Averaging and passage through resonances in two-frequency systems near separatrices
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 PhysicsPublisher
SpringerVersion
- 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
2024-12-30ISSN
0010-3616eISSN
1432-0916Publisher version
Language
- en