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

Abstract

Abstract. A simple example is considered of Hill's equation Jc + (a 2 + bp@))x = O,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 Ib], 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.

The geometry of resonance tongues: a singularity theory approach

“…We like to mention related results in the reversible and symplectic settings regarding parametric resonance with periodic and quasi-periodic forcing terms by Afsharnejad [1] and Broer et al [3,4,[7][8][9][10][12][13][14]16]. Here the methods use Floquet theory, obtained by averaging, as a function of the parameters.…”

Resonance and Singularities

“…The geometry of the individual tongues for small |ε| is exactly as in the periodic case. For larger values of |ε| the situation is more complicated also involving non-reducible quasi-periodic tori, compare with [34].…”

Resonance and Fractal Geometry

“…The geometry of the individual tongues for small |ε| is exactly as in the periodic case. For larger values of |ε| the situation is more complicated also involving non-reducible quasi-periodic tori, compare with [16]. Equation (14) happens to be the eigenvalue equation of the 1-dimensional Schrödinger operator with quasi-periodic potential.…”