Hopf Bifurcations of Functional Differential Equations with Dihedral Symmetries

We discuss the joint impact of temporal delay and spatial dihedral symmetries on the occurence and multiplicity of Hopf bifurcations for a system of FDEs. By applying the equivariant degree theory we establish a result on the existence of multiple branches of nonconstant periodic solutions and classify their symmetries. General results are illustrated by a ring of identical oscillators with identical coupling between adjacent cells.