Nonlinear stability of phase-locked states for the Kuramoto model with finite inertia

TL;DR

This paper analyzes the nonlinear stability of phase-locked states in a modified Kuramoto model with finite inertia, demonstrating orbital stability and phase shifts determined by initial conditions, supported by numerical simulations.

Contribution

It establishes orbital $ ext{l}^{ ext{o}}_{ ext{infty}}$-stability of phase-locked states in the Kuramoto model with inertia, including numerical validation and relaxed conditions.

Findings

Phase-locked states are orbitally stable under small perturbations.

The phase shift depends on initial phases, frequencies, and inertia.

Numerical simulations confirm stability, uniqueness, and phase shift predictions.

Abstract

We discuss the {\it nonlinear stability} of phase-locked states for globally coupled nonlinear oscillators with finite inertia, namely the modified Kuramoto model, in the context of the robust $\ell^{\infty}$-norm. We show that some classes of phase-locked states are orbitally $\ell{\infty}$-stable in the sense that its small perturbation asymptotically leads to only the phase shift of the phase-locked state from the original one without changing its fine structures as keeping the same suitable coupling strength among oscillators and the same natural frequencies. The phase shift is uniquely determined by the average of initial phases, the average of initial frequencies, and the strength of inertia. We numerically confirm the stability of the phase-locked state as well as its uniqueness and the phase shift, where various initial conditions are considered. Finally, we argue that some restricted conditions employed in the mathematical proof are not necessary, based on numerical simulation results.