Lindstedt series for perturbations of isochronous systems. II. KAM theorem and stability of the upside-down pendulum

We consider the planar pendulum with support point oscillating in the vertical direction, and we study its motion around the equilibrium point corresponding to the upside-down position. We prove that the equilibrium point is stable for the projection of the motion on the pendulum phase space (for a full measure subset of the stability region of the linearized system inside the two-dimensional space of parameters), by proving the persistence of invariant KAM tori for the two-dimensional Hamiltonian system describing the model.