\(QS8_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(8\) | \(4421\) | \(41\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y \, (1-2 \, x) \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+4 \, x \, y+3 \, y^{2} \end{cases}\]
with parameters: \(e = -0.8\)
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 name \(Fig 5.102 S^2_{3,1}\) 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 \(Fig 5.102 S^2_{3,1}\)): The system has limit cycles with distribution \((0,1)\).
- With name \(S^2_{3,1}\) 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 \(QS37_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS37_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS37_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS8_{2}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS8_{1}^{(0)}\).
- Through the border \(QS8_{1}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS8_{2}^{(0)}\).
- Through the border \(QS8_{6}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS73_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS11_{1}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{1}^{(0)}\).
- Through the border \(QS11_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{1}^{(0)}\).
- Through the border \(QS11_{4}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{3}^{(0)}\).
- Through the border \(QS11_{6}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{5}^{(0)}\).