Synchronization conditions in the Kuramoto model and their relationship to seminorms

TL;DR

This paper investigates the conditions for synchronization in the Kuramoto model, introducing the Kuramoto seminorm to characterize stable solutions and analyzing the probability of synchronization across various frequency distributions.

Contribution

It establishes bounds on frequency sets supporting synchronization and introduces the Kuramoto seminorm, linking convex geometry and extreme value theory to synchronization analysis.

Findings

Bounds on frequency sets supporting stable synchronization

Introduction of the Kuramoto seminorm for phase-locked solutions

Exact limiting distribution for synchronization probability

Abstract

In this paper we address two questions about the synchronization of coupled oscillators in the Kuramoto model with all-to-all coupling. In the first part we use some classical results in convex geometry to prove bounds on the size of the frequency set supporting the existence of stable, phase locked solutions and show that the set of such frequencies can be expressed by a seminorm which we call the Kuramoto norm. In the second part we use some ideas from extreme order statistics to compute upper and lower bounds on the probability of synchronization for very general frequency distributions. We do so by computing exactly the limiting extreme value distribution of a quantity that is equivalent to the Kuramoto norm.