Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems

We study a one-parameter family of symmetric piecewise linear differential systems in R^3 which is relevant in control theory. The family, which has some intersection points with the adimensional family of Chua's circuits, exhibits more than one attractor even when the two matrices defining its...

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Autores: Llibre, Jaume|||0000-0002-9511-5999, Ponce, Enrique|||0000-0003-0467-5032, Ros Padilla, Javier|||0000-0002-6396-1461
Formato: artículo
Fecha de publicación:2011
País:España
Recursos:Universitat Autònoma de Barcelona
Repositorio:Dipòsit Digital de Documents de la UAB
Idioma:inglés
OAI Identifier:oai:ddd.uab.cat:150450
Acesso em linha:https://ddd.uab.cat/record/150450
https://dx.doi.org/urn:doi:10.1016/j.na.2011.06.051
Access Level:acceso abierto
Palavra-chave:Harmonic balance
Kalman's conjecture
Limit cycles
Periodic orbit
Piecewise linear differential systems
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spelling Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systemsLlibre, Jaume|||0000-0002-9511-5999Ponce, Enrique|||0000-0003-0467-5032Ros Padilla, Javier|||0000-0002-6396-1461Harmonic balanceKalman's conjectureLimit cyclesPeriodic orbitPiecewise linear differential systemsWe study a one-parameter family of symmetric piecewise linear differential systems in R^3 which is relevant in control theory. The family, which has some intersection points with the adimensional family of Chua's circuits, exhibits more than one attractor even when the two matrices defining its dynamics in each zone are stable, in an apparent contradiction with the 3-dimensional Kalman's conjecture. For these systems we characterize algebraically their symmetric periodic orbits and obtain a partial view of the one-parameter unfolding of its triple-zero degeneracy. Having at our disposal exact information about periodic orbits of a family of nonlinear systems, which is rather unusual, the analysis allows us to assess the accuracy of the corresponding harmonic balance predictions. Also, it is shown that certain conditions in Kalman's conjecture can be violated without losing the global asymptotic stability of the origin. 22011-01-0120112011-01-01Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleapplication/pdfhttps://ddd.uab.cat/record/150450https://dx.doi.org/urn:doi:10.1016/j.na.2011.06.051reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengMinisterio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-410open accesshttp://purl.org/coar/access_right/c_abf2Aquest material està protegit per drets d'autor i/o drets afins. Podeu utilitzar aquest material en funció del que permet la legislació de drets d'autor i drets afins d'aplicació al vostre cas. Per a d'altres usos heu d'obtenir permís del(s) titular(s) de drets.https://rightsstatements.org/vocab/InC/1.0/info:eu-repo/semantics/openAccessoai:ddd.uab.cat:1504502026-06-06T12:50:31Z
dc.title.none.fl_str_mv Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
title Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
spellingShingle Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
Llibre, Jaume|||0000-0002-9511-5999
Harmonic balance
Kalman's conjecture
Limit cycles
Periodic orbit
Piecewise linear differential systems
title_short Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
title_full Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
title_fullStr Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
title_full_unstemmed Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
title_sort Algebraic determination of limit cycles in 3-dimensional piecewise linear differential systems
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
Ponce, Enrique|||0000-0003-0467-5032
Ros Padilla, Javier|||0000-0002-6396-1461
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
Ponce, Enrique|||0000-0003-0467-5032
Ros Padilla, Javier|||0000-0002-6396-1461
author_role author
author2 Ponce, Enrique|||0000-0003-0467-5032
Ros Padilla, Javier|||0000-0002-6396-1461
author2_role author
author
dc.subject.none.fl_str_mv Harmonic balance
Kalman's conjecture
Limit cycles
Periodic orbit
Piecewise linear differential systems
topic Harmonic balance
Kalman's conjecture
Limit cycles
Periodic orbit
Piecewise linear differential systems
description We study a one-parameter family of symmetric piecewise linear differential systems in R^3 which is relevant in control theory. The family, which has some intersection points with the adimensional family of Chua's circuits, exhibits more than one attractor even when the two matrices defining its dynamics in each zone are stable, in an apparent contradiction with the 3-dimensional Kalman's conjecture. For these systems we characterize algebraically their symmetric periodic orbits and obtain a partial view of the one-parameter unfolding of its triple-zero degeneracy. Having at our disposal exact information about periodic orbits of a family of nonlinear systems, which is rather unusual, the analysis allows us to assess the accuracy of the corresponding harmonic balance predictions. Also, it is shown that certain conditions in Kalman's conjecture can be violated without losing the global asymptotic stability of the origin.
publishDate 2011
dc.date.none.fl_str_mv 2
2011-01-01
2011
2011-01-01
dc.type.none.fl_str_mv 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
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dc.identifier.none.fl_str_mv https://ddd.uab.cat/record/150450
https://dx.doi.org/urn:doi:10.1016/j.na.2011.06.051
url https://ddd.uab.cat/record/150450
https://dx.doi.org/urn:doi:10.1016/j.na.2011.06.051
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Ministerio de Ciencia e Innovación https://doi.org/10.13039/501100004837 MTM2008-03437
Agència de Gestió d'Ajuts Universitaris i de Recerca https://doi.org/10.13039/501100003030 2009/SGR-410
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
https://rightsstatements.org/vocab/InC/1.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
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https://rightsstatements.org/vocab/InC/1.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
instname_str Universitat Autònoma de Barcelona
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