Family of quadratic differential systems with invariant parabolas: a complete classification in the space ${\mathbb R}^{12}$

Abstract

Consider the class ${\bf QS}$ of all non-degenerate planar quadratic differential systems and its subclass ${\bf QSP}$ of all its systems possessing an invariant parabola. This is an interesting family because on one side it is defined by an algebraic geometric property and on the other, it is a family where limit cycles occur. Note that each quadratic differential system can be identified with a point of ${\mathbb R}^{12}$ through its coefficients. In this paper, we provide necessary and sufficient conditions for a system in ${\bf QS}$ to have at least one invariant parabola. We give the global “bifurcation” diagram of the family ${\bf QS}$ which indicates where a parabola is present or absent and in case it is present, the diagram indicates how many parabolas there could be, their reciprocal position and what kind of singular points at infinity (simple or multiple) as well as their multiplicities are the points at infinity of the parabolas. The diagram is expressed in terms of affine invariant polynomials and it is done in the 12-dimensional space of parameters.

Keywords

• quadratic vector fields
• affine invariant polynomials
• invariant algebraic curve
• invariant parabola