Hopf Bifurcation of Relative Periodic Solutions: Case Study of a Ring of Passively Mode-Locked Lasers

TL;DR

This paper analyzes Hopf bifurcations leading to relative periodic solutions in symmetric delay differential systems, exemplified by a laser ring model with dihedral symmetry, using equivariant degree theory.

Contribution

It introduces a method to establish the existence and classification of relative periodic solutions in symmetric delay systems via equivariant degree theory.

Findings

Existence of branches of relative periodic solutions in symmetric delay systems.

Symmetric classification of these solutions using equivariant degree.

Application to a laser ring model with dihedral symmetry.

Abstract

In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting $\Gamma \times S^1$-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted $\Gamma\times S^1$-degree with one free parameter. As a case study, we consider a delay differential model of coupled identical passively mode-locked semiconductor lasers with the dihedral symmetry group $\Gamma=D_8$.