$QS73_{1}^{(1)}$

Description

Topological configuration of singularities: $s,a,a;(1,1)SN$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$73$	$421$	$21$

The quadratic differential system

\[\begin{cases} \dot{x} = x+a \, y+e \, y^{2}+d \, (m \, x+g \, y+x \, y) \\ \dot{y} = m \, x+g \, y+x \, y \end{cases}\]

with parameters: $a = 0.1, \quad e = -1, \quad d = 1, \quad m = 1, \quad g = 0.3$

has the following phase portrait done with P4. If you want, you may download the P4 file here.

With name $V18$ in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).
With name $U^1_{C2}$ in {J. C. Artés, J. Llibre and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension one, Birkhäuser/Springer, Cham, 2018, vi+267 pp.
With names $bs06 Fig. 2,30$ and $bs09 Fig. 2,30$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With names $bs06 Fig. 4$ and $bs09 Fig. 4$ in {J. W. Reyn and X. H. Huang}, Separatrix configurations of quadratic systems with finite multiplicity three and a $M^0_{1,1$ type of critical point at infinity}, Report U. Delft (1997?).
With name $1S10$ in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662.

Bifurcation interface between the codimension 0 portrait $QS8_{1}^{(0)}$ and $QS3_{1}^{(0)}$.

This phase portrait appears in J. C. Artés and L. Cairó ({Preprint} (2026)) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to $QS73_{1}^{(1)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.