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

  • Henk Broer
  • Carles Simó
Article

DOI: 10.1007/BF01237651

Cite this article as:
Broer, H. & Simó, C. Bol. Soc. Bras. Mat (1998) 29: 253. doi:10.1007/BF01237651

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.

Keywords

Schrödinger equation with quasi-periodic potential (non-) reducibility to Floquet form quasiperiodic resonance tongues and unstability pockets positive Lyapunov exponent collapse of resonance tongues and breakdown of tori 

Copyright information

© Sociedade Brasileira de Matemática 1998

Authors and Affiliations

  • Henk Broer
    • 1
  • Carles Simó
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Dept. de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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