Phase portraits for quadratic systems possessing an infinite elliptic-saddle or an infinite nilpotent saddle

This paper presents a global study of the class \(\bf{Q}{\widehat{ES}}\) of all real quadratic polynomial differential systems possessing exactly one elemental infinite singular point and one triple infinite singular point, which is either an infinite nilpotent elliptic-saddle or a nilpotent saddle. This class can be divided into three different families, namely, \(\bf{Q}{{\widehat{ES}(A)}}\) of phase portraits possessing three real finite singular points, \(\bf{Q}{{\widehat{ES}(B)}}\) of phase portraits possessing one real and two complex finite singular points, and \(\bf{Q}{{\widehat{ES}(C)}}\) of phase portraits possessing one real triple finite singular point. Here we provide the complete study of the geometry of these three families. Modulo the action of the affine group and time homotheties, families \(\bf{Q}{{\widehat{ES}(A)}}\) and \(\bf{Q}{{\widehat{ES}(B)}}\) are three-dimensional and family \(\bf{Q}{{\widehat{ES}(C)}}\) is two-dimensional. We study the respective bifurcation diagrams of their closures with respect to specific normal forms, in subsets of real Euclidean spaces. The bifurcation diagram of family \(\bf{Q}{{\widehat{ES}(A)}}\) (respectively, \(\bf{Q}{{\widehat{ES}(B)}}\) and \(\bf{Q}{{\widehat{ES}(C)}}\)) yields 1274 (respectively, 89 and 14) subsets with 91 (respectively, 27 and 12) topologically distinct phase portraits for systems in the closure \(\overline{\bf{Q}{{\widehat{ES}(A)}}}\) (respectively, \(\overline{\bf{Q}{{\widehat{ES}(B)}}}\) and \(\overline{\bf{Q}{{\widehat{ES}(C)}}}\)) within the representatives of \(\bf{Q}{{\widehat{ES}(A)}}\) (respectively, \(\bf{Q}{{\widehat{ES}(B)}}\) and \(\bf{Q}{{\widehat{ES}(C)}}\)) given by a specific normal form.