\(QS123_{1}^{(4)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(123\) | \(44\) | \(2121\) |
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 name \(Fig 7 c\) in {Y. Bolaños, J. Llibre and C. Valls}, Phase portraits of quadratic Lotka-Volterra systems with a Darboux invariant in the Poincaré disc, Commun. Contemp. Math. { bf 16} (2014), no.~6, 1350041, 23 pp.; MR3277950
- With name \(B10\) in {C. A. Buzzi and D. J. Tonon}, Quadratic planar systems with two parallel invariant straight lines, Qual. Theory Dyn. Syst. { bf 7} (2009), no.~2, 295--316; MR2486677
- With names \(Fig 1 1\) and \(Fig 4 2\) in {P. C. Carri\~ao, M. E. S. Gomes and A. A. G. Ruas}, Planar quadratic vector fields with two or three finite singularities and a finite saddle connection on a straight line, Qual. Theory Dyn. Syst. { bf 8} (2009), no.~1, 25--44; MR2575806
- With name \(70\) 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 \(4.12c\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of total multiplicity four, emph{Bul. Acad. c{S}tiin c{t}e Repub. Mold. Mat.}, { bf 1 (56)} (2008), 27--83.
- With name \(4,12c\) in {D. Schlomiuk and N. Vulpe}, Global classification of the planar Lotka--Volterra differential systems according to their configurations of invariant straight lines, emph{J. Fixed Point Theory Appl.}, { bf 8}, no. 1 (2010), 177--245.
- With names \(77\) and \(84\) in {A. Ferragut, J. D. García-Saldaña and C. Valls}, Phase portraits of Abel quadratic differential systems of second kind with symmetries, Dyn. Syst. { bf 34} (2019), no.~2, 301--333; MR3941199Note (for name \(77\)): two elliptic sectors needed
- With name \(1\) in {B. García, J. Llibre and J. S. Pérez del Río}, Phase portraits of the quadratic vector fields with a polynomial first integral, Rend. Circ. Mat. Palermo (2) { bf 55} (2006), no.~3, 420--440; MR2287071
- With names \(29\) and \(42\) in {M. Ndiaye and H. J. Giacomini}, Quadratic systems equivalent by domains to a linear one: global phase portraits, Extracta Math. { bf 15} (2000), no.~1, 97--119; MR1792982
- With name \(Ric. 54\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Global analysis of Riccati quadratic differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 34} (2024), no.~1, Paper No. 2450004, 46 pp.; MR4701478
- With name \(21\) in {J. C. Artés and J. Llibre}, Quadratic Hamiltonian vector fields, emph{J. Differential Equations}, { bf 107} (1994), 80--95.
- With name \(Ham 21\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with an integrable saddle: A complete classification in the coefficient space $ mathbb{R^{12}$}, emph{Nonlinear Anal.}, textbf{75}, no. 14 (2012), 5416--5447.
- With name \(Fig 1.36 a\) in {J. W. Reyn and R. E. Kooij}, Phase portraits of non-degenerate quadratic systems with finite multiplicity two, Differential Equations Dynam. Systems { bf 5} (1997), no.~3-4, 355--414; MR1660222
- With name \(QS123_{1}^{(4)}\) 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).
- With name \(Fig4.5 VIII\) in {J. W. Reyn}, Phase portraits of a quadratic system of differential equations occurring frequently in applications, emph{Nieuw Arch. Wisk. (4)}, textbf{5}, no. 2 (1987), 107--151.