Resonance tongues in Hill's equations: a geometric approach

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

The geometry of resonance tongues is considered in, mainly reversible, versions
of Hill's equation, close to the classical Mathieu case. Hill's map assigns
to each value of the multiparameter the corresponding Poincar\'e matrix.
By an averaging method, the geometry of Hill's map locally can be understood
in terms of cuspoid Whitney singularities. This adds robustness to the result.
The algorithmic nature of the averaging method enables a pull--back to the
resonance tongues of the original system.
   
