The inverted pendulum

It is a surprising fact that an inverted pendulum can be stabilised in the upright position through vertical oscillation of its pivot at a suitable amplitude and (high) frequency. This also works for an inverted double pendulum and in fact much more generally too (see D. Acheson, "A Pendulum Theorem", 1993).

It would be nice to have an animation which allows one to explore the region of stability for the inverted single or double pendulum by adjusting the amplitude and frequency of the pivot's oscillation (and perhaps also some other parameters, such as gravity) using sliders.