A note on the path formulation for (O(2),SO(2))-forced symmetry breaking bifurcation

Abstract

In this note we deal with the liftability of vector fields over the projection onto the parameter space for SO(2)-equivariant universal unfoldings (under contact equivalence) of O(2)-equivariant problems of corank 2 and the application of those results to the path formulation for bifurcation problems with the forced symmetry breaking from O(2) to SO(2). In the non equivariant case it is well known that vector fields are liftable over the projection if and only if they are tangent to its discriminant and that they form a free module. Previously we discussed equivariant cases under finite groups where the two modules are different, although both are free. Here we have continuous Lie groups, with a simple action, but both modules are again equal and free.