On solutions to the continuum version of the Kuramoto model with identical oscillators

TL;DR

This paper proves the existence and uniqueness of solutions for the continuum Kuramoto model with identical oscillators using a novel combination of the method of characteristics and iterative techniques, enabling global analysis.

Contribution

It introduces a new approach combining characteristics and iteration to analyze the continuum Kuramoto model, providing a foundation for studying more complex variants.

Findings

Proves existence and uniqueness of solutions

Characterizes oscillator density via limiting projected characteristic

Enables global asymptotic analysis of the system

Abstract

The Kuramoto model provides a concrete mathematical realization of emergent synchrony in a population of phase-coupled oscillators. Since Kuramoto's publication, \textit{Oscillations, Waves, and Turbulence}, researchers have worked to better characterize solution dynamics. In this paper, we combine the method of characteristics with an iterative technique to prove existence and uniqueness of solutions to the continuum version of the Kuramoto model in the special case where oscillators are identical and characterize the oscillator density in terms of a limiting projected characteristic. The model characterization, in turn, provides for global asymptotic analysis of the system via sub and super solutions. We believe the unique approach to model analysis developed here has the potential to yield novel results on other more complex versions of the extensively studied Kuramoto model.