Quadratic vector fields in class I.
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| Title: | Quadratic vector fields in class I. |
|---|---|
| Authors: | Carles Artés, Joan1 (AUTHOR), Chen, Hebai2 (AUTHOR), Manel Ferrer, Lluc1 (AUTHOR), Jia, Man2 (AUTHOR) jiaman9305@163.com |
| Source: | Dynamical Systems: An International Journal. Jun2025, Vol. 40 Issue 2, p191-222. 32p. |
| Subjects: | Vector fields, Quadratic fields, Symbolic computation, Projective spaces, Numerical calculations, Limit cycles, Bifurcation diagrams |
| Abstract: | In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. [ABSTRACT FROM AUTHOR] |
| Copyright of Dynamical Systems: An International Journal is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.) | |
| Database: | Engineering Source |
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| Header | DbId: egs DbLabel: Engineering Source An: 185229483 AccessLevel: 6 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: Quadratic vector fields in class I. – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Carles+Artés%2C+Joan%22">Carles Artés, Joan</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Chen%2C+Hebai%22">Chen, Hebai</searchLink><relatesTo>2</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Manel+Ferrer%2C+Lluc%22">Manel Ferrer, Lluc</searchLink><relatesTo>1</relatesTo> (AUTHOR)<br /><searchLink fieldCode="AR" term="%22Jia%2C+Man%22">Jia, Man</searchLink><relatesTo>2</relatesTo> (AUTHOR)<i> jiaman9305@163.com</i> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="JN" term="%22Dynamical+Systems%3A+An+International+Journal%22">Dynamical Systems: An International Journal</searchLink>. Jun2025, Vol. 40 Issue 2, p191-222. 32p. – Name: Subject Label: Subjects Group: Su Data: <searchLink fieldCode="DE" term="%22Vector+fields%22">Vector fields</searchLink><br /><searchLink fieldCode="DE" term="%22Quadratic+fields%22">Quadratic fields</searchLink><br /><searchLink fieldCode="DE" term="%22Symbolic+computation%22">Symbolic computation</searchLink><br /><searchLink fieldCode="DE" term="%22Projective+spaces%22">Projective spaces</searchLink><br /><searchLink fieldCode="DE" term="%22Numerical+calculations%22">Numerical calculations</searchLink><br /><searchLink fieldCode="DE" term="%22Limit+cycles%22">Limit cycles</searchLink><br /><searchLink fieldCode="DE" term="%22Bifurcation+diagrams%22">Bifurcation diagrams</searchLink> – Name: Abstract Label: Abstract Group: Ab Data: In [Ye et al., Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections. [ABSTRACT FROM AUTHOR] – Name: AbstractSuppliedCopyright Label: Group: Ab Data: <i>Copyright of Dynamical Systems: An International Journal is the property of Taylor & Francis Ltd and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.) |
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| RecordInfo | BibRecord: BibEntity: Identifiers: – Type: doi Value: 10.1080/14689367.2024.2436223 Languages: – Code: eng Text: English PhysicalDescription: Pagination: PageCount: 32 StartPage: 191 Subjects: – SubjectFull: Vector fields Type: general – SubjectFull: Quadratic fields Type: general – SubjectFull: Symbolic computation Type: general – SubjectFull: Projective spaces Type: general – SubjectFull: Numerical calculations Type: general – SubjectFull: Limit cycles Type: general – SubjectFull: Bifurcation diagrams Type: general Titles: – TitleFull: Quadratic vector fields in class I. Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Carles Artés, Joan – PersonEntity: Name: NameFull: Chen, Hebai – PersonEntity: Name: NameFull: Manel Ferrer, Lluc – PersonEntity: Name: NameFull: Jia, Man IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 06 Text: Jun2025 Type: published Y: 2025 Identifiers: – Type: issn-print Value: 14689367 Numbering: – Type: volume Value: 40 – Type: issue Value: 2 Titles: – TitleFull: Dynamical Systems: An International Journal Type: main |
| ResultId | 1 |
