  We study bifurcations associated with stability of the lowest stationary
point (SP) of a damped parametrically forced pendulum by varying $\omega_0$
(the natural frequency of the pendulum) and $A$ (the amplitude of the external
driving force). As $A$ is increased, the SP will restabilize after its
instability, destabilize again, and so {\it ad infinitum} for any given
$\omega_0$. Its destabilizations (restabilizations) occur via alternating
supercritical (subcritical) period-doubling bifurcations (PDB's) and pitchfork
bifurcations, except the first destabilization at which a supercritical or
subcritical bifurcation takes place depending on the value of $\omega_0$. 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.
