Quadratic Systems Possessing an Infinite Elliptic-Saddle or an Infinite Nilpotent Saddle

Abstract

This paper presents a global study of the class [Formula: see text] 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, [Formula: see text] of phase portraits possessing three real finite singular points, [Formula: see text] of phase portraits possessing one real and two complex finite singular points, and [Formula: see text] of phase portraits possessing one real triple finite singular point. Here, we provide a comprehensive study of the geometry of these three families. Modulo the action of the affine group and time homotheties, families [Formula: see text] and [Formula: see text] are three-dimensional and family [Formula: see text] is two-dimensional. We study the respective bifurcation diagrams of their closures with respect to specific normal forms, in sub-sets of real Euclidean spaces. The bifurcation diagram of family [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) yields 1274 (resp., 89 and 14) sub-sets with 91 (resp., 27 and 12) topologically distinct phase portraits for systems in the closure [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) within the representatives of [Formula: see text] (resp., [Formula: see text] and [Formula: see text]) given by a specific normal form.