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Stability for Quasi-Periodically Perturbed Hill's Equations

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Abstract

We consider a perturbed Hill's equation of the form +(p0(t)+ɛp1(t))ϕ=0, where p0 is real analytic and periodic, p1 is real analytic and quasi-periodic and ɛ ∈ℝ is ``small''. Assuming Diophantine conditions on the frequencies of the decoupled system, i.e. the frequencies of the external potentials p0 and p1 and the proper frequency of the unperturbed (ɛ=0) Hill's equation, but without making any assumptions on the perturbing potential p1 other than analyticity, we prove that quasi-periodic solutions of the unperturbed equation can be continued into quasi-periodic solutions if ɛ lies in a Cantor set of relatively large measure in where ɛ0 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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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Roma Tre, Roma, 00146, Italy

    Guido Gentile

  2. Instituto de Física, Universidade de São Paulo, Caixa Postal 66 318, São Paulo, 05315 970, SP, Brasil

    Daniel A Cortez & João C. A. Barata

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  1. Guido Gentile
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  2. Daniel A Cortez
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  3. João C. A. Barata
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Correspondence to João C. A. Barata.

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Communicated by G. Gallavotti

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Gentile, G., Cortez, D. & Barata, J. Stability for Quasi-Periodically Perturbed Hill's Equations. Commun. Math. Phys. 260, 403–443 (2005). https://doi.org/10.1007/s00220-005-1413-7

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