Escaping orbits are rare in the quasi-periodic Littlewood boundedness problem

Abstract

We study the superlinear oscillator equation $\ddot{x}+|x{|}^{\alpha -1}x=p(t)$ for $\alpha \ge 3$, where $p$ is a quasi-periodic forcing with no Diophantine condition on the frequencies and show that typically the set of initial values leading to solutions $x$ such that ${\lim }_{t\to \infty }(|x(t)|+|\dot{x}(t)|)=\infty$ has Lebesgue measure zero, provided the starting energy $|x(t_0)|+|\dot{x}(t_0)|$ is sufficiently large.