Classification of codimension-one bifurcations in a tetrad of lasers with feed forward coupling

Abstract

We consider a model of a tetrad of semiconductor lasers with unidirectional coupling. This system is described by Lang-Kobayashi (LK) rate equations, which is a system of delay differential equations with one fixed delay. Basic solutions to this system are called compound laser modes (CLMs). We classify the CLMs of this four-laser system according to their symmetry. The symmetric CLMs are identified by looking at the isotropy subgroups of the symmetry group of this system. Numerical continuation in DDE-Biftool generates a branch of symmetric CLMs. We then find steady-state and Hopf bifurcations along the branch of fully symmetric CLMs and identify those that are symmetry-breaking and symmetry-preserving. Finally, we use the Equivariant Branching Lemma and Equivariant Hopf Theorem to establish the existence of emanating branches of solutions from symmetry-breaking bifurcations.