Saddle-node canard cycles in slow-fast planar piecewise linear differential systems

By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercrit...

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Autores: Carmona Centeno, Victoriano, Fernández García, Soledad, Teruel Aguilar, Antonio Esteban
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Institución:Universidad de Sevilla (US)
Repositorio:idUS. Depósito de Investigación de la Universidad de Sevilla
OAI Identifier:oai:idus.us.es:11441/155336
Acceso en línea:https://hdl.handle.net/11441/155336
https://doi.org/10.1016/j.nahs.2024.101472
Access Level:acceso abierto
Palabra clave:Piecewise linear systems
Bifurcations
Slow–fast differential systems
Canard orbits
Saddle–node canard orbits
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spelling Saddle-node canard cycles in slow-fast planar piecewise linear differential systemsCarmona Centeno, VictorianoFernández García, SoledadTeruel Aguilar, Antonio EstebanPiecewise linear systemsBifurcationsSlow–fast differential systemsCanard orbitsSaddle–node canard orbitsBy applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.ElsevierMatemática Aplicada IIEcuaciones Diferenciales y Análisis NuméricoTIC130: Investigación en Sistemas Dinámicos en IngenieríaFQM120: Modelado Matemático y Simulación de Sistemas MedioambientalesJunta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project TIC-0130Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project P12-FQM-1658Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project P20-01160Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project US-1380740Ministerio de Ciencia, Innovación y Universidades (MCIU), España project PGC2018-096265-B-I00Ministerio de Ciencia, Innovación y Universidades (MCIU), España project PID2021- 123200NB-I00Ministerio de Ciencia e Innovación, Spain through the project PID2021-123153OB-C212024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/11441/155336https://doi.org/10.1016/j.nahs.2024.101472reponame:idUS. Depósito de Investigación de la Universidad de Sevillainstname:Universidad de Sevilla (US)InglésNonlinear Analysis: Hybrid Systems, 52 (101472).TIC-0130P12-FQM-1658P20-01160US-1380740PGC2018-096265-B-I00PID2021- 123200NB-I00PID2021-123153OB-C21https://www.sciencedirect.com/science/article/pii/S1751570X24000098info:eu-repo/semantics/openAccessoai:idus.us.es:11441/1553362026-06-17T12:51:07Z
dc.title.none.fl_str_mv Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
title Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
spellingShingle Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
Carmona Centeno, Victoriano
Piecewise linear systems
Bifurcations
Slow–fast differential systems
Canard orbits
Saddle–node canard orbits
title_short Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
title_full Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
title_fullStr Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
title_full_unstemmed Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
title_sort Saddle-node canard cycles in slow-fast planar piecewise linear differential systems
dc.creator.none.fl_str_mv Carmona Centeno, Victoriano
Fernández García, Soledad
Teruel Aguilar, Antonio Esteban
author Carmona Centeno, Victoriano
author_facet Carmona Centeno, Victoriano
Fernández García, Soledad
Teruel Aguilar, Antonio Esteban
author_role author
author2 Fernández García, Soledad
Teruel Aguilar, Antonio Esteban
author2_role author
author
dc.contributor.none.fl_str_mv Matemática Aplicada II
Ecuaciones Diferenciales y Análisis Numérico
TIC130: Investigación en Sistemas Dinámicos en Ingeniería
FQM120: Modelado Matemático y Simulación de Sistemas Medioambientales
Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project TIC-0130
Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project P12-FQM-1658
Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project P20-01160
Junta de Andalucía (Consejería de Economía, Conocimiento, Empresas y Universidad) project US-1380740
Ministerio de Ciencia, Innovación y Universidades (MCIU), España project PGC2018-096265-B-I00
Ministerio de Ciencia, Innovación y Universidades (MCIU), España project PID2021- 123200NB-I00
Ministerio de Ciencia e Innovación, Spain through the project PID2021-123153OB-C21
dc.subject.none.fl_str_mv Piecewise linear systems
Bifurcations
Slow–fast differential systems
Canard orbits
Saddle–node canard orbits
topic Piecewise linear systems
Bifurcations
Slow–fast differential systems
Canard orbits
Saddle–node canard orbits
description By applying a singular perturbation approach, canard explosions exhibited by a general family of singularly perturbed planar Piecewise Linear (PWL) differential systems are analyzed. The performed study involves both hyperbolic and non-hyperbolic canard limit cycles appearing after both, a supercritical and a subcritical Hopf bifurcation. The obtained results are comparable with those obtained for smooth vector fields. In some sense, the manuscript can be understood as an extension towards the PWL framework of the results obtained for smooth systems by Dumortier and Roussarie in Mem. Am. Math. Soc. 1996, and Krupa and Szmolyan in J. Differ. Equ. 2001. In addition, some novel slow–fast behaviors are obtained. In particular, in the supercritical case, and under suitable conditions, it is proved that the limit cycles are organized along a curve exhibiting two folds. Each of these folds corresponds to a saddle–node bifurcation of canard limit cycles, one involving headless canard cycles, and the other involving canard cycles with head. This configuration also occurs in smooth systems with N-shaped fast nullcline. However, it has not been previously reported in the Van der Pol system. Our results provide justification for this observation.
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/11441/155336
https://doi.org/10.1016/j.nahs.2024.101472
url https://hdl.handle.net/11441/155336
https://doi.org/10.1016/j.nahs.2024.101472
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Nonlinear Analysis: Hybrid Systems, 52 (101472).
TIC-0130
P12-FQM-1658
P20-01160
US-1380740
PGC2018-096265-B-I00
PID2021- 123200NB-I00
PID2021-123153OB-C21
https://www.sciencedirect.com/science/article/pii/S1751570X24000098
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Elsevier
publisher.none.fl_str_mv Elsevier
dc.source.none.fl_str_mv reponame:idUS. Depósito de Investigación de la Universidad de Sevilla
instname:Universidad de Sevilla (US)
instname_str Universidad de Sevilla (US)
reponame_str idUS. Depósito de Investigación de la Universidad de Sevilla
collection idUS. Depósito de Investigación de la Universidad de Sevilla
repository.name.fl_str_mv
repository.mail.fl_str_mv
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score 15.812429

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