Pulsating and rotating spirals in a delayed feedback diffractive nonlinear optical system

TL;DR

This paper investigates the formation and stability of pulsating and rotating spiral waves in a nonlinear optical system with delayed feedback, using mathematical modeling, bifurcation analysis, and numerical simulations.

Contribution

It derives a limiting equation on a circle from a delayed diffusion model and predicts spiral wave behaviors, verified through numerical simulations.

Findings

Existence of pulsating and rotating spiral waves confirmed.

Normal form analysis predicts spiral stability.

Numerical simulations validate theoretical predictions.

Abstract

We study spiral waves in a mathematical model of a nonlinear optical system with a feedback loop. Starting from a delayed scalar diffusion equation in a thin annulus with oblique derivative boundary conditions, we shrink the annulus and derive the limiting equation on a circle. Based on the explicitly constructed normal form of the Hopf bifurcation for the one-dimensional delayed scalar diffusion equation, we make predictions about the existence and stability of two-dimensional spirals that we verify in direct numerical simulations, observing pulsating and rotating spiral waves.