  A generalization of the classical Kapitza pendulum is considered: an inverted
planar mathematical pendulum with a vertically vibrating pivot point in a
time-periodic horizontal force field. We study the existence of forced
oscillations in the system. It is shown that there always exists a periodic
solution along which the rod of the pendulum never becomes horizontal, i.e. the
pendulum never falls, provided the period of vibration and the period of
horizontal force are commensurable. We also present a sufficient condition for
the existence of at least two different periodic solutions without falling. We
show numerically that there exist stable periodic solutions without falling.
