Resurrection of An Inverted Pendulum

We study the bifurcations associated with stability of the inverted (stationary) state in the parametrically forced pendulum by varying the driving amplitude $\epsilon$ and frequency $\omega$. We first note that the inverted state is unstable for $\epsilon=0$. However, as $\epsilon$ is increased, the inverted state exhibits a cascade of ``resurrections,'' i.e., it becomes stabilized after its instability, destabilize again, and so forth ad infinitum for any given $\omega$. Its stabilizations (destabilizations) occur via alternating subcritical (supercritical) pitchfork and period-doubling bifurcations. An infinite sequence of period-doubling bifurcations, leading to chaos, also follows each destabilization of the inverted state. The critical behaviors in the period-doubling cascades are also discussed