Abstract
The main aim of this thesis has been to investigate the distinction between a nonresonant and resonant Hopf bifurcation from relative equilibria. Resonant and nonresonant 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 initially investigate Hopf bifurcations from relative equilibria posed in systems with compact S0(3) symmetry where S0(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 nonresonant case, but not the resonant case. New methods developed in this thesis combined with the normal form theory resolves the resonant case. We find that the resonant Hopf bifurcation produces a motion which is strikingly different from the nonresonant case. By considering the geometric properties of the underlying relative equilibrium we give a definition of resonance directly related to the Hopf bifurcation phenomena. This yields conditions for the occurrence of a resonant Hopf bifurcation from a relative equilibrium in a system with general compact symmetry. By extending the approach used to resolve the S0(3) case we then solve the skew product equations for a general compact symmetry group. The specific case of a Hopf bifurcation from a relative equilibria with symmetry of an irregular dodecahedron is also considered. Furthermore we look at nonresonant and resonant Hopf Bifurcations from relative equilibria posed in a system with noncompact Euclidean symmetry in three spatial dimensions.