\(QS10_{4}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4411\) | \(312211\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+a \, x^{2}+b \, x \, y \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+2 \, c \, x \, y+(c^{2}-1) \, y^{2} \end{cases}\]
with parameters: \(e = \frac{1}{1000}, \quad a = 20, \quad c = 20, \quad b = 500\)
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 \(Fig 5.205 S^2_{10,04}\) and \(Fig 5.207 S^2_{10,04}\) 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 \(Fig 5.205 S^2_{10,04}\)): The system has limit cycles with distribution \((0,1)\).Note (for name \(Fig 5.207 S^2_{10,04}\)): The system has limit cycles with distribution \((1,0)\).
- With name \(S^2_{10,04}\) 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 \(QS11_{5}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{2}^{(0)}\).
- Through the border \(QS38_{8}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
- Through the border \(QS38_{9}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS10_{22}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{2}^{(0)}\).
- Through the border \(QS10_{12}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{8}^{(0)}\).
- Through the border \(QS10_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{11}^{(0)}\).
- Through the border \(QS74_{15}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{7}^{(0)}\).