Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node

Planar quadratic differential systems occur in many areas of applied mathematics. Although more than 1000 papers were written on these systems, a complete understanding of this class is still missing. Classical problems, and in particular, Hilbert’s 16th problem, are still open for this class. One of the goals of recent researchers is the topological classification of quadratic systems. As this attempt is not possible for the whole class due to the large number of parameters, which generally is 12 — although can be reduced to five after affine transformations and time rescaling, many subclasses are considered and studied. Specific characteristics are taken into account and this implies a decrease in the number of parameters, which makes possible the study. In this paper, we perform a global study (modulo islands) of the class [Formula: see text], which is the closure, within quadratic differential systems, of the family [Formula: see text] of all such systems that have a weak focus of first order and a finite saddle-node. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space [Formula: see text], is quite rich in its complexity since it yields 399 subsets with 192 topologically distinct phase portraits for [Formula: see text], 146 of which have a representative in [Formula: see text]. It can be shown that some of these parts have at least two limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by a set of bifurcations which we suspect to be analytic corresponding to global separatrices that have connections, or double limit cycles. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.