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 in 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.
Keywords:
Hamiltonian systems, Kolmogorov theorem, aperiodic time dependence.
\Bibitem{ForWig14}
\by Alessandro~Fortunati, Stephen~Wiggins
\paper Persistence of Diophantine Flows for Quadratic Nearly Integrable Hamiltonians under Slowly Decaying Aperiodic Time Dependence
\jour Regul. Chaotic Dyn.
\yr 2014
\vol 19
\issue 5
\pages 586--600
\mathnet{http://mi.mathnet.ru/rcd184}
\crossref{https://doi.org/10.1134/S1560354714050062}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3266829}
\zmath{https://zbmath.org/?q=an:1308.70029}
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Linking options:
https://www.mathnet.ru/eng/rcd184
https://www.mathnet.ru/eng/rcd/v19/i5/p586
This publication is cited in the following 13 articles:
Donato Scarcella, “Weakly asymptotically quasiperiodic solutions for time-dependent Hamiltonians with a view to celestial mechanics”, Journal of Differential Equations, 2025
Donato Scarcella, “Asymptotic motions converging to arbitrary dynamics for time-dependent Hamiltonians”, Nonlinear Analysis, 243 (2024), 113528
Cheng H. Xu L., “The Existence of Exponentially Decreasing Solutions to Time Dependent Hyperbolic Systems”, J. Math. Anal. Appl., 501:2 (2021), 125199
A. Fortunati, S. Wiggins, “Transient invariant and quasi-invariant structures in an example of an aperiodically time dependent fluid flow”, Int. J. Bifurcation Chaos, 28:5 (2018), 1830015
A. Fortunati, “Travelling waves over an arbitrary bathymetry: a local stability result”, Dyn. Partial Differ. Equ., 15:1 (2018), 81–94
Alessandro Fortunati, Stephen Wiggins, “Transient Invariant and Quasi-Invariant Structures in an Example of an Aperiodically Time Dependent Fluid Flow”, Int. J. Bifurcation Chaos, 28:05 (2018), 1830015
A. Fortunati, S. Wiggins, “Normal forms a la Moser for aperiodically time-dependent Hamiltonians in the vicinity of a hyperbolic equilibrium”, Discret. Contin. Dyn. Syst.-Ser. S, 9:4 (2016), 1109–1118
A. Fortunati, S. Wiggins, “Negligibility of small divisor effects in the normal form theory for nearly-integrable Hamiltonians with decaying non-autonomous perturbations”, Celest. Mech. Dyn. Astron., 125:2 (2016), 247–262
Alessandro Fortunati, Stephen Wiggins, Essays in Mathematics and its Applications, 2016, 89
Alessandro Fortunati, Stephen Wiggins, “A Kolmogorov Theorem for Nearly Integrable Poisson Systems with Asymptotically Decaying Time-dependent Perturbation”, Regul. Chaotic Dyn., 20:4 (2015), 476–485
M. Canadell, R. Llave, “KAM tori and whiskered invariant tori for non-autonomous systems”, Physica D, 310 (2015), 104–113
Citation in format AMSBIB
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