Stability for quasi-periodically perturbed Hill's equations

Details Stability for quasi-periodically perturbed Hill's equations Stability for quasi-periodically perturbed Hill's equations

2004-10-11

arxiv.org/abs/math-ph/0410030v1

Physics

Mathematical Physics

40 pages, 4 figures

Scientific paper

10.1007/s00220-005-1413-7

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.

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