KAM theory, Lindstedt series and the stability of the upside-down pendulum (Q1865966)

The present paper deals with a planar pendulum with support point oscillating with a frequency \(\omega\) large enough in the vertical direction. It is well known that the upside-down position of the pendulum corresponds to an equilibrium point for the projection of the motion on the pendulum phase space. The stability of the upside-down position has been proven by the averaging method. In contrast to the averaging method, the authors in this paper show that such an equilibrium point is stable by using the Lindstedt series method and by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model. The authors give four appendices for explaining the proof.