Global configurations of singularities for quadratic differential systems with exactly two finite singularities of total multiplicity four

Abstract

In this article we obtain the geometric classification of singularities, finite and infinite, for the three subclasses of quadratic differential systems with $m_f=4$ possessing exactly two finite singularities, namely: (i) systems with two double complex singularities (18 configurations); (ii) systems with two double real singularities (33 configurations) and (iii) systems with one triple and one simple real singularities (123 configurations). We also give here the global bifurcation diagrams of configurations of singularities, both finite and infinite, with respect to the geometric equivalence relation, for these subclasses of systems. The bifurcation set of this diagram is algebraic. The bifurcation diagram is done in the 12-dimensional space of parameters and it is expressed in terms of polynomial invariants, which give an algorithm for determining the geometric configuration of singularities for any quadratic system.

Keywords

• quadratic vector fields
• infinite and finite singularities
• affine invariant polynomials
• poincaré compactification
• configuration of singularities
• geometric equivalence relation