\(QS203_{1}^{(7)}\)
Description
Phase Portrait
Topological Invariants
| TCSP |
|---|
| \(203\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = a + c \, x \\ \dot{y} = b - x^{2} \end{cases}\]
with parameters: \(a = 1, \quad c = 1, \quad b = 1\)
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 \(Fig 2 C-P1\) 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 \(b\) in {J. Llibre and M. F. da Silva}, Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line, C. R. Math. Acad. Sci. Paris { bf 357} (2019), no.~2, 143--166; MR3927021
- With name \(a\) in {J. Llibre and T. Salhi}, Planar quadratic differential systems with invariants of the form $ax^2+bxy+cy^2+d x+ey+c_1t$, Bull. Iranian Math. Soc. { bf 50} (2024), no.~4, Paper No. 49, 16 pp.; MR4755552Note (for name \(a\)): degenerate straight line missed
- With name \(83\) in {J. Llibre and X. Zhang}, Topological phase portraits of planar semi-linear quadratic vector fields, Houston J. Math. { bf 27} (2001), no.~2, 247--296; MR1874098
- With name \(Ric. D22\) 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 \(QS203_{1}^{(7)}\) 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 \(8.9.9L1\) in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic differential systems with a finite saddle-node and an infinite saddle-node $(1,1)SN$-$( roman{A)$}, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 31} (2021), no.~2, Paper No. 2150026, 24 pp.; MR4221748
Comments
The red/blue curve drawn by P4 is not a separatrix but a borsec of the elliptic singularity. The parameter a in this system affects only the position of the symmetry. So, the only possible border is when b=0, but then we jump directly to \(QS206_{1}^{(9)}\).