Loughborough University
Browse

Averaging and passage through resonances in two-frequency systems near separatrices

journal contribution
posted on 2025-01-09, 14:46 authored by Anatoly NeishtadtAnatoly Neishtadt, Alexey Okunev

In this paper we obtain sharp asymptotic estimates for the accuracy of the averaging method for time-periodic perturbations of one-frequency Hamiltonian systems while passing through a separatrix. The Hamiltonian depends on a parameter that slowly changes for the perturbed system (thus, slow-fast Hamiltonian systems with two and a half degrees of freedom are included in our class). Let ε be the small parameter of the system, then under certain genericity conditions we prove that the accuracy of averaging is O(√ ε| ln ε|) for times of order ε−1 (such times correspond to a change of slow variables of order 1) for all initial data outside an exceptional set with the measure O(√ ε| ln5 ε|). The main novelty of the paper lies in estimating the scattering amplitude and the measure of captured orbits while passing through resonances near separatrices. Our results can also be applied to perturbations of generic two-frequency integrable systems near separatrices, as they can be reduced to periodic perturbations of one-frequency systems.

Funding

Leverhulme Trust, Grant No. RPG-2018-143

History

School

  • Science

Department

  • Mathematical Sciences

Published in

Communications in Mathematical Physics

Publisher

Springer

Version

  • AM (Accepted Manuscript)

Publisher statement

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/[insert DOI]

Acceptance date

2024-12-30

ISSN

0010-3616

eISSN

1432-0916

Language

  • en

Depositor

Prof Anatoly Neishtadt. Deposit date: 30 December 2024

Usage metrics

    Loughborough Publications

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC