\(QS11_{20}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(11\) | \(4433\) | \(2121\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -e+x^{2}+2 \, h \, x \, y+(n-1-2 \, h) \, y^{2} \\ \dot{y} = -e \, c+y+c \, x^{2}+(2+2 \, h-2 \, c-n) \, x \, y+(c-2-2 \, h+2 \, n) \, y^{2} \end{cases}\]
with parameters: \(h = \frac{11}{5}, \quad c = \frac{9}{10}, \quad n = 10, \quad e = 0.00001\)
has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.
The phase portrait appears in the following papers
- With names \(PP44\) and \(PP46\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(2\) in {A. Belfar and R. Benterki}, Qualitative dynamics of five quadratic polynomial differential systems exhibiting five classical cubic algebraic curves, Rend. Circ. Mat. Palermo (2) { bf 72} (2023), no.~1, 393--420; MR4543844
- With name \(4.11b\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four, emph{Bul. Acad. c{S}tiin c{t}e Repub. Mold. Mat.}, { bf 1 (56)} (2008), 27--83.
- With names \(4,11b\) and \(3,9(c3)\) 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 \(32\) 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 \(PP16\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(Fig. 54 c\) in {J. Llibre and C. Pantazi}, Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 1, Paper No. 2350003, 54 pp.
- With name \(Fig. 1 c\) in {J. Llibre, W. F. Pereira and C. G. Pessoa}, Phase portraits of Bernoulli quadratic polynomial differential systems, Electron. J. Differential Equations { bf 2020}, Paper No. 48, 19 pp.; MR4102990
- With name \(Ric. 37\) 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 \(U^1_{B28}\) 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 names \(Fig 27 P2O\) and \(Fig 29 P1P\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With names \(Fig 2B X-XI\) and \(Fig 2D VII-VIII\) 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.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS8_{4}^{(0)}\) and \(QS10_{16}^{(0)}\).