The Full Study of Planar Quadratic Differential Systems Possessing a Line of Singularities at Infinity

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

“…Due to ii) and iii) the system is Darboux integrable and it has a rational quadratic first integral of degree two. Due to the properties ii) and iii) we can apply to this system the results on the moduli spaces constructed in [59] and obtain the following result.…”

“…For a number of families of polynomial vector fields this problem was completely solved by using polynomial invariants (see for example [56], [57], [59], [62]). For these as well as for other families, the polynomial invariants were sufficient to obtain the topological classification.…”

“…the system dx/dt = 1+y +x 2 , dy/dt = xy with phase portrait Picture C 2 .2(b) below which was constructed in [59, p. 755]. We give here the full picture in [59] which shows that the systems with the configuration of invariant lines Config. C 2 .2 (having one real and two (dotted) complex invariant lines) correspond to two topologically distinct phase portraits, one of them being Picture C 2 .2(b), the phase portrait of S and of S ′ .…”

“…Much work on geometric classifications of families of polynomial vector fields followed, some done by the author in collaboration with Vulpe ([54], [56], [57], [58], [59], [61]), other work done by Vulpe together with Artés and Llibre ([7], [9]) or by the author together with Artés, Llibre, and Vulpe ( [6]). …”

“…Our aim was also to write in a self-contained manner, as much as possible, so as to make this article accessible to anyone interested in these problems, to encourage new work to be done by people not necessarily belonging to the Sibirsky school, or to the author's collaborators. With the exception of the proof of the theorem in the section on bifurcations, where we had to rely on results in [59] in order to show how invariant polynomials work, we think this aim was attained.…”