\(QS38_{33}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(38\) | \(543\) | \(111111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = (1-e) \, x^{2}+2 \, h \, x \, y+(n-1-2 \, h) \, y^{2} \\ \dot{y} = y+c \, (1-e) \, x^{2}+(2+2 \, h-n) \, (1-e) \, x \, y-2 \, c \, x \, y+(c-2-2 \, h+2 \, n) \, y^{2} \end{cases}\]
with parameters: \(h = \frac{11}{5}, \quad c = \frac{9}{10}, \quad n = 10, \quad e = 0.001\)
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 name \(PP31\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(10\) in {R. Benterki and A. Belfar}, Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves, Demonstr. Math. { bf 56} (2023), no.~1, Paper No. 20220218, 16 pp.; MR4592893
- With name \(A06\) 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 \(4.10b\) 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 \(4,10b\), \(3,2(f4)\) and \(3,3(f4)\) 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 \(PP08\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(Fig. 1 g\) 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 \(2L2\) 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 \(Ric. 6\) 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 \(U^1_{A55}\) 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 \(Fig3.2 D4\) and \(Fig3.4 II\) 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 \(QS25_{1}^{(0)}\) and \(QS10_{16}^{(0)}\).