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

Abstract

Let QS be the class of non-degenerate planar quadratic differential systems and QS3 its subclass formed by the systems possessing an invariant cubic ƒ(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 QS3. Working with these normal forms we complete the characterization of the phase portraits in QS3 having a Darboux invariant of the form ƒ(x, y) est, with s ∈ ℝ.