J. Crawford, M. Golubitsky and W. Langford
Modulated Rotating Waves in 0(2) Mode Interactions
Dynamics and Stability of Systems 3 Nos. 3 & 4 (1988).
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.