Hopf bifurcations from relative equilibria in spherical geometry

Abstract

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 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 nonresonant case, but not the resonant case. New methods developed in this paper combined with the normal form theory resolves the resonant case.