Bifurcation and branching of equilibria in reversible equivariant vector fields

We study steady-state bifurcation in reversible equivariant vector fields. We assume an action on the phase space of a compact Lie group G with a normal subgroup H of index two, and study vector fields that are H-equivariant and have all elements of the complement G \ H as time-reversal symmetries. We focus on separable bifurcation problems that can be reduced to equivariant steady-state bifurcation problems, possibly with parameter symmetries. We describe both bifurcations of equilibria that arise when external parameters are varied, and branching of families of equilibria that may arise in the phase space when external parameters are fixed. We also show how our results apply to bifurcation problems for reversible relative equilibria and reversible (relative) periodic orbits.