Stability for quasi-periodically perturbed Hill's equations

TL;DR

This paper proves the persistence of quasi-periodic solutions in a perturbed Hill's equation with quasi-periodic forcing, under Diophantine conditions, using a resummation of Lindstedt series without non-degeneracy assumptions.

Contribution

It extends the stability analysis of Hill's equations to quasi-periodic perturbations without requiring non-degeneracy conditions, employing a novel resummation method.

Findings

Quasi-periodic solutions persist for small perturbations in a large measure set.

The method applies to equations with quasi-periodic external potentials.

Solutions are continued via a resummed Lindstedt series approach.

Abstract

We consider a perturbed Hill's equation of the form $\ddot \phi + (p_{0}(t) + \epsilon p_{1}(t)) \phi = 0$, 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 ($\epsilon=0$) 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 $\epsilon$ lies in a Cantor set of relatively large measure in $[-\epsilon_0,\epsilon_0]$, where $\epsilon_0$ 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.