$QS101_{1}^{(3)}$

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$101$	$22$	$11$

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 $1.9L1$ in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}
With name $2.3L2$ 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 names $bs02 Fig. 2,30$ and $bs15 Fig. 2,30$ in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
With names $FIg18 2$, $FIg18 4$, $FIg18 13$ and $Fig20 30$ in {P. de Jager}, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, emph{J. Differential Equations}, textbf{87} (1990), 169--204.Note (for name $FIg18 13$): The system has 1 limit cycle.
With names $bs02 Fig. 4$ and $bs15 Fig. 4$ 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 names $2S2$ and $2S11$ in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic differential systems with a finite saddle-node and an infinite saddle-node $(1, 1)SN$ - $({ rm B)$}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 31} (2021), no.~9, Paper No. 2130026, 110 pp.; MR4291723Note (for name $2S11$): The system has 1 limit cycle.

This phase portrait appears in J. C. Artés and C. Trullàs ({International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to $QS101_{1}^{(3)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.