Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariantq

Abstract

Let \(QS\) be the class of non-degenerate planar quadratic differential ystems and \(QS_3\) its subclass formed by the systems possessing an invariant cubic \(f(x,y)=0\). In this article, using the action of the group of real affine transformations and time rescaling on \(QS\), we obtain all the possible normal forms for the quadratic systems in \(QS_3\). Working with these normal forms we complete the characterization of the phase portraits in \(QS_3\) having a Darboux invariant of the form \(f(x, y)e^{st}\), with \(s \in \mathbb R\).
 For more information see https://ejde.math.txstate.edu/Volumes/2021/69/abstr.html