$QS76_{3}^{(1)}$

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$76$	$442$	$312110$

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: $a = -2, \quad e = 1, \quad d = -2, \quad m = 2, \quad g = -1$

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

With name $V45$ 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 $U^1_{C7}$ 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 name $cp01 Fig 2.65$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With name $cp01 Fig. 36$ 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?).

Bifurcation interface between the codimension 0 portrait $QS1_{2}^{(0)}$ and $QS10_{5}^{(0)}$.

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 $QS76_{3}^{(1)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.