Electron. J. Differential Equations, Vol. 2024 (2024), No. 50, pp. 1-38.
Abel quadratic differential systems of second kind
Joan C. Artes, Jaume Llibre, Dana Schlomiuk, Nicolae Vulpe
Abstract:
The Abel differential equations of second kind,
named after Niels Henrik Abel, are a class of ordinary
differential equations studied by many authors.
Here we consider the Abel quadratic polynomial differential equations
of second kind denoting this class by \(QS_{Ab}\). Firstly we split the whole
family of non-degenerate quadratic systems in four subfamilies
according to the number of infinite singularities. Secondly for
each one of these four subfamilies we determine necessary and sufficient
affine invariant conditions for a quadratic system in this subfamily
to belong to the class \(QS_{Ab}\).
Thirdly we classify all the phase portraits in the
Poincaré disc of the systems in \(QS_{Ab}\) in the case when they have
at infinity either one triple singularity (21 phase portraits) or
an infinite number of singularities (9 phase portraits). Moreover
we determine the affine invariant criteria for the realization of
each one of the 30 topologically distinct phase portraits.
Submitted July 10, 2024. Published September 4, 2024.
Math Subject Classifications: 58K45, 34C23, 34A34.
Key Words: Quadratic differential systems; phase portraits;
second kind of Abel differential equations; affine invariant polynomials.
DOI: 10.58997/ejde.2024.50
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Joan C. Artés
Departament de Matemátiques
Universitat Autónoma de Barcelona
08193 Bellaterra, Barcelona, Spain
email: joancarles.artes@uab.cat
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Jaume Llibre
Departament de Matemátiques
Universitat Autónoma de Barcelona
08193 Bellaterra, Barcelona, Spain
email: jaume.llibre@uab.cat
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Dana Schlomiuk
Département de Mathématiques et de Statistiques
Université de Montréal
Montreal, Canada
email: dana.schlomiuk@umontreal.ca
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Nicolae Vulpe
Institute of Mathematics and Computer Science
Academy of Science of Moldova
5 Academiei str, Chisnau, MD-2028, Moldova
email: nvulpe@gmail.com
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