A variation of the classical Taylor–Couette system is studied where, with the outer cylinder stationary, the inner cylinder rotates at constant angular velocity while executing harmonic oscillations in the axial direction. Experiments reveal a Hopf (Neimark–Sacker) bifurcation from a limit cycle to a torus. Alternating bands of frequency-locked and quasi-periodic flow are observed and identified. Power spectral plots and (delay reconstructed) Poincaré maps are used to characterize the temporal dynamics. Results are presented on the rotation number variation across parameter space, the shape and growth of frequency-locked resonance horns, and the spatial development of the flow considerably beyond the primary transition surface.