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 ǫ and frequency ω. We first note that the inverted state is unstable for ǫ = 0. However, as ǫ 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 ω. 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. PACS numbers: 05.45.+b, 03.20.+i, 05.70.Jk Typeset using REVTEX ∗Electronic address: [email protected]