04-321 Guido Gentile, Daniel A. Cortez, Joao C. A. Barata

Stability for quasi-periodically perturbed Hill's equations (836K, postscript) Oct 11, 04

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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.

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