The quasibiennial oscillation (QBO) of equatorial winds on Earth is the clearest example of the spontaneous emergence of a periodic phenomenon in geophysical fluids. In recent years, observations have revealed intriguing disruptions of this regular behavior, and different QBO-like regimes have been reported in a variety of systems. Here, we show that part of the variability in mean-flow reversals can be attributed to the intrinsic dynamics of wave-mean-flow interactions in stratified fluids. Using a constant-in-time monochromatic wave forcing, bifurcation diagrams are mapped for a hierarchy of simplified models of the QBO, ranging from a quasilinear model to fully nonlinear simulations. The existence of new bifurcations associated with faster and shallower flow reversals, as well as a quasiperiodic route to chaos are reported in these models. The possibility for periodicity disruptions is investigated by probing the resilience of regular wind reversals to external perturbations.