SYMMETRY-BREAKING FOR LEADING ORDER GRADIENT MAPS IN R(2) WITH APPLICATIONS TO O(3)

In steady-state bifurcation theory, spontaneous symmetry-breaking is i nvestigated for quadratic or cubic gradients as the leading nonlineari ty on the two-dimensional fixed point subspace of a maximal isotropy s ubgroup. In each case, sufficient conditions, including a transversali ty condition, are established for the existence of symmetry-breaking b ifurcation branches, and stability of the non-trivial branches is disc ussed. An algorithm for establishing the transversality condition is d eveloped using Grobner bases and implemented in the symbolic algebra l anguage Maple. As an application, it is established that the octahedra l group breaks the symmetry of O(3) in its 25 and 31 dimensional repre sentations with at least 1 and 2 smooth branches respectively.