Phase Portraits of the Family IV of the Quadratic Polynomial Differential Systems

Abstract In the $$\mathbb {R}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> plane, the simplest non-linear differential systems are the quadratic polynomial differential systems. This type of differential systems have been studied intensively because of its non-linearity and wide range of applications. These systems have been classified into ten classes. In this article, we characterize in the Poincarè disk all topologically different phase portraits for one of these classes. Concretely we provide the complete study of the geometry of family IV. The family IV is six-dimensional and is reduced to several subfamilies which are three-dimensional. We give the bifurcation diagram of each specific normal form using invariant polynomials. The split of all quadratic systems in ten subfamilies could have been a good way to find all possible phase portraits of quadratic systems, if all ten families could have been studied completely. All families are six-dimensional and some of them can even be divided in several subfamilies with fewer parameters. However, certain families (I, II, and III) are too generic and cannot be reduced enough to allow its complete study. Regardless, this division of quadratic systems has been useful to study several families of quadratic systems having some other properties, and it is worth trying to get as many of them as possible studied in a complete way.