\(QS85_{3}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(85\) | \(2\) | \(110110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x+a \, y+e \, y^{2}+d \, (m \, x+g \, y+x \, y) \\ \dot{y} = m \, x+g \, y+x \, y \end{cases}\]
with parameters: \(a = -2, \quad e = -1, \quad d = -3, \quad m = 0, \quad g = 3\)
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 \(1S1\) in {J. C. Artés, Hebai Chen, L. M. Ferrer and Man Jia}, Quadratic vector fields in class $I$, Dyn. Syst. { bf 40} (2025), no.~2, 191--222; MR4906437
- With name \(A1S4\) in {J. C. Artés, L. Cairó and J. Llibre}, Phase portraits of the family IV of the quadratic polynomial differential systems, Qual. Theory Dyn. Syst. { bf 24} (2025), no.~2, Paper No. 66, 34 pp.; MR4860323
- With name \(47\) in {A. Ferragut and C. Valls}, Phase portraits of Abel quadratic differential systems of the second kind, Dyn. Syst. { bf 33} (2018), no.~4, 581--601; MR3869849
- With name \(74\) in {J. Llibre and X. Zhang}, Topological phase portraits of planar semi-linear quadratic vector fields, Houston J. Math. { bf 27} (2001), no.~2, 247--296; MR1874098
- With name \(V17\) 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 names \(Fig1b (1)\), \(Fig2b (IV)\) and \(Fig2b (V)\) in {J. W. Reyn}, Phase portraits of quadratic systems with finite multiplicity one, Nonlinear Anal. { bf 28} (1997), no.~4, 755--778; MR1420390
- With name \(U^1_{C17}\) 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 \(hn06 Fig 2.49\), \(in07 Fig 2.50\) and \(hn10 Fig 2.56\) in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
- With names \(E31=e1\) and \(e5\) in {B. Coll, A. Gasull and J. Llibre}, Quadratic systems with a unique finite rest point, emph{Publ. Mat.}, textbf{32} (1988), 199--259.Note (for name \(e5\)): The system has 1 limit cycle.
- With names \(hn06 Fig. 20\), \(in07 Fig. 21\) and \(hn10 Fig. 27\) 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 \(1S1\) in {J. C. Artés, A. C. Rezende and R. Oliveira}, Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node, emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, { bf 23}, no. 8 (2013), 1350140, 21 pp.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS25_{1}^{(0)}\) and \(QS19_{3}^{(0)}\).
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 \(QS85_{3}^{(1)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.