  In this paper we study the global dynamics of the inverted spherical pendulum
with a vertically vibrating suspension point in the presence of an external
horizontal periodic force field. We do not assume that this force field is weak
or rapidly oscillating. We prove that there always exists a non-falling
periodic solution, i.e., there exists an initial condition such that the rod of
the pendulum never becomes horizontal along the corresponding solution. We also
show numerically that there exists an asymptotically stable non-falling
solution for a wide range of parameters of the system.
