Phase Portraits of Quadratic Differential Systems with a Weak Focus and an 11SN

As part of the project to obtain all topologically distinct phase portraits of the quadratic differential systems, we present here another family of codimension-2, Qwf1SN11, which corresponds to cases with a weak focus of order 1 and an infinite saddle-node formed by the coalescence of a finite singularity with an infinite singularity. As usual in these cases, the normal form that contains these systems also contains other more degenerate ones: we therefore study the complete closure [Formula: see text]. The bifurcation diagram for this class, constructed in the appropriate parameter space, which is the 3-dimensional real projective space, is quite rich in its complexity and gives 304 subsets with 165 phase portraits for [Formula: see text], 119 for Qwf1SN11, 32 having limit cycles and 13 with centers. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of finite or infinite singularity bifurcations and by a nonalgebraic set of points corresponding to global separators that have connections or double limit cycles. Algebraic invariants are needed to construct the algebraic part of the bifurcation set; symbolic calculations are needed to process certain fairly complex invariants; and numerical calculations are needed to determine the position of the nonalgebraic bifurcation set of connections.