J.D. Crawford, M. Golubitsky and W.F. Langford
Modulated rotating waves in O(2) mode interactions
Dyn. Stab. Sys. 3 No. 3-4 (1988) 159-175.
The interaction of steady-state and Hopf bifurcations in the presence of 0(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 S0(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 reaction-diffusion problems.