**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