$QS74_{10}^{(1)}$

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$74$	$411$	$311111$

Example

The quadratic differential system

\[\begin{cases} \dot{x} = y+a \, x^{2}+b \, x \, y \\ \dot{y} = d+e \, y+x^{2}+2 \, c \, x \, y+(c^{2}-1) \, y^{2} \end{cases}\]

with parameters: $c = 10, \quad e = 0.01, \quad d = -0.00000000001, \quad a = 8, \quad b = a*(c+1)$

has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.

The phase portrait appears in the following papers

With names $V24$, $V25$ and $V27$ in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).Note (for name $V25$): The system has limit cycles with distribution $(1,0)$.Note (for name $V27$): The system has limit cycles with distribution $(0,1)$.
With name $U^1_{C27}$ 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 $an37 Fig 2.43$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).

Missed 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?).

Bifurcations in codimension 0

Bifurcation interface between the codimension 0 portrait $QS10_{8}^{(0)}$ and $QS5_{5}^{(0)}$.

Comments

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 $QS74_{10}^{(1)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.