**Author:**
Guido Gentile, Daniel A. Cortez, João C. A. Barata

**Title:** *
Stability for quasi-periodically perturbed Hill's equations
*

**Abstract: **
We consider a perturbed Hill's equation of the form $\ddot \phi +
\left( p_{0}(t) + \varepsilon p_{1}(t) \right) \phi = 0$,
where $p_{0}$ is real analytic and periodic,
$p_{1}$ is real analytic and
quasi-periodic and $\eps$ is a ``small'' real parameter.
Assuming Diophantine conditions on the frequencies of the
decoupled system, i.e. the frequencies of the external potentials
$p_{0}$ and $p_{1}$ and the proper frequency of the unperturbed
($\varepsilon=0$) Hill's equation, but without making
non-degeneracy assumptions on the perturbing potential $p_{1}$,
we prove that quasi-periodic solutions
of the unperturbed equation can be continued into quasi-periodic
solutions if $\varepsilon$ lies in a Cantor set of relatively large measure
in $[-\varepsilon_0,\varepsilon_0]$, where $\varepsilon_0$ is small enough.
Our method is based on a resummation procedure of a formal Lindstedt
series obtained as a solution of a generalized Riccati equation
associated to Hill's problem.

**Keywords:**
Hill's equation; Quasi-periodic motion; Lindstedt series;
Renormalization group; Reducibility; Cantor set; Divergent series

Guido Gentile

Dipartimento di Matematica

Università di Roma Tre

Largo San Leonardo Murialdo 1, 00146 Roma, Italy

e-mail: gentile@mat.uniroma3.it

Daniel A. Cortez

Instituto de Física

Universidade de São Paulo

Caixa Postal 66 318

e-mail: dacortez@fma.if.usp.br

João C. A. Barata

Instituto de Física

Universidade de São Paulo

Caixa Postal 66 318

e-mail: jbarata@fma.if.usp.br