Sudden stratospheric warmings (SSWs) are large scale events that are of first-order importance to understanding the wintertime dynamical variability in the polar stratosphere. These events are dynamically driven by wave-mean flow interaction of tropospherically-generated, vertically propagating planetary waves. These waves break at critical levels and subsequently force reversal of the zonal wind and a warming of the polar cap. Understanding the interaction between these planetary waves and the stratospheric zonal flow is of first-order importance in forcing SSWs. Characteristics of these planetary waves are investigated in simple models of stratospheric wave-mean flow interaction. A linear stability analysis of the equilibrium states of these systems as functions of the incoming wave forcing amplitude is performed. Bifurcations of the zonal wind exist within the models, such that for sufficiently large quasi-stationary wave amplitudes, the zonal wind is forced into a SSW-like state. Transient momentum forcing due to quasi-random gravity wave activity is also taken into account by means of an additive noise term in the zonal momentum equation. This small-scale forcing needs to be parameterized in climate models and is usually considered to be of little importance in driving SSWs. It is shown that this noisy small-scale forcing can strongly affect the nature of stratospheric wave-mean flow interaction and the occurrence of SSWs in the truncated models. This study also examines the zonal wind response to transient forcing of tropospheric planetary waves, motivated by the fact that as the forcing amplitude in the real atmosphere is observed to undergo strong temporal fluctuations. It is shown that for wave forcing periods less than the internal time scales of the model, modeled sudden warmings are highly sensitive to the time scale of the prescribed tropospheric forcing in both systems. Forcing of SSWs in this transient case can only occur for wave amplitudes that are large relative to the quasi-stationary wave amplitudes necessary to force SSWs. Simple statistics derived from the ERA-40 and ERA-Interim datasets are shown to largely support this result, obtained with simple models, although more work is required to confirm this observational support.