Inertia perturbation theory for the inertial Kuramoto model

TL;DR

This paper introduces an inertia perturbation approach to the inertial Kuramoto model, providing new synchronization results and bounds in the small inertia regime, and exploring phase velocity determinability.

Contribution

It develops a Tikhonov theorem for the inertial Kuramoto model and derives a new synchronization condition with strong bounds, extending understanding of the model's behavior.

Findings

New synchronization statement with bounds on the order parameter

Limitations of phase velocity determinability in small inertia regime

Complementary to previous work on phase-locking in low inertia-high coupling regime

Abstract

In this work, we study the inertial Kuramoto model, which is a second-order extension of the classical first-order Kuramoto model, as an inertial perturbation of the first-order Kuramoto model. We develop a quantitative Tikhonov theorem, from which we derive a new synchronization statement in the small inertia regime, with strong bounds on the limiting order parameter. We also explore the determinability of phase velocities from phase positions, which shows that the perturbation viewpoint must be limited to the small inertia regime. This paper complements our recent work (2025), where we established asymptotic phase-locking of inertial Kuramoto oscillators under generic initial conditions in the low inertia-high coupling regime.