We study the superlinear oscillator equationẍ + |x| α−1 x = p(t) for α ≥ 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→∞ (|x(t)| + |ẋ(t)|) = ∞ has Lebesgue measure zero, provided the starting energy |x(t 0 )| + |ẋ(t 0 )| is sufficiently large.

We study the one-dimensional Fermi-Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

We study the one-dimensional Fermi–Ulam ping-pong problem with a Bohr almost periodic forcing function and show that the set of initial condition leading to escaping orbits typically has Lebesgue measure zero.

We study the oscillator $$\ddot{x} + n^2 x + h(x) = p(t)$$ x ¨ + n 2 x + h ( x ) = p ( t ) , where h is a piecewise linear saturation function and p is a continuous $$2\pi $$ 2 π -periodic forcing. It is shown that there is recurrence if and only if p satisfies the Lazer–Leach condition. This condition relates the n-th Fourier coefficient of p(t) with the maximum of h and was first introduced to characterize the existence of periodic solutions.