J. Differential Equations 226 (2006) 118-134
David Chan
Abstract
Resonant and non-resonant Hopf bifurcations from relative
equilibria posed in two spatial dimensions, in systems
with Euclidean SE(2) symmetry, have been extensively studied in the context
of spiral waves in a plane and are now well understood. We investigate Hopf
bifurcations from relative equilibria posed in systems with compact SO(3)
symmetry where SO(3) is the group of rotations in three dimensions/on a
sphere. Unlike the SE(2) case the skew product equations cannot be solved directly
and we use the normal form theory due to Fiedler and Turaev to simplify these systems.
We show that the normal form theory resolves the non-resonant case, but not the resonant
case. New methods developed in this paper combined with the normal form theory resolves
the resonant case.