Resonance tongues in the quasi-periodic Hill-Schrödinger equation with three frequencies

Abstract

In this paper we investigate numerically the following Hill’s equation x″ + (a + bq(t))x = 0 where \( q(t) = \cos t + \cos \sqrt {2t} + \cos \sqrt {3t} \) is a quasi-periodic forcing with three rationally independent frequencies. It appears, also, as the eigenvalue equation of a Schrödinger operator with quasi-periodic potential.Massive numerical computations were performed for the rotation number and the Lyapunov exponent in order to detect open and collapsed gaps, resonance tongues. Our results show that the quasi-periodic case with three independent frequencies is very different not only from the periodic analogs, but also from the case of two frequencies. Indeed, for large values of b the spectrum contains open intervals at the bottom. From a dynamical point of view we numerically give evidence of the existence of open intervals of a, for large b, where the system is nonuniformly hyperbolic: the system does not have an exponential dichotomy but the Lyapunov exponent is positive. In contrast with the region with zero Lyapunov exponents, both the rotation number and the Lyapunov exponent do not seem to have square root behavior at endpoints of gaps. The rate of convergence to the rotation number and the Lyapunov exponent in the nonuniformly hyperbolic case is also seen to be different from the reducible case.