A combined experimental and numerical investigation is presented of the multiple oscillatory states that exist in the flows produced in a completely filled, enclosed, circular cylinder driven by the constant rotation of one of its endwalls. The flow in a cylinder of height to radius ratio 2.5 is interrogated experimentally using flow visualization and digitized images to extract quantitative temporal information. Numerical solutions of the axisymmetric Navier–Stokes equations are used to study the same flow over a range of Reynolds numbers where the flow is observed to remain axisymmetric. Three oscillatory states have been identified, two of them are periodic and the third is quasi-periodic with a modulation frequency much smaller than the base frequency. The range of Reynolds numbers for which the quasi-periodic flow exists brackets the switch between the two periodic states. The results from the combined experimental and numerical study agree both qualitatively and quantitatively, providing unambiguous evidence of the existence and robustness of these multiple time-dependent states.