\(QS8_{5}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(8\) | \(4431\) | \(31\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y \, (1-2 \, x) \\ \dot{y} = eps^{2} \, x/5-eps \, y+x^{2}+4 \, x \, y+3 \, y^{2} \end{cases}\]
has the following phase portrait done with P4. If you want, you may download the P4 file here.
The phase portrait appears in the following papers
- With name \(A V15\) in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).
- With names \(Fig. 30\) and \(Fig. 31\) 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 \(Fig 5.102 S^2_{3,5}\) 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 \(Fig 5.102 S^2_{3,5}\)): The system has limit cycles with distribution \((0,1)\).
- With name \(S^2_{3,5}\) 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 \(V45\) 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 \(QS37_{7}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS37_{8}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS37_{9}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS8_{5}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS8_{1}^{(0)}\).
- Through the border \(QS8_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS8_{4}^{(0)}\).
- Through the border \(QS73_{2}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS11_{8}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{7}^{(0)}\).
- Through the border \(QS11_{9}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{7}^{(0)}\).
- Through the border \(QS11_{12}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{10}^{(0)}\).
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 \(QS8_{5}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.