Guido Gentile, Daniel A. Cortez, Jo o C. A. Barata
Stability for quasi-periodically perturbed Hill's equations
(836K, postscript)

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.