Hill's equation with quasi-periodic forcing: resonance tongues, instability pockets and global phenomena

Abstract

A simple example is considered of Hill's equation $$\ddot x + (a^2 + bp(t))x = 0$$ , where the forcing termp, instead of periodic, is quasi-periodic with two frequencies. A geometric exploration is carried out of certain resonance tongues, containing instability pockets. This phenomenon in the perturbative case of small |b|, can be explained by averaging. Next a numerical exploration is given for the global case of arbitraryb, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.