$QS196_{1}^{(8)}$

Description

Topological configuration of singularities: $∅,[ |2 , ∅ ];N,[ |2, ∅ ]$

Phase Portrait

Topological Invariants

TCSP
$196$

The quadratic differential system

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

with parameters: $g = 1$

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

With name $LV_d.12$ in {D. Schlomiuk and N. Vulpe}, Global classification of the planar Lotka--Volterra differential systems according to their configurations of invariant straight lines, emph{J. Fixed Point Theory Appl.}, { bf 8}, no. 1 (2010), 177--245.
With name $Fig 1 B1-P3$ in {J. C. Artés, L. Cairó and J. Llibre}, New Exploration of Phase portraits Classification of QuadraticPolynomial Differential Systems based on Invariant Theory. Applied Math. No. 1(0), (2025), 24pp.
With name $Ric. D18$ in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Global analysis of Riccati quadratic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 34} (2024), no.~1, Paper No. 2450004, 46 pp.; MR4701478
With name $QS196_{1}^{(8)}$ in {J. C. Artés and N. Vulpe}, The codimension of the phase portraits for degenerate quadratic differential systems, Bul. Acad. c Stiin c te Repub. Mold. Mat. { bf 2024}, no.~3(106), 29--53; MR4967334
With name $2.5.8L1$ in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic differential systems with a finite saddle-node and an infinite saddle-node $(1,1)SN$-$( roman{A)$}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 31} (2021), no.~2, Paper No. 2150026, 24 pp.; MR4221748

Through the border $QS206_{1}^{(9)}$, by means of a bifurcation of type $deg$, we reach the neighbor $QS196_{1}^{(8)}$.

Bifurcation interface between the portrait $QS137_{1}^{(3)}$ and $QS194_{1}^{(7)}$.
Bifurcation interface between the portrait $QS195_{1}^{(7)}$ and $QS195_{1}^{(7)}$.
Bifurcation interface between the portrait $QS202_{1}^{(7)}$ and $QS202_{1}^{(7)}$.

As we explain, it can bifurcate in several portraits of codimension 3, but by perturbation of invariant K_2, it can also perturbate into the codimension 3 $QS137_{1}^{(3)}$ but in its geometric version with configuration ∅,[ ||^c , ∅ ];N^*,[ ||^c,∅ ]