FM 00-21 ; mp_arc 00-410

Author: Michele V. Bartuccelli, Guido Gentile, Kyriakos V. Georgiou

Title: Lindstedt series for perturbations of isochronous systems. II. KAM theorem and stability of the upside-down pendulum
 
Abstract: 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 system describing the model.

Keywords: KAM invariant tori, isochronuos systems, perturbation theory, Lindstedt series, Mathieu's equation, vertically driven pendulum

Michele Bartuccelli
Department of Mathematics and Statistics,
University of Surrey
Guildford, GU2 7XH
e-mail: m.bartuccelli@eim.surrey.ac.uk

Guido Gentile
Dipartimento di Matematica
Università di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy
e-mail: gentile@matrm3.mat.uniroma3.it

Kyriakos G. Georgiou
Department of Mathematics and Statistics,
University of Surrey
Guildford, GU2 7XH
e-mail: k.georgiou@eim.surrey.ac.uk