Resonance tongues in Hill's equations: A geometric approach

The geometry of resonance tongues is considered in, mainly reversible, vers ions of Hill's equation, close to the classical Mathieu case. Hill's map as signs to each value of the multiparameter the corresponding Poincare matrix . Dy an averaging method, the geometry of Hill's map locally can be underst ood 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. (C) 2000 Academic Press.