  We study the structure of symmetric vortices in a Ginzburg-Landau model based
on S. C. Zhang's SO(5) theory of high temperature superconductivity and
antiferromagnetism. We consider both a full Ginzburg-Landau theory (with
Ginzburg-Landau scaling parameter kappa) and a high-kappa limiting model. In
all cases we find that the usual superconducting vortices (with normal phase in
the central core region) become unstable (not energy minimizing) when the
chemical potential crosses a threshold level, giving rise to a new vortex
profile with antiferromagnetic ordering in the core region. We show that this
phase transition in the cores is due to a bifurcation from a simple eigenvalue
of the linearized equations. In the limiting large kappa model we prove that
the antiferromagnetic core solutions are always nondegenerate local energy
minimizers and prove an exact multiplicity result for physically relevent
solutions.
