\(QS208_{1}^{(8)}\)
Description
Phase Portrait
Topological Invariants
| TCSP |
|---|
| \(208\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x^{2} \\ \dot{y} = x \, y \end{cases}\]
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 name \(B 2.5L3\) in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).
- With name \(24\) 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 \(20\) 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 \(DI5\) 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 \(P4\) 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 \(LV_d.14\) 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 \(Fig 1 C-P2\) in {J. C. Artés, L. Cairó and J. Llibre}, New Exploration of Phase portraits Classification of QuadraticPolynomial Differential Systems based on Invariant Theory. Applied Math. No. 1(0), (2025), 24pp.
- With name \(E\) 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 \(Ric. D26\) 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 \(Fig 42 p1Y\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(QS208_{1}^{(8)}\) 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 \(Fig5.3 I-II\) 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 7
- Bifurcation interface between the portrait \(QS170_{1}^{(7)}\) and \(QS170_{1}^{(7)}\).
- Bifurcation interface between the portrait \(QS207_{1}^{(7)}\) and \(QS207_{1}^{(7)}\).