability in the Kuramoto model, highlighting recent rigorous results, providing insights into proofs, and discussing extensions and examples in the context of coupled oscillators.

Contribution

It offers a comprehensive review of recent mathematical results on stability in the Kuramoto model, including original insights and extensions to variations of the model.

Findings

Rigorous validation of stability in the continuum Kuramoto model

Insights into proof techniques for asymptotic stability

Extensions to model variations and additional examples

Abstract

Now a standard in Nonlinear Sciences, the Kuramoto model is the perfect example of the transition to synchrony in heterogeneous systems of coupled oscillators. While its basic phenomenology has been sketched in early works, the corresponding rigorous validation has long remained problematic and was achieved only recently. This paper reviews the mathematical results on asymptotic stability of stationary solutions in the continuum limit of the Kuramoto model, and provides insights into the principal arguments of proofs. This review is complemented with additional original results, various examples, and possible extensions to some variations of the model in the literature.