The topological classification of a family of quadratic differential systems in terms of affine invariant polynomials

Dana Schlomiuk^a, Nicolae Vulpe<sup>b

a</sup> Département de Mathématiques et de Statistiques Université de Montréal
^b Institute of Mathematics and Computer Science, Academy of Science of Moldova

Abstract: In this paper we provide affine invariant necessary and sufficient conditions for a non-degenerate quadratic differential system to have an invariant conic $f(x, y)=0$ and a Darboux invariant of the form $f(x, y)^\lambda e^{st}$ with $\lambda,s\in \mathbb{R}$ and $s\ne0$. The family of all such systems has a total of seven topologically distinct phase portraits. For each one of these seven phase portraits we provide necessary and sufficient conditions in terms of affine invariant polynomials for a non-degenerate quadratic system in this family to possess this phase portrait.

Keywords and phrases: quadratic differential system, invariant conic, darboux invariant, affine invariant polynomial, group action, phase portrait.

MSC: 58K45, 34C05, 34C23, 34A34

Received: 10.07.2019

Language: English