\(QS5_{5}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(5\) | \(4211\) | \(311111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+3 \, x^{2}+31 \, x \, y/10 \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+4 \, x \, y+3 \, y^{2} \end{cases}\]
with parameters: \(e = -0.1\)
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 names \(Fig 5.140 S^2_{12,5}\) and \(Fig 5.159 S^2_{12,5}\) 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.Note (for name \(Fig 5.140 S^2_{12,5}\)): The system has limit cycles with distribution \((0,1,0)\).Note (for name \(Fig 5.159 S^2_{12,5}\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(S^2_{12,5}\) 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
Neighbours of Codimension 1
- Through the border \(QS7_{5}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS74_{9}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{1}^{(0)}\).
- Through the border \(QS74_{10}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{8}^{(0)}\).
- Through the border \(QS31_{10}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS31_{11}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS5_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{1}^{(0)}\).
- Through the border \(QS5_{2}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{7}^{(0)}\).