\(QS1_{4}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(1\) | \(4441\) | \(321111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+x \, y \\ \dot{y} = x^{2}-y^{2}+d+e \, y \end{cases}\]
with parameters: \(e = 0.1, \quad d = -0.000001\)
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 \(U^1_{D18}\) 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 \(V12\) 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 \(Fig. 1 03\), \(Fig. 1 05\), \(Fig. 1 06\), \(Fig. 1 08\), \(Fig. 4 3\), \(Fig. 4 5\), \(Fig. 4 6\), \(Fig. 5 3\), \(Fig. 5 5\), \(Fig. 6 3\), \(Fig. 6 4\) and \(Fig. 6 6\) 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 \(Fig. 1 05\)): The system has 1 limit cycle.Note (for name \(Fig. 1 08\)): The system has 1 limit cycle.Note (for name \(Fig. 4 5\)): The system has 1 limit cycle.
- With name \(c^b_2\) 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 \(QS1_{3}^{(0)}\) 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 \(QS1_{4}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.