Hopf bifurcation with SN-symmetry

We study Hopf bifurcation with SN-symmetry for the standard absolutely irreducible action of SN obtained from the action of SN by permutation of N coordinates. Stewart (1996 Symmetry methods in collisionless many-body problems, J. Nonlinear Sci. 6 543–63) obtains a classification theorem for the C-axial subgroups of SN × S1. We use this classification to prove the existence of branches of periodic solutions with C-axial symmetry in systems of ordinary differential equations with SN-symmetry posed on a direct sum of two such SN-absolutely irreducible representations, as a result of a Hopf bifurcation occurring as a real parameter is varied. We determine the (generic) conditions on the coefficients of the fifth order SN × S1-equivariant vector field that describe the stability and criticality of those solution branches. We finish this paper with an application to the cases N = 4 and N = 5