Stability of phase difference trajectories of networks of Kuramoto oscillators with time-varying couplings and intrinsic frequencies

TL;DR

This paper analyzes the stability of phase difference trajectories in networks of Kuramoto oscillators with time-varying couplings and frequencies, providing conditions for stability and illustrating various dynamic behaviors through numerical examples.

Contribution

It extends stability analysis of Kuramoto oscillators to time-varying couplings and frequencies, including cases with negative couplings and periodic variations.

Findings

PDs are asymptotically stable under certain graph connectivity conditions

Time-varying PDs tend to follow dynamic patterns rather than fixed points

Numerical examples confirm theoretical stability and dynamic behaviors

Abstract

We study dynamics of phase-differences (PDs) of coupled oscillators where both the intrinsic frequencies and the couplings vary in time. In case the coupling coefficients are all nonnegative, we prove that the PDs are asymptotically stable if there exists T>0 such that the aggregation of the time-varying graphs across any time interval of length $T$ has a spanning tree. We also consider the situation that the coupling coefficients may be negative and provide sufficient conditions for the asymptotic stability of the PD dynamics. Due to time-variations, the PDs are asymptotic to time-varying patterns rather than constant values. Hence, the PD dynamics can be regarded as a generalisation of the well-known phase-locking phenomena. We explicitly investigate several particular cases of time-varying graph structures, including asymptotically periodic PDs due to periodic coupling coefficients and intrinsic frequencies, small perturbations, and fast-switching near constant coupling and frequencies, which lead to PD dynamics close to a phase-locked one. Numerical examples are provided to illustrate the theoretical results.