\(QS5_{7}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(5\) | \(4311\) | \(211111\) |
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 name \(Fig 5.160 S^2_{12,7}\) 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 \(Fig 5.160 S^2_{12,7}\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(S^2_{12,7}\) 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 \(QS7_{8}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS74_{15}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{4}^{(0)}\).
- Through the border \(QS74_{14}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{7}^{(0)}\).
- Through the border \(QS31_{14}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS31_{15}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS5_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{4}^{(0)}\).
- Through the border \(QS5_{3}^{(1)}\), by means of a bifurcation of type \(D(b)\), we reach the neighbor \(QS5_{5}^{(0)}\).
- Through the border \(QS5_{7}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{6}^{(0)}\).
Comments
It is known that this phase portrait admits a limit cycle on the finite antisaddle that receives just one separatrix from infinity, but there is the challenge to see if the other antidaddle which receives just one finite separatrix is possible to have a limit cycle.