$QS19_{1}^{(1)}$

Description

Topological configuration of singularities: $a,a;S,S,N$

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
$19$	$11$	$110110$

The quadratic differential system

\[\begin{cases} \dot{x} = 1+x^{2}-4 \, x \, y-y^{2} \\ \dot{y} = 2 \, x \, y \end{cases}\]

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

With name $28$ in {A. Ferragut, J. D. García-Saldaña and C. Valls}, Phase portraits of Abel quadratic differential systems of second kind with symmetries, Dyn. Syst. { bf 34} (2019), no.~2, 301--333; MR3941199
With name $P12$ in {J. Llibre and C. Valls}, Global phase portraits for the Abel quadratic polynomial differential equations of second kind with $Z_2$-symmetries, Canad. Math. Bull. { bf 61} (2018), no.~1, 149--165; MR3746481
With name $U^1_{D27}$ 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 $Fig 1.33 a$ in {J. W. Reyn and R. E. Kooij}, Phase portraits of non-degenerate quadratic systems with finite multiplicity two, Differential Equations Dynam. Systems { bf 5} (1997), no.~3-4, 355--414; MR1660222
With name $V21$ 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 $QS19_{2}^{(0)}$ and $QS19_{3}^{(0)}$.

This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to $QS19_{1}^{(1)}$ could potentially exhibit an additional limit cycle bifurcating from the focus.