\(QS10_{12}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4421\) | \(321201\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+x^{2}+4 \, x \, y \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}-4 \, x \, y+3 \, y^{2} \end{cases}\]
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 \(Fig 5.209 S^2_{10,12}\) 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 \(Fig 5.209 S^2_{10,12}\)): The system has limit cycles with distribution \((1,0)\).
- With name \(S^2_{10,12}\) 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
- With name \(V2\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.
Neighbours of Codimension 1
- Through the border \(QS11_{14}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{3}^{(0)}\).
- Through the border \(QS76_{7}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS1_{4}^{(0)}\).
- Through the border \(QS38_{24}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS38_{25}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{3}^{(0)}\).
- Through the border \(QS10_{15}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{6}^{(0)}\).
- Through the border \(QS10_{18}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{9}^{(0)}\).
- Through the border \(QS10_{6}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{14}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS10_{12}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.