Algebraic Limit Cycles

In the qualitative theory of differential equations in the plane ℝ2, one of the most difficult objects to study is the existence of limit cycles. Here, we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems. More precisely, (i) we stud...

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Author: Llibre, Jaume|||0000-0002-9511-5999
Format: article
Publication Date:2025
Country:España
Institution:Universitat Autònoma de Barcelona
Repository:Dipòsit Digital de Documents de la UAB
Language:English
OAI Identifier:oai:ddd.uab.cat:326012
Online Access:https://ddd.uab.cat/record/326012
https://dx.doi.org/urn:doi:10.1142/S0218127425400085
Access Level:Embargoed access
Keyword:Algebraic limit cycle
Limit cycle
Polynomial vector field
Quadratic polynomial vector field
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spelling Algebraic Limit CyclesLlibre, Jaume|||0000-0002-9511-5999Algebraic limit cycleLimit cyclePolynomial vector fieldQuadratic polynomial vector fieldIn the qualitative theory of differential equations in the plane ℝ2, one of the most difficult objects to study is the existence of limit cycles. Here, we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems. More precisely, (i) we study the algebraic limit cycles of the polynomial differential systems of degree 2; (ii) we provide the maximum number of generic algebraic limit cycles that the polynomial differential systems of degree n can exhibit; (iii) we show using the algebraic limit cycles that any finite configuration of limit cycles can be realized by some polynomial differential system; and (iv) we provide the maximum number of algebraic limit cycles formed by circles that a polynomial differential system of degree n can exhibit. 220252025-01-0120262026-11-30Articlehttp://purl.org/coar/resource_type/c_6501AMhttp://purl.org/coar/version/c_ab4af688f83e57aainfo:eu-repo/semantics/articleimage/vnd.microsoft.iconhttps://ddd.uab.cat/record/326012https://dx.doi.org/urn:doi:10.1142/S0218127425400085reponame:Dipòsit Digital de Documents de la UABinstname:Universitat Autònoma de BarcelonaInglésengAgencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00Generalitat de Catalunya https://doi.org/10.13039/501100002809 2021/SGR-00113embargoed accesshttp://purl.org/coar/access_right/c_f1cfAquest 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/embargoedAccessoai:ddd.uab.cat:3260122026-06-06T12:50:31Z
dc.title.none.fl_str_mv Algebraic Limit Cycles
title Algebraic Limit Cycles
spellingShingle Algebraic Limit Cycles
Llibre, Jaume|||0000-0002-9511-5999
Algebraic limit cycle
Limit cycle
Polynomial vector field
Quadratic polynomial vector field
title_short Algebraic Limit Cycles
title_full Algebraic Limit Cycles
title_fullStr Algebraic Limit Cycles
title_full_unstemmed Algebraic Limit Cycles
title_sort Algebraic Limit Cycles
dc.creator.none.fl_str_mv Llibre, Jaume|||0000-0002-9511-5999
author Llibre, Jaume|||0000-0002-9511-5999
author_facet Llibre, Jaume|||0000-0002-9511-5999
author_role author
dc.subject.none.fl_str_mv Algebraic limit cycle
Limit cycle
Polynomial vector field
Quadratic polynomial vector field
topic Algebraic limit cycle
Limit cycle
Polynomial vector field
Quadratic polynomial vector field
description In the qualitative theory of differential equations in the plane ℝ2, one of the most difficult objects to study is the existence of limit cycles. Here, we summarize some results and open problems on the algebraic limit cycles of the planar polynomial differential systems. More precisely, (i) we study the algebraic limit cycles of the polynomial differential systems of degree 2; (ii) we provide the maximum number of generic algebraic limit cycles that the polynomial differential systems of degree n can exhibit; (iii) we show using the algebraic limit cycles that any finite configuration of limit cycles can be realized by some polynomial differential system; and (iv) we provide the maximum number of algebraic limit cycles formed by circles that a polynomial differential system of degree n can exhibit.
publishDate 2025
dc.date.none.fl_str_mv
2
2025
2025-01-01
2026
2026-11-30
dc.type.none.fl_str_mv Article
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https://dx.doi.org/urn:doi:10.1142/S0218127425400085
url https://ddd.uab.cat/record/326012
https://dx.doi.org/urn:doi:10.1142/S0218127425400085
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación https://doi.org/10.13039/501100011033 PID2022-136613NB-I00
Generalitat de Catalunya https://doi.org/10.13039/501100002809 2021/SGR-00113
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dc.source.none.fl_str_mv reponame:Dipòsit Digital de Documents de la UAB
instname:Universitat Autònoma de Barcelona
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