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A simple example is considered of Hill's equation x" + (a^2 + bp(t))x = 0, where the forcing term p, 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 arbitrary b, where some interesting phenomena occur. Regarding these, a detailed numerical investigation and tentative explanations are presented.

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