$QS86_{1}^{(3)}$

Description

Topological configuration of singularities: $a;(1,1)SN,(0,2)SN$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$86$	$1$	$1110$

The quadratic differential system

\[\begin{cases} \dot{x} = P_x(x,y) \\ \dot{y} = P_y(x,y) \end{cases}\]

has the following phase portrait done with P4.

With name $4.5L1$ 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 $in02 Fig 2.39$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With names $E51$ and $fig 5.09(b)=e11$ in {B. Coll, A. Gasull and J. Llibre}, Quadratic systems with a unique finite rest point, emph{Publ. Mat.}, textbf{32} (1988), 199--259.Note (for name $fig 5.09(b)=e11$): The system has 1 limit cycle.
With name $in02 Fig. 11$ 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 $5.9L1$ 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.

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 $QS86_{1}^{(3)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.