Frequency locking of modulated waves

TL;DR

This paper analyzes how a modulated wave in an $ ext{S}^1$-equivariant system can lock its modulation frequency to an external forcing, with applications to self-pulsating lasers under periodic optical signals.

Contribution

It characterizes the parameter domain where stable frequency locking occurs for modulated waves under external forcing in a simplified model.

Findings

Identifies the domain of stable frequency locking in parameter space.

Describes the influence of forcing amplitude and frequencies on locking behavior.

Provides a simplified scenario relevant to laser dynamics.

Abstract

We consider the behavior of a modulated wave solution to an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far from that of the modulated wave solution. We describe the domain in the three-dimensional control parameter space (of frequencies and amplitude of the forcing) where stable locking of the modulation frequencies of the forcing and the modulated wave solution occurs. Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external periodically modulated optical signals.