\(QS1_{2}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(1\) | \(4441\) | \(311211\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x \, (-1+x+8 \, y/3) \\ \dot{y} = 3 \, y \, (1-9 \, x/4-y)+e \end{cases}\]
with parameters: \(e = -0.1\)
has the following phase portrait done with P4. If you want, you may download the P4 file here.
The phase portrait appears in the following papers
- With name \(U^1_{D16}\) 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 \(V13\) 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 \(Fig. 1 12\), \(Fig. 1 15\), \(Fig. 1 17\), \(Fig. 1 18\), \(Fig. 7 3\), \(Fig. 7 4\), \(Fig. 7 6\), \(Fig. 8 3\), \(Fig. 8 5\), \(Fig. 9 3\), \(Fig. 9 5\) and \(Fig. 9 6\) 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 (non-convex case), Qual. Theory Dyn. Syst. { bf 7} (2009), no.~2, 417--433; MR2486684Note (for name \(Fig. 7 4\)): The system has 1 limit cycle.Note (for name \(Fig. 9 5\)): The system has 1 limit cycle.
- With name \(c^b_1\) in {A. Zegeling}, Quadratic systems with three saddles and one antisaddle, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report 80 (1989).
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS1_{2}^{(0)}\) and \(QS1_{4}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS1_{2}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.