Stability for quasi-periodically perturbed Hill's equations

Abstract

We consider a perturbed Hill's equation of the form $\ddot ϕ+ (p_{0}(t) + εp_{1}(t)) ϕ= 0$, 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 ($ε=0$) 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 $ε$ lies in a Cantor set of relatively large measure in $[-ε_0,ε_0]$, where $ε_0$ 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.