$QS206_{1}^{(9)}$

Description

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

Phase Portrait

Topological Invariants

TCSP
$206$

The quadratic differential system

\[\begin{cases} \dot{x} = 0 \\ \dot{y} = -x^{2} \end{cases}\]

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

With name $16$ in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $ R^{12$}, emph{Rend. Circ. Mat. Palermo}, textbf{56}, no. 3 (2007), 417--444.
With name $PL3$ in {L. Cairó and J. Llibre}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2. Nonlinear Anal. 67 (2007), no. 2, 327–348.
With name $QS206_{1}^{(9)}$ 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 $16$ in {J. C. Artés and J. Llibre}, Quadratic Hamiltonian vector fields, emph{J. Differential Equations}, { bf 107} (1994), 80--95.
With name $P4$ 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

Bifurcation interface between the portrait $QS12_{1}^{(0)}$ and $QS205_{1}^{(8)}$.
Bifurcation interface between the portrait $QS196_{1}^{(8)}$ and $QS196_{1}^{(8)}$.

This phase portrait has the biggest codimension. However, it does not border every other region in the parameter space. All the rellevant comitants are zero except C_2. This means that systems with the infinite line fulfilled with singularities cannot have it as border. So, it cannot bifurcate into $QS208_{1}^{(8)}$. They occupy non adjacent parts in the parameter space. As we explain, it can bifurcate in $QS196_{1}^{(8)}$ and $QS205_{1}^{(8)}$ by perturbation of invariant L_3, and it can also perturbate into the codimension 0 $QS12_{1}^{(0)}$ but in its geometric version with configuration ∅,[ ||^c , ∅ ];∅,[ ||^c, N^* ]