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

Abstract

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.

Escaping Orbits Are also Rare in the Almost Periodic Fermi–Ulam Ping-Pong

“…There, the question is whether solutions of an equation ẍ + G (x) = p(t) stay bounded in the (x, ẋ)-phase space if the potential G satisfies some superlinearity condition. In [24] it is shown that the associated escaping set E typically has Lebesgue measure zero for G (x) = |x| α−1 x with α ≥ 3 and a quasi-periodic forcing function p(t). Indeed, this result can be improved to the almost periodic case in a way analogous to the one presented here (for the ping-pong problem).…”

Escaping orbits are also rare in the almost periodic Fermi-Ulam ping-pong

“…There, the question is whether solutions of an equation ẍ + G ′ (x) = p(t) stay bounded in the (x, ẋ)-phase space if the potential G satisfies some superlinearity condition. In [Sch19] it is shown that the associated escaping set E typically has Lebesgue measure zero for G ′ (x) = |x| α−1 x with α ≥ 3 and a quasi-periodic forcing function p(t). Indeed, this result can be improved to the almost periodic case in a way analogous to the one presented here (for the ping-pong problem).…”

Growth rates of orbits in non-periodic twist maps and a theorem by Neishtadt

“…Over the last years we have examined the dynamics of twist maps with non-periodic angles [2][3][4][5]. Motivated by the Fermi-Ulam ping-pong model, and also by the Littlewood boundedness problem, we have obtained results on the role of the bounded orbits in the general dynamics [6] and also on the improbability of escaping orbits [7,11,12]. However, the first result for this class of maps is older and due to Neishtadt [9].…”