\(QS74_{4}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(74\) | \(431\) | \(310110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+3 \, x \, (x-y)/2 \\ \dot{y} = d+e \, y+x^{2}-y^{2} \end{cases}\]
with parameters: \(d = -0.000001, \quad e = 0.1\)
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 \(V16\) 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 \(U^1_{C21}\) 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 names \(bn09 Fig 2.45\), \(fn07 Fig 2.47\), \(an40 Fig 2.51\) and \(fn12 Fig 2.54\) in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
- With names \(an17 Fig. 15\), \(bn09 Fig. 16\), \(fn07 Fig. 18\), \(an24 Fig. 22\) and \(fn12 Fig. 25\) 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?).
- With name \(1S4\) 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.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS10_{3}^{(0)}\) and \(QS5_{2}^{(0)}\).
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 \(QS74_{4}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.