Stability of an [N/2]-dimensional invariant torus in the Kuramoto model at small coupling

TL;DR

This paper investigates the stability of an invariant torus in the Kuramoto model with symmetric natural frequencies, revealing stability conditions depend on whether the population size is odd or even, especially at small coupling.

Contribution

It provides a perturbative analysis demonstrating the asymptotic stability of the invariant torus for odd N and the conditional stability for even N in the small coupling regime.

Findings

Invariant torus is asymptotically stable for odd N at small coupling.

For even N, the torus's stability depends on natural frequencies.

Both stable and unstable cases coexist for even N in the small coupling limit.

Abstract

When the natural frequencies are allocated symmetrically in the Kuramoto model there exists an invariant torus of dimension [N/2]+1 (N is the population size). A global phase shift invariance allows to reduce the model to $N-1$ dimensions using the phase differences, and doing so the invariant torus becomes [N/2]-dimensional. By means of perturbative calculations based on the renormalization group technique, we show that this torus is asymptotically stable at small coupling if N is odd. If N is even the torus can be stable or unstable depending on the natural frequencies, and both possibilities persist in the small coupling limit.