\(QS89_{1}^{(2)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(89\) | \(1\) | \(1110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = P_x(x,y) \\ \dot{y} = P_y(x,y) \end{cases}\]
has the following phase portrait done with P4.
The phase portrait appears in the following papers
- With names \(U^2_{BC,23}\), \(U^2_{BC,23lc}\) and \(U^2_{BC,26}\) in Missing reference in BC1Note (for name \(U^2_{BC,23lc}\)): The system has 1 limit cycle.
- With name \(1.9L1\) 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 \(40\) in {B. Coll, A. Ferragut and J. Llibre}, Phase portraits of the quadratic systems with a polynomial inverse integrating factor, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 19} (2009), no.~3, 765--783; MR2533481
- With name \(A1.5L2\) 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 \(Fig 11 (b)\) in {M. R. A. Gouveia, J. Llibre and L. A. F. Roberto}, Phase portraits of the quadratic polynomial Liénard differential systems, Proc. Roy. Soc. Edinburgh Sect. A { bf 151} (2021), no.~1, 202--216; MR4202639
- With names \(B V5\), \(B V9\) and \(C S3\) in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic systems possessing an infinite elliptic-saddle or an infinite nilpotent saddle, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 34} (2024), no.~11, Paper No. 2430023, 43 pp.; MR4801966Note (for name \(B V9\)): The system has 1 limit cycle.
- With name \(6\) 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 \(5S25\) 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 \(Fig2g (2)\) in {J. W. Reyn}, Phase portraits of quadratic systems with finite multiplicity one, Nonlinear Anal. { bf 28} (1997), no.~4, 755--778; MR1420390
- With names \(E12\) and \(fig 5.03(b)\) 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 \(fig 5.03(b)\)): The system has 1 limit cycle.
- With names \(Fig10.2 (1)\), \(Fig10.2 (4)\), \(Fig11.1 (1)\), \(Fig11.2 (3)\), \(Fig11.2 (10)\) and \(Fig11.2 (13)\) in {J. W. Reyn and X. H. Huang}, Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity, Rocky Mountain J. Math. { bf 27} (1997), no.~3, 929--978; MR1490285Note (for name \(Fig11.2 (10)\)): The system has 1 limit cycle.
- With names \(QS89_{1}^{(2)}\) and \(QS89_{1}^{(2)}\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Phase portraits of a family of real quadratic differential systemspossessing a nilpotent or intricate singularity at infinity, {Preprint} (2026).Note (for name \(QS89_{1}^{(2)}\)): The system has 1 limit cycle.
- With name \(1.3L2\) 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.
- With name \(2.8L2\) in {J. C. Artés, A. C. Rezende and R. D. S. Oliveira}, The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C), emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{25}, no. 3 (2015), 1530009, 111 pp.
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 \(QS89_{1}^{(2)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.