Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation

Detalhes bibliográficos
Autor(a) principal: da Silva, P. R. [UNESP]
Data de Publicação: 2020
Outros Autores: Meza-Sarmiento, I. S. [UNESP], Novaes, D. D.
Tipo de documento: Artigo
Idioma: eng
Título da fonte: Repositório Institucional da UNESP
Texto Completo: http://dx.doi.org/10.1007/s12591-018-0439-1
http://hdl.handle.net/11449/200147
Resumo: We consider piecewise smooth vector fields (PSVF) defined in open sets M⊆ Rn with switching manifold being a smooth surface Σ. We assume that M\ Σ contains exactly two connected regions, namely Σ + and Σ -. Then, the PSVF are given by pairs X= (X+, X-) , with X= X+ in Σ + and X= X- in Σ -. A regularization of X is a 1-parameter family of smooth vector fields Xε, ε> 0 , satisfying that Xε converges pointwise to X on M\ Σ , when ε→ 0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization Xε. While the linear regularization requires that for every ε> 0 the regularized field Xε is in the convex combination of X+ and X-, the nonlinear regularization requires only that Xε is in a continuous combination of X+ and X-. We prove that, for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R2 and in R3.
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spelling Nonlinear Sliding of Discontinuous Vector Fields and Singular PerturbationNon-smooth vector fieldsRegularizationSingular perturbationSliding vector fieldsVector fieldsWe consider piecewise smooth vector fields (PSVF) defined in open sets M⊆ Rn with switching manifold being a smooth surface Σ. We assume that M\ Σ contains exactly two connected regions, namely Σ + and Σ -. Then, the PSVF are given by pairs X= (X+, X-) , with X= X+ in Σ + and X= X- in Σ -. A regularization of X is a 1-parameter family of smooth vector fields Xε, ε> 0 , satisfying that Xε converges pointwise to X on M\ Σ , when ε→ 0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization Xε. While the linear regularization requires that for every ε> 0 the regularized field Xε is in the convex combination of X+ and X-, the nonlinear regularization requires only that Xε is in a continuous combination of X+ and X-. We prove that, for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R2 and in R3.Departamento de Matemática IBILCE-UNESP, Rua C. Colombo, 2265Departamento de Matemática Universidade Estadual de Campinas, Rua Sérgio Buarque de Holanda, 651, Cidade Universitária Zeferino VazDepartamento de Matemática IBILCE-UNESP, Rua C. Colombo, 2265Universidade Estadual Paulista (Unesp)Universidade Estadual de Campinas (UNICAMP)da Silva, P. R. [UNESP]Meza-Sarmiento, I. S. [UNESP]Novaes, D. D.2020-12-12T01:58:56Z2020-12-12T01:58:56Z2020-01-01info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/articlehttp://dx.doi.org/10.1007/s12591-018-0439-1Differential Equations and Dynamical Systems.0974-68700971-3514http://hdl.handle.net/11449/20014710.1007/s12591-018-0439-12-s2.0-85081330630Scopusreponame:Repositório Institucional da UNESPinstname:Universidade Estadual Paulista (UNESP)instacron:UNESPengDifferential Equations and Dynamical Systemsinfo:eu-repo/semantics/openAccess2021-10-23T12:23:59Zoai:repositorio.unesp.br:11449/200147Repositório InstitucionalPUBhttp://repositorio.unesp.br/oai/requestopendoar:29462024-08-05T21:03:28.968420Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)false
dc.title.none.fl_str_mv Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
title Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
spellingShingle Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
da Silva, P. R. [UNESP]
Non-smooth vector fields
Regularization
Singular perturbation
Sliding vector fields
Vector fields
title_short Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
title_full Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
title_fullStr Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
title_full_unstemmed Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
title_sort Nonlinear Sliding of Discontinuous Vector Fields and Singular Perturbation
author da Silva, P. R. [UNESP]
author_facet da Silva, P. R. [UNESP]
Meza-Sarmiento, I. S. [UNESP]
Novaes, D. D.
author_role author
author2 Meza-Sarmiento, I. S. [UNESP]
Novaes, D. D.
author2_role author
author
dc.contributor.none.fl_str_mv Universidade Estadual Paulista (Unesp)
Universidade Estadual de Campinas (UNICAMP)
dc.contributor.author.fl_str_mv da Silva, P. R. [UNESP]
Meza-Sarmiento, I. S. [UNESP]
Novaes, D. D.
dc.subject.por.fl_str_mv Non-smooth vector fields
Regularization
Singular perturbation
Sliding vector fields
Vector fields
topic Non-smooth vector fields
Regularization
Singular perturbation
Sliding vector fields
Vector fields
description We consider piecewise smooth vector fields (PSVF) defined in open sets M⊆ Rn with switching manifold being a smooth surface Σ. We assume that M\ Σ contains exactly two connected regions, namely Σ + and Σ -. Then, the PSVF are given by pairs X= (X+, X-) , with X= X+ in Σ + and X= X- in Σ -. A regularization of X is a 1-parameter family of smooth vector fields Xε, ε> 0 , satisfying that Xε converges pointwise to X on M\ Σ , when ε→ 0. Inspired by the Fenichel Theory, the sliding and sewing dynamics on the discontinuity locus Σ can be defined as some sort of limit of the dynamics of a nearby smooth regularization Xε. While the linear regularization requires that for every ε> 0 the regularized field Xε is in the convex combination of X+ and X-, the nonlinear regularization requires only that Xε is in a continuous combination of X+ and X-. We prove that, for both cases, the sliding dynamics on Σ is determined by the reduced dynamics on the critical manifold of a singular perturbation problem. We apply our techniques in the description of the nonlinear regularization of normal forms of PSVF in R2 and in R3.
publishDate 2020
dc.date.none.fl_str_mv 2020-12-12T01:58:56Z
2020-12-12T01:58:56Z
2020-01-01
dc.type.status.fl_str_mv info:eu-repo/semantics/publishedVersion
dc.type.driver.fl_str_mv info:eu-repo/semantics/article
format article
status_str publishedVersion
dc.identifier.uri.fl_str_mv http://dx.doi.org/10.1007/s12591-018-0439-1
Differential Equations and Dynamical Systems.
0974-6870
0971-3514
http://hdl.handle.net/11449/200147
10.1007/s12591-018-0439-1
2-s2.0-85081330630
url http://dx.doi.org/10.1007/s12591-018-0439-1
http://hdl.handle.net/11449/200147
identifier_str_mv Differential Equations and Dynamical Systems.
0974-6870
0971-3514
10.1007/s12591-018-0439-1
2-s2.0-85081330630
dc.language.iso.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv Differential Equations and Dynamical Systems
dc.rights.driver.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.source.none.fl_str_mv Scopus
reponame:Repositório Institucional da UNESP
instname:Universidade Estadual Paulista (UNESP)
instacron:UNESP
instname_str Universidade Estadual Paulista (UNESP)
instacron_str UNESP
institution UNESP
reponame_str Repositório Institucional da UNESP
collection Repositório Institucional da UNESP
repository.name.fl_str_mv Repositório Institucional da UNESP - Universidade Estadual Paulista (UNESP)
repository.mail.fl_str_mv
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