In a recent study, Sotiropoulos et al. (2001) studied numerically the chaotic particle paths in the interior of stationary vortex-breakdown bubbles that form in a closed cylindrical container with a rotating lid. Here we report the first experimental verification of these numerical findings along with new insights into the dynamics of vortex-breakdown bubbles. We visualize the Lagrangian transport within the bubbles using planar laser-induced fluorescence (LIF) and show that even though the flow fields are steady - from the Eulerian standpoint - the spatial distribution of the dye tracer varies continuously, and in a seemingly random manner, over very long observation intervals. This finding is consistent with the arbitrarily long Šil'nikov transients of upstream-originating orbits documented numerically by Sotiropoulos et al. (2001). Sequences of instantaneous LIF images also show that the steady bubbles exchange fluid with the outer flow via random bursting events during which blobs of dye exit the bubble through the spiral-in saddle. We construct experimental Poincaré maps by time-averaging a sufficiently long sequence of instantaneous LIF images. Ergodic theory concepts (Mezić and Sotiropoulos 2002) can be used to formally show that the level sets of the resulting time-averaged light intensity field reveal the invariant sets (unmixed islands) of the flow. The experimental Poincaré maps are in good agreement with the numerical computations. We apply this method to visualize the dynamics in the interior of the vortex-breakdown bubble that forms in the wake of the first bubble for governing parameters in the steady, two-bubble regime. In striking contrast with the asymmetric image obtained for the first bubble, the time-averaged light intensity field for the second bubble is remarkably axisymmetric. Numerical computations confirm this finding and further reveal that the apparent axisymmetry of this bubble is due to the fact that orbits in its interior exhibit quasi-periodic dynamics. We argue that this stark contrast in dynamics should be attributed to differences in the swirl-to-axial velocity ratio in the vicinity of each bubble. By studying the bifurcations of a simple dynamical system, with manifold topology resembling that of a vortex-breakdown bubble, we show that sufficiently high swirl intensities can stabilize the chaotic orbits, leading to quasi-periodic dynamics.