Bifurcation behavior of standing waves

Two different types of standing waves (SW0 and SWπ) can appear instead of spiral vortices from a supercritical Hopf bifurcation in counter-rotating Taylor-Couette flow for sufficiently small aspect ratios [1,2]. The bifurcation sequence from basic flow to spiral vortices via SW0 can include modulated waves, homoclinic bifurcations, and hysteresis as a consequence of broken translational invariance [3]. Here we show that the same kind of sequence can also occur for the other type of standing wave, i.e., SWπ. Furthermore we show that SWπ can exist also up to much larger inner Reynolds numbers than is has been found for SW0. Far from onset SWπ can undergo bifurcation sequences that differs qualitatively from those close to onset. These sequences involve a supercritical symmetry breaking as well as a supercritical Hopf bifurcation towards a new type of modulated wave.