Filippov trajectories and clustering in the Kuramoto model with singular couplings

TL;DR

This paper investigates the dynamics of a generalized Kuramoto model with singular couplings, analyzing well-posedness, phase sticking, and synchronization, especially in regimes where classical solutions are not well-defined, using Filippov's framework.

Contribution

It introduces a comprehensive analysis of singular weighted Kuramoto systems, including well-posedness, phase sticking, and the connection between regular and singular solutions.

Findings

Solutions in the most singular cases are understood via Filippov's solutions.

Characterization of phase sticking in subcritical and critical regimes.

Emergence of synchronization in both singular and regular models.

Abstract

We study the synchronization of a generalized Kuramoto system in which the coupling weights are determined by the phase differences between oscillators. We employ the fast-learning regime in a Hebbian-like plasticity rule so that the interaction between oscillators is enhanced by the approach of phases. First, we study the well-posedness problem for the singular weighted Kuramoto systems in which the Lipschitz continuity is deprived. We present the dynamics of the system equipped with singular weights in all the subcritical, critical and supercritical regimes of the singularity. A key fact is that solutions in the most singular cases must be considered in Filippov's sense. We characterize sticking of phases in the subcritical and critical case and we exhibit a continuation criterion of classical solutions after any collision state in the supercritical regime. Second, we prove that strong solutions to these systems of differential inclusions can be recovered as singular limits of regular weights. We also provide the emergence of synchronous dynamics for the singular and regular weighted Kuramoto models.