**Author:**
Michele V. Bartuccelli, Jonathan H.B. Deane, and Guido Gentile,

**Title:** *
Globally and locally attractive solutions
for quasi-periodically forced systems.
*

**Abstract: **
We consider a class of differential equations, $\ddot x +
\gamma \dot x + g(x) = f(\omega t)$, with $\omega \in {\bf R}^{d}$,
describing one-dimensional dissipative systems subject to a periodic
or quasi-periodic (Diophantine) forcing. We study existence and
properties of the limit cycle described by the trajectory with the same
quasi-periodicity as the forcing. For $g(x)=x^{2p+1}$, $p\in {\bf N}$,
we show that, when the dissipation coefficient is large enough,
there is only one limit cycle and that it is a global attractor.
In the case of other forces, including $g(x)=x^{2p}$
(with $p=1$ describing the varactor equation), we find estimates
for the basin of attraction of the limit cycle.

**Keywords:**
Dissipative systems; Quasi-periodically forced systems;
Varactor equation; Attractor; Basin of attraction;
Global attractivity; Invariant set.

Michele Bartuccelli

Department of Mathematics and Statistics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: m.bartuccelli@surrey.ac.uk

Jonathan Deane

Department of Mathematics and Statistics

University of Surrey

Guildford, GU2 7HX, UK

e-mail: j.deane@surrey.ac.uk

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it