Symmetry breaking and period doubling on a torus in the VLF regime in Taylor-Couette flow

We present an extensive experimental study of the very-low-frequency (VLF) mode, a very slow time-periodic oscillation with azimuthal wave number m=0 in axisymmetric Taylor-Couette flow. The VLF mode appears as a secondary or higher time-dependent instability in the entire wavelength range for flow systems with radius ratio 0.5. We focus on measurements which cover a parameter range that reaches from the onset of time dependence to the transition to chaos in the wavelength range λ<1.78d (d is the gap width of the cylinder) appearing in flow systems having 10–50 vortices. It was found that, increasing the Reynolds number, one observes—independently of the number of vortices of the flow system—always the same ‘‘sequence’’ of states. This is, first, the transition from Taylor vortex flow to the onset of the time-periodic small-jet mode via a Hopf bifurcation going along with a simultaneous breaking of the axial symmetry of the flow, second, the onset of the VLF mode via a homoclinic bifurcation for smaller cylinders where the underlying wavy Taylor vortex flow is still the small-jet mode (therefore we have a ${\mathit{T}}^2$ torus); and finally, the transitions to chaos, which were found to occur as period-doubling routes on ${\mathit{T}}^2$ tori. Additionally a quantitative description of this transition to chaos is given, calculating the correlation dimension on the basis of a proper reconstructed phase space. A model of interacting time-dependent Taylor vortex flow is discussed and compared to the appearance of VLF-mode oscillations in the flow. © 1996 The American Physical Society.