Non-axisymmetric simulations verify and extend the results from a previous linear Floquet stability analysis of the wakes of rings. The wakes corresponding to the saturated state of each predicted non-axisymmetric instability mode over the entire aspect ratio range are successfully computed, and isosurface plots are presented elucidating the vortical wake structures. The existence of three non-axisymmetric flow regimes (Modes I, II and III) for the flow past rings with aspect ratios $\textit{Ar\/} \lesssim 3.9$ is verified, as is the existence of non-axisymmetric instabilities of vortex streets in the flow past rings with $\textit{Ar\/} \gtrsim 3.9$. Wakes are computed which correspond to the Mode A and B instabilities found in the flow past a circular cylinder, and a wake is computed which develops from a subharmonic Mode C instability. This wake features an azimuthal wavelength of approximately 1.7 ring cross-section diameters, which is between the azimuthal wavelengths of the Mode A and B instabilities. This mode cannot occur, at least in a pure state, in the flow past a circular cylinder.

Nonlinear transition characteristics are predicted by evaluating coefficients of the truncated Landau equation, and transition hysteresis is verified by studying the mode amplitude variation with Reynolds number in the vicinity of the transitions. The regular Mode I and Mode III transitions are found to occur through supercritical and subcritical bifurcations, respectively, and the secondary Hopf bifurcations to these transitions, as well as the Mode II Hopf transition, are found to be supercritical. We verify that the Mode A and Mode B transitions are subcritical and supercritical, respectively, and we determine that the nature of the Mode C transition is dependent on aspect ratio. Landau constants are evaluated for the Hopf transitions throughout the aspect ratio range studied.