The size of the sync basin resolved

TL;DR

This paper investigates the basin sizes of twisted states in Kuramoto oscillators on cycle graphs, providing numerical and analytical evidence supporting a Gaussian scaling law related to the winding number.

Contribution

It offers the first analytical derivation of the basin size scaling law for twisted states, revealing the dynamical mechanism behind Gaussian scaling in high-dimensional oscillator networks.

Findings

Basin sizes of twisted states scale as e^{-kq^2} with the winding number q.

Rapid stabilization of winding number occurs before long-range correlations develop.

A Central Limit Theorem explains the Gaussian scaling of basin sizes.

Abstract

Sparsely coupled Kuramoto oscillators offer a fertile playground for exploring high-dimensional basins of attraction due to their simple yet multistable dynamics. For $n$ identical Kuramoto oscillators on cycle graphs, it is well known that the only attractors are twisted states, whose phases wind around the circle with a constant gap between neighboring oscillators ($\theta_j = 2\pi q j/n$). It was conjectured in 2006 that basin sizes of these twisted states scale as $e^{-kq^2}$ to the winding number $q$. Here, we provide new numerical and analytical evidence supporting the conjecture and uncover the dynamical mechanism behind the Gaussian scaling. The key idea is that, when starting with a random initial condition, the winding number of the solution stabilizes rapidly at $t \propto \log n$, before long-range correlation can develop among oscillators. This timescale separation allows us to calculate the winding number as a sum of weakly-dependent random variables, leading to a Central Limit Theorem derivation of the basin scaling.