Persistence of Diophantine flows for quadratic nearly-integrable Hamiltonians under slowly decaying aperiodic time dependence

TL;DR

This paper proves the persistence of certain invariant tori in nearly-integrable quadratic Hamiltonian systems with aperiodic, exponentially decaying time dependence, extending classical results to more general time-dependent perturbations.

Contribution

It establishes a Kolmogorov-type theorem for Hamiltonians with aperiodic, decaying time dependence, allowing for arbitrarily small decay rates.

Findings

Existence of invariant tori with Diophantine frequencies in perturbed systems.

Persistence of these tori under exponentially decaying perturbations.

Flexibility in choosing decay coefficients relative to perturbation size.

Abstract

The aim of this paper is to prove a Kolmogorov-type result for a nearly-integrable Hamiltonian, quadratic in the actions, with an aperiodic time dependence. The existence of a torus with a prefixed Diophantine frequency is shown in the forced system, provided that the perturbation is real-analytic and (exponentially) decaying with time. The advantage consists of the possibility to choose an arbitrarily small decaying coefficient, consistently with the perturbation size.