\(QS10_{21}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4432\) | \(111110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = 3 \, x \, (-1+x-y/2) \\ \dot{y} = y \, (1+x/2-y) \end{cases}\]
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 names \(PP10a\) and \(PP10b\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(Chap 3 8\) in {B. Imane and B. Souad}, Global phase portraits of quadratic differential systems exhibiting an invariant algebraic curve or an algebraic cubic first integral, {Ph.D. Universite Mohamed el Bachir}, (2020).
- With name \(23\) 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 \(4.9a\) 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 \(4,9a\) and \(3,1(c1)\) 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 \(PP09\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant parabola, Electron. J. Qual. Theory Differ. Equ. { bf 2025}, Paper No. 66, 54 pp.; MR5018064
- With name \(PP05\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(Fig. 24 g\) 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 \(Fig. 1 e\) 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 \(4L33\) in {M. C. Mota, R. D. S. Oliveira and A. M. Travaglini}, The interplay among the topological bifurcation diagram, integrability and geometry for the family { bf QSH(D)}, Geom. Dedicata { bf 217} (2023), no.~6, Paper No. 95, 42 pp.; MR4631488
- With name \(Ric. 5\) 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 \(U^1_{D60}\) 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 \(Fig 21 p1K\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With names \(Fig 2A II\), \(Fig 2A IV\), \(Fig 2B III\), \(Fig 2B XII\), \(Fig 2D III\) and \(Fig 2D VI\) 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 \(QS10_{14}^{(0)}\) and \(QS10_{16}^{(0)}\).