Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

Abstract

A quadratic polynomial differential system can be identified with a single point of R 12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in R 12 having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.

Global Phase Portraits of the Quadratic Systems Having a Singular and Irreducible Invariant Curve of Degree 3

“…For a good survey see the book of Reyn [Reyn, 1994] or the book of Artés et.al [Artés et al, 2021], and references therein. For example, the following families of quadratic systems have been studied: homogeneous [Coll et al, 1987], semi-homogeneous [Cairó & Llibre, 1997], bounded [Dickson et al, 1970], reversible [Gavrilov & Iliev, 2000;Coll et al, 2009], Hamiltonian [Artés & Llibre, 1994a;Chow et al, 2002], Lienard [Dumortier & Li, 1997], integrable using Carleman and Painlevé tools [Hua et al, 1996], rational integrable [Artés et al, 2007[Artés et al, , 2010[Artés et al, , 2009, the ones having a star nodal point [Berlinski, 1966], a center [Vulpe, 1983;Lunkevich & Sibirskii, 1982;Coll et al, 2009;Vulpe, 1983], one focus and one antisaddle [Artés & Llibre, 1994b], with a semi-elementary triple node [Artés et al, 2013], chordal [Gasull et al, 1986;Gasull & Llibre, 1988], with four infinite singular points and one invariant straight line [Roset, 1991], with invariant lines [Schlomiuk & Vulpe, 2008a], and so on.…”