FM 2005-2 (mp_arc 04-321; arXiv math-ph/0410030)

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