\(QS10_{1}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(10\) | \(4421\) | \(311110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -y-m \, x^{2}-g \, x \, y-n \, y^{2} \\ \dot{y} = x \, (1+a \, y) \end{cases}\]
with parameters: \(m = 1, \quad g = sqrt(5), \quad n = \frac{1}{2}, \quad a = 1\)
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 \(U^1_{D40}\) 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.
- With name \(V3\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662.
- With names \(Fig. 2 01\), \(Fig. 2 02\), \(Fig. 2 10\) and \(Fig. 2 11\) in {P. C. Carri\~ao, M. E. S. Gomes and A. A. G. Ruas}, Planar quadratic vector fields with finite saddle connection on a straight line (convex case), Qual. Theory Dyn. Syst. { bf 6} (2005), no.~2, 187--204; MR2420856Note (for name \(Fig. 2 02\)): The system has limit cycles with distribution \((0,1)\).Note (for name \(Fig. 2 10\)): The system has limit cycles with distribution \((0,1)\).
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS10_{3}^{(0)}\) and \(QS10_{5}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS10_{1}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.