\(QS10_{7}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4431\) | \(411101\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+3 \, x^{2}+11 \, x \, y/8 \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+3 \, x \, y+5 \, y^{2}/4 \end{cases}\]
with parameters: \(e = 0.2\)
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 \(CLS\) in {M. Alberich-Carramiñana, A. Ferragut and J. Llibre}, Quadratic planar differential systems with algebraic limit cycles via quadratic plane, Cremona maps, Adv. Math. { bf 389} (2021), Paper No. 107924, 38 pp.; MR4290137Note (for name \(CLS\)): The system has limit cycles with distribution \((0,1)\).
- With name \(Fig 5.202 S^2_{10,07}\) 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 \(Fig 5.202 S^2_{10,07}\)): The system has limit cycles with distribution \((0,1)\).
- With name \(S^2_{10,07}\) 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 \(QS11_{8}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS11_{9}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS38_{17}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
- Through the border \(QS38_{15}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS38_{16}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS10_{7}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{1}^{(0)}\).
- Through the border \(QS10_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{13}^{(0)}\).
- Through the border \(QS74_{14}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{7}^{(0)}\).