Multiple transitions to chaos in a damped parametrically forced pendulum

We study bifurcations associated with stability of the lowest stationary point of a damped parametrically forced pendulum by varying ${\mathrm{\omega }}_0$ (the natural frequency of the pendulum) and A (the amplitude of the external driving force). As A is increased, the stationary point will restabilize after its instability, destabilize again, and so ad infinitum for any given ${\mathrm{\omega }}_0$. Its destabilizations (restabilizations) occur via alternating supercritical (subcritical) period-doubling bifurcations (PDB’s) and pitchfork bifurcations, except for the first destabilization, at which a supercritical or subcritical bifurcation takes place depending on the value of ${\mathrm{\omega }}_0$. For each case of the supercritical destabilizations, an infinite sequence of PDB’s follows and leads to chaos. Consequently, an infinite series of period-doubling transitions to chaos appears with increasing A. The critical behaviors at the transition points are also discussed. © 1996 The American Physical Society.