Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 50, pp. 1-38.

Title: Abel quadratic differential systems of second kind

Authors: Joan C. Artes (Univ. Autonoma  de Barcelona, Spain)
         Jaume Llibre  (Univ. Autonoma  de Barcelona, Spain)
         Dana Schlomiuk (Univ. de Montreal, Canada)
         Nicolae Vulpe (Academy of Science of Moldova, Moldova)

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
Poincar\'e 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 04, 2024.
Math Subject Classifications: 58K45, 34C23, 34A34.
Key Words: Quadratic differential systems; phase portraits;
    second kind of Abel differential equations; affine invariant polynomials.