Perturbations of the Mathieu equation within the class of Hill's equation (Q923195)

In this paper perturbations of Mathieu's equation \((1)\quad \ddot x+(\lambda +\mu \cos t)x=0\) are considered which retain the form of Hill's equation (2) ẍ\(+Q(t)x=0\). By considering universal unfoldings of the Poincaré map and by applying Takens' lemma the nature of the stability transition regions in (\(\lambda\),\(\mu\)) space is discussed. It is shown that not all unfoldings can be obtained by perturbing (1) within the class of systems (2). Similar results are obtained for the symmetric Hill type systems.