Equivariant degree method for analysis of Hopf bifurcation of relative periodic solutions: Case study of a ring of oscillators

The goal of this paper is to develop the Equivariant Degree based method for studying relative periodic solutions in the setiings with lack of smoothness and/or genericity. In this paper, we consider an equivariant Hopf bifurcation of relative periodic solutions from relative equilibria in systems of functional differential equations respecting Gamma x S-1-spatial symmetries. The existence of branches of relative periodic solutions together with their symmetric classification is established using the equivariant twisted Gamma x 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; and, a system of hysterestic electro-mechanical oscillators coupled in the same symmetric fashion. (C) 2018 Published by Elsevier Inc.