The "not quite" inverted pendulum

The planar pendulum has been used as an example of a simple oscillator for over 300 years with notable historical contributions by Galileo and Huygens. Initially interest was focused on small displacements from the stable hanging position specifically to determine the period of oscillation. More recently attention has switched to larger amplitude and chaotic motions as a consequence of periodic forcing to the pivot point. To explain experimentally observed behavior, we investigate here the existence of stable inverted solutions of a pendulum sinusoidally driven, in the first instance by a purely vertical force and then by an almost vertical force by giving a small tilt to the system. An effective potential function is considered to provide analytical justification of the numerical simulations.