Stability of entrainment of a continuum of coupled oscillators

TL;DR

This paper investigates how a continuum of coupled oscillators can be entrained by a periodic signal, analyzing the stability of this process and the effects of coupling versus driving forces.

Contribution

It provides new stability results for an infinite collection of oscillators under periodic driving, especially when coupling destabilizes the entrainment.

Findings

Derived stability conditions for entrainment in coupled oscillators

Compared effects of coupling and external driving on phase stability

Identified trade-offs between coupling strength and entrainment robustness

Abstract

We examine the design of the entrainment process for an uncountably infinite collection of coupled phase oscillators that are all subject to the same periodic driving signal. In the absence of coupling, an appropriately designed input can result in each oscillator attaining the frequency of the driving signal, with a phase offset determined by its natural frequency. We consider a special case of interacting oscillators in which the coupling tends to destabilize the phase configuration to which the driving signal would send the collection in the absence of coupling. In this setting we derive stability results that characterize the trade-off between the effects of driving and coupling, and compare these results to the well-known Kuramoto model of a collection of free-running coupled oscillators.