In this study we present a numerical investigation of steady states, onset of oscillatory instability, and slightly supercritical oscillatory states of an axisymmetric swirling flow of a Newtonian incompressible fluid in a cylinder, with independently rotating top and bottom. The first part of the study is devoted to the influence of co- and counter-rotation of the bottom on the steady vortex breakdown, which takes place in the well-known problem of flow in a cylinder with a rotating top. It is shown that weak counter-rotation of the bottom may suppress the vortex breakdown. Stronger counter-rotation may induce a stable steady vortex breakdown at relatively large Reynolds numbers for which a vortex breakdown does not appear in the case of the stationary bottom. Weak corotation may promote the vortex breakdown at lower Reynolds numbers than in the cylinder with the stationary bottom. Stronger corotation leads to the detachment of the recirculation zone from the axis and the formation of an additional vortex ring. The second part of the study is devoted to the investigation of the onset of oscillatory instability of steady flows. It is shown that the oscillatory instability sets in due to a Hopf bifurcation. The critical Reynolds number and the critical frequency of oscillations, were calculated as a function of the rotation ratio (ξ=Ωbottom/Ωtop) for a fixed value of the aspect ratio γ (height/radius) of the cylinder γ=1.5. The stability analysis showed that there are several most unstable linear modes of the perturbation that become successively dominant with a continuous change of ξ. It is shown that the oscillatory instability may lead to an appearance and coexistence of more than one oscillating separation vortex bubble.