Poincaré-Melnikov-Arnold method for twist maps

The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Mel...

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Detalles Bibliográficos
Autores: Delshams Valdés, Amadeu|||0000-0003-4134-8882, Ramírez Ros, Rafael|||0000-0002-2127-2940
Tipo de recurso: artículo
Fecha de publicación:1997
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/860
Acceso en línea:https://hdl.handle.net/2117/860
Access Level:acceso abierto
Palabra clave:Dynamical systems
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
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spelling Poincaré-Melnikov-Arnold method for twist mapsDelshams Valdés, Amadeu|||0000-0003-4134-8882Ramírez Ros, Rafael|||0000-0002-2127-2940Dynamical systemsSistemes dinàmicsSistemes dinàmicsClassificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systemsClassificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systemsClassificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theoryThe Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a $2d$-dimensional twist map given by~R. McLachlan, for $d\ge 2$. This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map.19971997-01-0120072007-05-03journal articlehttp://purl.org/coar/resource_type/c_6501NAhttp://purl.org/coar/version/c_be7fb7dd8ff6fe43info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/860reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)Inglésengopen accesshttp://purl.org/coar/access_right/c_abf2Attribution-NonCommercial-NoDerivs 2.5 Spainhttp://creativecommons.org/licenses/by-nc-nd/2.5/es/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/8602026-05-27T15:37:01Z
dc.title.none.fl_str_mv Poincaré-Melnikov-Arnold method for twist maps
title Poincaré-Melnikov-Arnold method for twist maps
spellingShingle Poincaré-Melnikov-Arnold method for twist maps
Delshams Valdés, Amadeu|||0000-0003-4134-8882
Dynamical systems
Sistemes dinàmics
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
title_short Poincaré-Melnikov-Arnold method for twist maps
title_full Poincaré-Melnikov-Arnold method for twist maps
title_fullStr Poincaré-Melnikov-Arnold method for twist maps
title_full_unstemmed Poincaré-Melnikov-Arnold method for twist maps
title_sort Poincaré-Melnikov-Arnold method for twist maps
dc.creator.none.fl_str_mv Delshams Valdés, Amadeu|||0000-0003-4134-8882
Ramírez Ros, Rafael|||0000-0002-2127-2940
author Delshams Valdés, Amadeu|||0000-0003-4134-8882
author_facet Delshams Valdés, Amadeu|||0000-0003-4134-8882
Ramírez Ros, Rafael|||0000-0002-2127-2940
author_role author
author2 Ramírez Ros, Rafael|||0000-0002-2127-2940
author2_role author
dc.subject.none.fl_str_mv Dynamical systems
Sistemes dinàmics
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
topic Dynamical systems
Sistemes dinàmics
Sistemes dinàmics
Classificació AMS::37 Dynamical systems and ergodic theory::37J Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
Classificació AMS::37 Dynamical systems and ergodic theory::37E Low-dimensional dynamical systems
Classificació AMS::37 Dynamical systems and ergodic theory::37G Local and nonlocal bifurcation theory
description The Poincar\'e--Melnikov--Arnold method is the standard tool for detecting splitting of invariant manifolds for systems of ordinary differential equations close to ``integrable'' ones with associated separatrices. This method gives rise to an integral (continuous sum) known as the Melnikov function (or Melnikov integral). If this function is not identically zero, the separatrices split. Moreover, the non-degenerate zeros of this function are associated to transversal intersections of the perturbed invariant (stable and unstable) manifolds. There exists a similar theory for planar maps, and in this case the Melnikov function is not a continuous sum anymore, but an infinite and (a priori) analytically uncomputable (discrete) sum. In a previous work, we have given a method to compute explicitly this kind of sums in terms of elliptic functions, under hypotheses of meromorphicity over the functions in the sum. This method allows us to obtain a strong non-integrability criterion and to apply it to perturbations of elliptic billiards and integrable standard-like maps like the McMillan map. Explicit estimates of the splitting angles are also given. Our aim is extend this method to the study of the splitting of doubly asymptotic manifolds (separatrices) associated to hyperbolic fixed points of twist maps in arbitrary dimensions. We work with maps generated globally by a generating function. Using the variational principle satisfied by these maps, we associate the non-degenerated critical points of a scalar function (here called Melnikov potential) to the transversal intersections of the perturbed asymptotic manifolds. We want to stress the difference of this point of view with the usual one in the literature, that is based in the study of non-degenerated zeros of a vectorial function. The symplectic structure and the variational principle play a fundamental role in our construction. As a first example where this theory can be applied, we study standard-like perturbations of a $2d$-dimensional twist map given by~R. McLachlan, for $d\ge 2$. This map is a multidimensional generalization of the McMillan map. We prove, among other results, that any entire perturbation destroys the separatrix of the McLachlan map.
publishDate 1997
dc.date.none.fl_str_mv 1997
1997-01-01
2007
2007-05-03
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
NA
http://purl.org/coar/version/c_be7fb7dd8ff6fe43
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/860
url https://hdl.handle.net/2117/860
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
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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
Attribution-NonCommercial-NoDerivs 2.5 Spain
http://creativecommons.org/licenses/by-nc-nd/2.5/es/
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
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