Determining the self-rotation number following a Naimark-Sacker bifurcation in the periodically forced Taylor-Couette flow

Systems which admit waves via Hopf bifurcations and even systems that do not undergo a Hopf bifurcation but which support weakly damped waves may, when parametrically excited, respond quasiperiodically. The bifurcations are from a limit cycle (the time-periodic basic flow) to a torus, i.e. Naimark-Sacker bifurcations. Floquet analysis detects such bifurcations, but does not unambiguously determine the second frequency following such a bifurcation. Here we present a technique to unambiguously determine the frequencies of such quasiperiodic flows using only results from Floquet theory and the un queness of the self-rotation number (the generalization of the rotation number for continuous systems). The robustness of the technique is illustrated in a parametrically excited Taylor-Couette flow, even in cases where the bifurcating solutions are subject to catastrophic jumps in their spatial/temporal structure.