**Author:**
M. V. Bartuccelli, G. Gentile , K.V. Georgiou

**Title:** *
On the Stability of the Upside-Down Pendulum with Damping}
*

**Abstract: **
A rigorous analysis is presented in order to show that,
in presence of friction, the upward equilibrium position
of the vertically driven pendulum,
with a small non-vanishing damping term,
becomes asymptotically stable when the period of the forcing
is below an appropriate threshold value.
As a byproduct we obtain an analytic expression of the solution
for initial data close enough to the equilibrium position.

**Keywords:**
Pendulum, Mathieu's equation, perturbation theory, stability,
basins of attraction

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