Abstract

The goal of this lecture is the study of the action of the group G of affine transformations and time homotheties on the class QW3 of all real quadratic differential systems which have a weak focus of third order. All elements of this class produce up to three limit cycles close to the weak focus in quadratic perturbations. For this reason the class is important for Hilbert's 16th problem for quadratic differential systems. As it is well known, this problem is still open. Modulo the action of the group, the class is two-dimensional. Normal forms were used to provide the bifurcation diagram of this class, given in a series of works by Andronova, by Artes and Llibre and by Llibre and Schlomiuk who realized it in the projective plane. It was observed that several phase portraits, appearing in distinct regions of the bifurcation diagram, coincide. Are there symmetries on the normal forms due to the action of the group, identifying these portraits? In this lecture we prove that there are exactly six families of such transformations which identify distinct points in the bifurcation diagram and we exhibit these transformations. This answer is used for the construction of the moduli space of QW3 under this group action.

The lecture is based on joint work with Myriam Demers.