Computing the stability regions of Hill's equation

A numerical scheme is presented for computing the boundaries between the stable and unstable regions of Hill's equation. The approach based on formulating an integral representation of Hill's equation. Thus the problem of finding the boundary between the stable and unstable regions in parameter space is then a spectral problem of an integral operator. Several properties of the operator are derived and an iteration scheme for numerical calculation of the stability boundary is presented. The numerical scheme has a number of advantages over the classical methods for locating the stability boundary. Several examples are presented.