Bifurcations and transitions to chaos in an inverted pendulum

We consider a parametrically forced pendulum with a vertically oscillating suspension point. It is well known that, as the amplitude of the vertical oscillation is increased, its inverted state (corresponding to the vertically-up configuration) undergoes a cascade of ``resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum. We make a detailed numerical investigation of the bifurcations associated with such resurrections of the inverted pendulum by varying the amplitude and frequency of the vertical oscillation. It is found that the inverted state stabilizes via alternating ``reverse'' subcritical pitchfork and period-doubling bifurcations, while it destabilizes via alternating ``normal'' supercritical period-doubling and pitchfork bifrucations. An infinite sequence of period-doubling bifurcations, leading to chaos, follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed.