\(QS10_{13}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4422\) | \(321101\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+3 \, x^{2}+11 \, x \, y/8 \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}+3 \, x \, y+5 \, y^{2}/4+a \, (y+3 \, x^{2}+11 \, x \, y/8) \end{cases}\]
with parameters: \(e = 0.2, \quad a = -0.017\)
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 \(S^2_{10,13}\) 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 \(QS11_{15}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
- Through the border \(QS11_{16}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{4}^{(0)}\).
- Through the border \(QS38_{26}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
- Through the border \(QS38_{27}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{2}^{(0)}\).
- Through the border \(QS38_{28}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS10_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{7}^{(0)}\).
- Through the border \(QS10_{20}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{14}^{(0)}\).
Comments
This is a phase portrait of which I am very fond. This was the last phase portrait of codimension 0 that was found. Moreover, it lives in a very small region of the parameter space, and some of its neighbors have also been quite complicated to deal with. Item more, it is the only non-trivial stable phase portrait which has not yet been proved to be possible to bifurcate from a system with a center.