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 Poincaré 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.
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