\(QS1_{2}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(1\) | \(4442\) | \(312211\) |
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 \(S^2_{7,2}\) 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 \(V12\) 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}.
- With name \(e^d\) 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 \(QS28_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS15_{1}^{(0)}\).
- Through the border \(QS28_{3}^{(1)}\), 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 \(QS1_{1}^{(0)}\).
- Through the border \(QS1_{3}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS1_{3}^{(0)}\).
- Through the border \(QS1_{2}^{(1)}\), by means of a bifurcation of type \(D(a)\), we reach the neighbor \(QS1_{4}^{(0)}\).
- Through the border \(QS76_{3}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{5}^{(0)}\).
- Through the border \(QS76_{8}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{14}^{(0)}\).
- Through the border \(QS76_{9}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{15}^{(0)}\).
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 \(QS1_{2}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.