The Spherical Kapitza-Whitney Pendulum

TL;DR

This paper investigates the dynamics of a spherical pendulum with a vibrating suspension point under a horizontal periodic force, proving the existence of non-falling solutions and demonstrating their stability through numerical analysis.

Contribution

It establishes the existence of non-falling periodic solutions in a spherical pendulum with arbitrary force strength, extending understanding beyond weak or rapid oscillation assumptions.

Findings

Existence of at least one non-falling periodic solution.

Numerical evidence of asymptotic stability for a range of parameters.

Non-falling solutions persist without the force being weak or rapidly oscillating.

Abstract

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.