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

Henk Broer, Dept. of Mathematics and Computing Science, University of 
   Groningen, Blauwborgje 3, 9747 AC Groningen, The Netherlands

Carles Sim\'o Dept. de Matem\`atica Aplicada i An\`alisi, Universitat de
   Barcelona, Gran Via 585, 08007 Barcelona, Spain

Abstract

A simple example is considered of Hill's equation
$\ddot{x}+(a^2+b\,p(t))x=0,$ where the forcing term $p,$ 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 arbitrary $b,$ where some interesting
phenomena occur. Regarding these, a detailed numerical investigation
and tentative explanations are presented.
