North-HoUand, Amsterdam CLASSIFICATION AND UNFOLDING OF DEGENERATE HOPF BIFURCATIONS WITH 0(2) SYMMETRY: NO DISTINGUISHED PARAMETER

The Hopf bifurcation in the presence of 0(2) symmetry is considered. When the bifurcation breaks the symmetry, the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously. In applications these correspond to rotating (traveling) waves and standing waves. Using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented. No distinguished parameter is assumed. The universal unfoldings reveal the existence of both 2-tori and 3-tori; corresponding to quasiperiodic waves with two and three independent frequencies, respectively.