Normal form and Nekhoroshev stability for nearly integrable hamiltonian systems with unconditionally slow aperiodic time dependence

Abstract

ABSTRACT. The aim of this paper is to extend the results of Giorgilli and Zehnder for aperiodic time dependent systems to a case of general nearly-integrable convex analytic Hamiltonians. The existence of a normal form and then a stability result are shown in the case of a slow aperiodic time dependence that, under some smallness conditions, is independent of the size of the perturbation.

The Kolmogorov-Arnold-Moser (KAM) and Nekhoroshev Theorems with Arbitrary Time Dependence

“…The aim of this paper was to give an overview of the Nekhoroshev and Kolmogorov stability-type results for integrable Hamiltonian systems subject to aperiodic timedependent perturbations, obtained in the papers [13] and [14]. These are recently added tesserae to the rich mosaic of the Stability Theory of Hamiltonian Systems, one of the several fields in which V.I.…”

“…More precisely, the system of recurrence equations arising from (7) forbids straightforward bounds as in [15] but requires an ad hoc analysis, carried out in this case with the use of the generating function method. See [13] for the details. The smallness condition of required by (5) turns out to be an essential ingredient in order to satisfy condition (7).…”

“…The form of the system they treated was somewhat different than the classical near-integrable Hamiltonian systems since their goals were somewhat different. The first papers to develop Nekhoroshev type results for systems with general time dependence in the classical setting were [12,13], and the only paper treating a KAM type result in the classical setting is [14]. The purpose of this paper is to describe the results in these latter papers dealing with aperiodic time dependence, with particular attention on the issues that arise for explicitly time-dependent Hamiltonians and the correspondent regularity hypotheses that the perturbation function is required to satisfy.…”

Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations

“…A first approach consists in keeping the terms involving the time derivative of the generating function (also called extra-terms) in the normal form and then providing a bound for them. This approach, originally suggested in [GZ92] then used in [FW14a], yields a normal form result for the case a of slow time dependence. This hypothesis provides a smallness condition for the mentioned extra-terms.…”

Quasi-effective stability for nearly integrable Hamiltonian systems

“…For examples, Guzzo, Chierchia and Benettin have announced that they obtained optimal stability exponents under the steepness [6]; Bounemoura and Fischler make use of geometry of numbers to relate two dual Diophantine problems which correspond to the situations of KAM and Nekhoroshev theorems, respectively [2]. For the others, see [3,5,8,19].…”