The paper proposes a method for studying symmetric global Hopf bifurcation problems in a parabolic system. The objective is to detect unbounded branches of non-constant periodic solutions that arise from an equilibrium point and describe their symmetric properties in detail. The method is based on the twisted equivariant degree theory, which counts orbits of solutions to symmetric equations, similar to the usual Brouwer degree, but on the report of their symmetric properties.
The dynamics of point vortices in a plane is a an example of a Hamiltonian system with symmetry SE(2). In this, the vortices do not have the attribute of mass but rather move according to a Hamiltonian and a symplectic form obtained from the two plane coordinates (rather than from position and momentum as in a Lagrangian system). The (momentum dependant) symplectic reduced spaces of the non-abelian and non-compact symmetry SE(2) have varying dimension. We find a regularized reduction and realize perturbation from non-generic to generic momentum as Hamiltonian SO(2) symmetry breaking on a fixed symplectic manifold. In the case of four vortices the reduced dimension is four and stability should follow from the usual normal form and Moser-twist theorem.