\(QS207_{1}^{(7)}\)

Description

Topological configuration of singularities: \(a,[ | , ∅ ];[inf, |, 3]\)

Phase Portrait

Topological Invariants

TCSP
\(207\)

Example

The quadratic differential system

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

with parameters: \(c = 1\)

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

The phase portrait appears in the following papers

With name \(19\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point, Rend. Circ. Mat. Palermo (2) { bf 70} (2021), no.~2, 923--945; MR4286006
With name \(Fig 5 c\) in {Y. Bolaños, J. Llibre and C. Valls}, Phase portraits of quadratic Lotka-Volterra systems with a Darboux invariant in the Poincaré disc, Commun. Contemp. Math. { bf 16} (2014), no.~6, 1350041, 23 pp.; MR3277950
With name \(D3\) in {C. A. Buzzi and D. J. Tonon}, Quadratic planar systems with two parallel invariant straight lines, Qual. Theory Dyn. Syst. { bf 7} (2009), no.~2, 295--316; MR2486677
With name \(P3\) in {D. Schlomiuk and X. Zhang}, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, emph{J. Differential Equations}, { bf 265}, no. 8 (2018), 3650--3684.
With name \(LV_d.13\) 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 \(C2\) in {A. Gasull and R. Prohens}, Quadratic and cubic systems with degenerate infinity, J. Math. Anal. Appl. { bf 198} (1996), no.~1, 25--34; MR1373524
With name \(Ric. D25\) 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 \(QS207_{1}^{(7)}\) 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 \(P57\) in {J. C. Artés, A. C. Rezende and R. D. S. Oliveira}, The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C), emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{25}, no. 3 (2015), 1530009, 111 pp.
With name \(Fig4.5 no name\) in {J. W. Reyn}, Phase portraits of a quadratic system of differential equations occurring frequently in applications, emph{Nieuw Arch. Wisk. (4)}, textbf{5}, no. 2 (1987), 107--151.

Neighbours of Codimension 8

Through the border \(QS208_{1}^{(8)}\), by means of a bifurcation of type \(deg\), we reach the neighbor \(QS207_{1}^{(7)}\).