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

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. The proof, based on the Lie series formalism, is a generalization of a work by A. Giorgilli.

Normal forms <em>à la Moser</em> for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium

“…aperiodic. In the same spirit of the aperiodic version of the Kolmogorov theorem of [FW14a], which we use as a guideline (see also [Giob]), the proof consists on the extension of the KAM approach of [CG94] and [Gal97]. Even in the original problem of Moser, despite the absence of "genuine" small divisors 1 , the well known property of superconvergence of the KAM schemes, turns out to be of crucial importance in order to compensate the accumulation of "artificial" divisors generated by the Cauchy estimates.…”

“…This feature is profitably used also in our case. The treatment of the class of time-dependent homological equations, naturally arising in the normalization algorithm, has been improved with respect to [FW14a]. Basically, the canonical transformation on which the single step of the mentioned algorithm is based, has the property to leave the time unchanged 2 .…”

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.…”

“…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

“…Alternatively, those terms can be removed by including them into the homological equation, which turns out to be, in this way, a linear ODE in time. This has been profitably used in [FW14b], [FW15a] and in [FW15b] but requires (except for a particular case described in [FW15b]) an important assumption. More precisely, it is necessary to suppose that the perturbation, as a function of t, belongs to the class of summable functions over the real semi-axis 2 .…”