\(QS10_{10}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4441\) | \(311110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+x^{2} \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}-y^{2} \end{cases}\]
with parameters: \(e = 0.5\)
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 names \(Fig. 32\) and \(Fig. 33\) in {J. Llibre and C. Pantazi}, Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 1, Paper No. 2350003, 54 pp.
- With name \(Fig 5.204 S^2_{10,10}\) 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 \(Fig 5.204 S^2_{10,10}\)): The system has limit cycles with distribution \((0,1)\).
- With name \(S^2_{10,10}\) 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 \(V29\) 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 \(QS11_{12}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS76_{6}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS1_{3}^{(0)}\).
- Through the border \(QS38_{21}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS38_{22}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS10_{11}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{3}^{(0)}\).
- Through the border \(QS10_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS10_{14}^{(0)}\).
- Through the border \(QS74_{3}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{2}^{(0)}\).
- Through the border \(QS74_{7}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS5_{4}^{(0)}\).
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 \(QS10_{10}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.