Phase change and order 2 averaging for one-frequency systems with separatrix crossing
We study the evolution of angular variable (phase) for general (not necessarily Hamiltonian) perturbations of Hamiltonian systems with one degree of freedom near separatrices of the unperturbed system. To this end, we use the averaged system of order 2. We obtain estimates for the accuracy of the order 2 averaged system near separatrices and use these estimates to prove a formula for the phase change when solutions of the perturbed system approach separatrices of the unperturbed system (such formula is known when the perturbation is Hamiltonian). As an application of this formula, we show that two natural de?nitions of probability of capture into different domains after separatrix crossing proposed by V.I. Arnold and D.V. Anosov lead to the same formula for this probability.
Funding
Leverhulme Trust (Grant No. RPG-2018-143).
History
School
- Science
Department
- Mathematical Sciences
Published in
NonlinearityVolume
35Issue
8Pages
4469 - 4516Publisher
IOP PublishingVersion
- VoR (Version of Record)
Rights holder
© IOP Publishing Ltd & London Mathematical SocietyPublisher statement
This is an Open Access Article. It is published by IOP Publishing under the Creative Commons Attribution 3.0 Unported Licence (CC BY). Full details of this licence are available at: https://creativecommons.org/licenses/by/3.0/Acceptance date
2022-06-20Publication date
2022-07-18Copyright date
2022ISSN
0951-7715eISSN
1361-6544Publisher version
Language
- en