Basin sizes depend on stable eigenvalues in the Kuramoto model

TL;DR

This paper demonstrates that in the Kuramoto model, the basin sizes and global dynamics can be estimated using the eigenvalues of stable synchronized states, linking local spectral properties to global behavior.

Contribution

It reveals that local eigenvalues of stable states encode global basin size information, connecting local spectral analysis with global dynamics in the Kuramoto model.

Findings

Basin sizes depend on stable eigenvalues.

Eigenvalues can estimate global statistics.

Numerical simulations support the analytical results.

Abstract

We show that for the Kuramoto model (with identical phase oscillators equally coupled) its global statistics and size of the basins of attraction can be estimated through the eigenvalues of all stable (frequency) synchronized states. This result is somehow unexpected since, by doing that, one could just use local analysis to obtain global dynamic properties. But recent works based on Koopman and Perron-Frobenius operators demonstrate that global features of a nonlinear dynamical system, with some specific conditions, are somehow encoded in the local eigenvalues of its equilibrium states. Recognized numerical simulations in the literature reinforce our analytical results.