Ivan Polekhin

We consider the problem of existence of forced oscillations on a Riemannian manifold, the metric on which is defined by the kinetic energy of a mechanical system. Under the assumption that the generalized forces are periodic functions of time, we find periodic solutions of the same period. We present sufficient conditions for the existence of such solutions, which essentially depend on the behavior of geodesics on the corresponding Riemannian manifold.

In this paper we study the global dynamics of the inverted spherical pendulum with a vertically rapidly 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. Provided that the period of the vertical motion and the period of the horizontal force are commensurate, we prove that there always exists a nonfalling periodic solution, i. e., there exists an initial condition such that, along the corresponding solution, the rod of the pendulum always remains above the horizontal plane passing through the pivot point. We also show numerically that there exists an asymptotically stable nonfalling solution for a wide range of parameters of the system.