\(QS31_{3}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(31\) | \(541\) | \(110110\) |
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: \(h = 6, \quad c = \frac{-1}{100}, \quad n = 10, \quad e = 0.001\)
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 \(V59\) in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}
- With name \(7\) 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 \(U^1_{A58}\) 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.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS19_{3}^{(0)}\) and \(QS5_{2}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés and C. Trullàs ({International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS31_{3}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.