P. Chossat and M. Golubitsky
Hopf Bifurcation in the Presence of Symmetry, Center Manifold and Liapunov-Schmidt Reduction
In: Oscillation, Bifurcation and Chaos (F.V. Atkinson, W.F. Langford
and A.B. Mingarelli, eds.) CMS-AMS Conf. Proc. Ser. 8 (1987),
AMS, Providence, 343-352.
Assume that the linear part of a vector field X is semisimple and
has eigenvalues at +wi/-wi. We show that if the quadratic terms
of X vanish when restricted to the center subspace, then to third
order the Liapunov-Schmidt reduction for finding periodic solutions
of X is already in Birkhoff normal form. Several examples of systems with
symmetry that satisfy this hypothesis are discussed.