  The transition from two-dimensional to three-dimensional flows in a finite
circular cylinder driven by an axially oscillating sidewall is explored in
detail. The complete symmetry group of this flow, including a spatio-temporal
symmetry related to the oscillating sidewall, is $Z_2\times O(2)$. Previous
studies in flows with the same symmetries, such as symmetric bluff-body wakes
and periodically forced rectangular cavities, were unable to obtain the
theoretically predicted bifurcation to modulated traveling waves. In the
simpler cylindrical geometry, where the azimuthal direction is physically
periodic, we have found these predicted modulated traveling waves as stable
fully saturated nonlinear solutions for the first time. A careful analysis of
the base states and their linear stability identifies different parameter
regimes where three-dimensional states that are either synchronous with the
forcing or quasiperiodic, corresponding to different symmetry-breaking
processes. These results are in good agreement with theoretical predictions and
previous results in similar flows. These different regimes are separated by
three codimension-two bifurcation points that are yet to be fully analyzed
theoretically. Finally, the saturated nonlinear states and their properties in
different parameter regimes are analyzed.
