Modulated rotating waves in O(2) mode interactions

The interaction of steady-state and Hopf bifurcations in the presence of O(2) symmetry generically gives a secondary Hopf bifurcation to a family of 2-tori, from the primary rotating wave branch. We present explicit formulas for the coefficients which determine the direction of bifurcation and the stability of the 2-tori. These formulas show that the tori are determined by third-degree terms in the normal-form equations, evaluated at the origin. The flow on the torus near criticality has a small second frequency, and is close to linear flow, without resonances. Existence of an additional SO(2) symmetry, as in the Taylor-Couette problem, forces the flow to be exactly linear; however, the tori are unstable at bifurcation in the Taylor-Couette case. More generally, these tori may reveal themselves physically as slowly modulated rotating waves, for example in reactiondiffusion problems.