\(QS5_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(5\) | \(4211\) | \(411011\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+2 \, x^{2}+3 \, x \, y \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+5 \, x \, y/2+9 \, y^{2}/16 \end{cases}\]
with parameters: \(e = -0.3\)
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 \(Fig 5.138 S^2_{12,1}\) 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 \(Fig 5.138 S^2_{12,1}\)): The system has limit cycles with distribution \((0,1,0)\).
- With name \(S^2_{12,1}\) 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 \(QS7_{1}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS74_{2}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{2}^{(0)}\).
- Through the border \(QS74_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{3}^{(0)}\).
- Through the border \(QS31_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{1}^{(0)}\).
- Through the border \(QS31_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{1}^{(0)}\).
- Through the border \(QS5_{8}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{5}^{(0)}\).
Comments
No limit cycle is known in this portrait. There is the challenge to see if the antisaddle which receives just one infinite separatrix may have a limit cycle.