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Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

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Abstract

A simple example is considered of Hill's equation\(\ddot x + (a^2 + bp(t))x = 0\), where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

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Authors and Affiliations

  1. Dept. of Mathematics and Computer Science, University of Groningen, PO Box 800, 9700 AV, Groningen, The Netherlands

    Henk Broer

  2. Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    Carles Simó

Authors
  1. Henk Broer
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  2. Carles Simó
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Dedicated to the memory of Ricardo Mañé

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Broer, H., Simó, C. Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena. Bol. Soc. Bras. Mat 29, 253–293 (1998). https://doi.org/10.1007/BF01237651

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  • DOI: https://doi.org/10.1007/BF01237651

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