Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

In this work we study the splitting distance of a rapidly perturbed pendulum with a -periodic function and . Systems of this kind undergo exponentially small splitting, and, when , it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided . Ou...

Full description

Bibliographic Details
Authors: Baldomá Barraca, Inmaculada|||0000-0002-4838-1186, Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717, Moreno González, Román|||0000-0001-7769-4942
Format: article
Publication Date:2024
Country:España
Institution:Universitat Politècnica de Catalunya (UPC)
Repository:UPCommons. Portal del coneixement obert de la UPC
Language:English
OAI Identifier:oai:upcommons.upc.edu:2117/426335
Online Access:https://hdl.handle.net/2117/426335
https://dx.doi.org/10.1137/23M1550992
Access Level:Open access
Keyword:Hamiltonian systems
Differentiable dynamical systems
Splitting of separatrices
Exponentially small phenomena
Sistemes dinàmics diferenciables
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
id ES_35cce88b50ffa9bd8fac15fa41d50252
oai_identifier_str oai:upcommons.upc.edu:2117/426335
network_acronym_str ES
network_name_str España
repository_id_str
spelling Splitting of separatrices for rapid degenerate perturbations of the classical pendulumBaldomá Barraca, Inmaculada|||0000-0002-4838-1186Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717Moreno González, Román|||0000-0001-7769-4942Hamiltonian systemsDifferentiable dynamical systemsSplitting of separatricesHamiltonian systemsExponentially small phenomenaSistemes dinàmics diferenciablesHamilton, Sistemes deClassificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theoryClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsÀrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmicsIn this work we study the splitting distance of a rapidly perturbed pendulum with a -periodic function and . Systems of this kind undergo exponentially small splitting, and, when , it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided . Our study focuses on the case , and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as . On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with where, for most , the perturbation satisfies . As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order , where is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.This work is part of the grants PGC2018-098676-B-100 and PID-2021-122954NB-100 funded by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe.” T. M. S. is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2019. R.M. is supported by grant PRE2019-088132 funded by MCIN/AEI/10.13039/501100011033 and “ESF Investing in your future”. This work is also supported by the Spanish State Research Agency, through the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R& D (CEX2020- 001084-M)Peer Reviewed20242024-06-3020252025-03-12journal articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/426335https://dx.doi.org/10.1137/23M1550992reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4263352026-05-27T15:37:01Z
dc.title.none.fl_str_mv Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
title Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
spellingShingle Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
Baldomá Barraca, Inmaculada|||0000-0002-4838-1186
Hamiltonian systems
Differentiable dynamical systems
Splitting of separatrices
Hamiltonian systems
Exponentially small phenomena
Sistemes dinàmics diferenciables
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
title_short Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
title_full Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
title_fullStr Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
title_full_unstemmed Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
title_sort Splitting of separatrices for rapid degenerate perturbations of the classical pendulum
dc.creator.none.fl_str_mv Baldomá Barraca, Inmaculada|||0000-0002-4838-1186
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Moreno González, Román|||0000-0001-7769-4942
author Baldomá Barraca, Inmaculada|||0000-0002-4838-1186
author_facet Baldomá Barraca, Inmaculada|||0000-0002-4838-1186
Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Moreno González, Román|||0000-0001-7769-4942
author_role author
author2 Martínez-Seara Alonso, M. Teresa|||0000-0001-8421-8717
Moreno González, Román|||0000-0001-7769-4942
author2_role author
author
dc.subject.none.fl_str_mv Hamiltonian systems
Differentiable dynamical systems
Splitting of separatrices
Hamiltonian systems
Exponentially small phenomena
Sistemes dinàmics diferenciables
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
topic Hamiltonian systems
Differentiable dynamical systems
Splitting of separatrices
Hamiltonian systems
Exponentially small phenomena
Sistemes dinàmics diferenciables
Hamilton, Sistemes de
Classificació AMS::37 Dynamical systems and ergodic theory::37C Smooth dynamical systems: general theory
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals::Sistemes dinàmics
description In this work we study the splitting distance of a rapidly perturbed pendulum with a -periodic function and . Systems of this kind undergo exponentially small splitting, and, when , it is known that the Melnikov function actually gives an asymptotic expression for the splitting function provided . Our study focuses on the case , and it is motivated by two main reasons. On the one hand, our study is motivated by the general understanding of the splitting, as current results fail for a perturbation as simple as . On the other hand, a study of the splitting of invariant manifolds of tori of rational frequency in Arnold’s original model for diffusion leads to the consideration of pendulum-like Hamiltonians with where, for most , the perturbation satisfies . As expected, the Melnikov function is not a correct approximation for the splitting in this case. To tackle the problem we use a splitting formula based on the solutions of the so-called inner equation and make use of the Hamilton–Jacobi formalism. The leading exponentially small term appears at order , where is an integer determined exclusively by the harmonics of the perturbation. We also provide an algorithm to compute it.
publishDate 2024
dc.date.none.fl_str_mv 2024
2024-06-30
2025
2025-03-12
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
AM
http://purl.org/coar/version/c_ab4af688f83e57aa
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/426335
https://dx.doi.org/10.1137/23M1550992
url https://hdl.handle.net/2117/426335
https://dx.doi.org/10.1137/23M1550992
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
_version_ 1869405911705976832
score 15.812429

Similar Items