\(QS76_{4}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(76\) | \(441\) | \(321021\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y \, (1+x)+a \, x^{2} \\ \dot{y} = d+e \, y+x^{2}-2 \, x \, y \end{cases}\]
with parameters: \(a = 0.1, \quad e = -0.1, \quad d = -0.000001\)
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 \(V48\) in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).
- With name \(U^1_{C8}\) 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 \(cp10 Fig 2.65\) in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
- With name \(cp10 Fig. 36\) in {J. W. Reyn and X. H. Huang}, Separatrix configurations of quadratic systems with finite multiplicity three and a $M^0_{1,1$ type of critical point at infinity}, Report U. Delft (1997?).
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS1_{3}^{(0)}\) and \(QS10_{6}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés and L. Cairó ({Preprint} (2026)) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS76_{4}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.