\(QS1_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(1\) | \(4443\) | \(212121\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x \, (4-4 \, x-5 \, y)+k \, e \, y \, (1-y) \\ \dot{y} = -y \, (1-6 \, x/5-y)+e \, x \, (1-x) \end{cases}\]
with parameters: \(k = 1, \quad e = 0.2\)
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 \(4.3b\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four, emph{Bul. Acad. c{S}tiin c{t}e Repub. Mold. Mat.}, { bf 1 (56)} (2008), 27--83.Note (for name \(4.3b\)): missed arrow
- With names \(4,3b\) and \(3.1(a3)\) in {D. Schlomiuk and N. Vulpe}, Global classification of the planar Lotka--Volterra differential systems according to their configurations of invariant straight lines, emph{J. Fixed Point Theory Appl.}, { bf 8}, no. 1 (2010), 177--245.
- With name \(8\) in {A. Ferragut, J. D. García-Saldaña and C. Valls}, Phase portraits of Abel quadratic differential systems of second kind with symmetries, Dyn. Syst. { bf 34} (2019), no.~2, 301--333; MR3941199
- With name \(S^2_{7,1}\) 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 names \(Fig 2B IX\), \(Fig 2C V\) and \(Fig 2D IX\) in {J. W. Reyn}, Phase portraits of a quadratic system of differential equations occurring frequently in applications, emph{Nieuw Arch. Wisk. (4)}, textbf{5}, no. 2 (1987), 107--151.
- With name \(d\) in {A. Zegeling}, Quadratic systems with three saddles and one antisaddle, Delft University of Technology, Faculty of Technical Mathematics and Informatics, Report 80 (1989).
Neighbours of Codimension 1
- Through the border \(QS28_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS15_{1}^{(0)}\).
- Through the border \(QS1_{1}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS1_{2}^{(0)}\).
- Through the border \(QS76_{10}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{16}^{(0)}\).
Comments
This is the first (and for the moment unique) phase portrait with no invariant straight lines which can have a finite focus and which has been proved that cannot have limit cycles (see Art-Llib-Med:2007). The fact that a phase portrait with such a particular property has received the first alphabetical name in this classification, is completely fortuite.