\(QS10_{15}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4422\) | \(211211\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x \, (1-x-4 \, y/3)-e \, y \, (1-x/2-y) \\ \dot{y} = -y \, (1-x/2-y) \end{cases}\]
with parameters: \(e = 1\)
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 names \(PP26a\), \(PP26b\), \(PP26c\) and \(PP26d\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(PP13\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(S8\) 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 \(Ric. 15\) 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 \(Fig 5 P1B\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(S^2_{10,15}\) in {J. C. Artés, R. E. Kooij and J. Llibre}, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. { bf 134} (1998), no.~639, viii+108 pp.; MR1432139
- With name \(V13\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.
Neighbours of Codimension 1
- Through the border \(QS11_{19}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{3}^{(0)}\).
- Through the border \(QS76_{9}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS1_{2}^{(0)}\).
- Through the border \(QS38_{32}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS10_{19}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{9}^{(0)}\).
- Through the border \(QS10_{5}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS10_{16}^{(0)}\).
- Through the border \(QS74_{11}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{6}^{(0)}\).
Comments
Even though this phase portrait was conjectured to have no limit cycles in resenhas, see next comment:
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS10_{15}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.