\(QS122_{1}^{(6)}\)
Description
Phase Portrait
Topological Invariants
| TCSP |
|---|
| \(122\) |
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 \(A 2.5L1\) in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).
- With name \(19\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree 2: A complete classification in the coefficient space $ R^{12$}, emph{Rend. Circ. Mat. Palermo}, textbf{56}, no. 3 (2007), 417--444.
- With name \(PP06\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(12\) in {A. Belfar and R. Benterki}, Qualitative dynamics of quadratic systems exhibiting reducible invariant algebraic curve of degree 3, Palest. J. Math. { bf 11} (2022), Special Issue II, 1--12; MR4447008
- With name \(DI2\) in {L. Cairó and J. Llibre}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 2. Nonlinear Anal. 67 (2007), no. 2, 327–348.
- With name \(31\) 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 \(C2.4\) in {D. Schlomiuk and N. Vulpe}, The full study of planar quadratic differential systems possessing a line of singularities at infinity, emph{J. Dynam. Differential Equations}, { bf 20}, no. 4 (2008), 737--775.
- With name \(65\) 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 \(65\)): Again, the elliptic sector is wished
- With name \(F\) in {A. Gasull and R. Prohens}, Quadratic and cubic systems with degenerate infinity, J. Math. Anal. Appl. { bf 198} (1996), no.~1, 25--34; MR1373524
- With name \(C P1\) 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.; MR4801966
- With name \(r\) in {J. Llibre and M. F. da Silva}, Phase portraits of integrable quadratic systems with an invariant parabola and an invariant straight line, C. R. Math. Acad. Sci. Paris { bf 357} (2019), no.~2, 143--166; MR3927021
- With name \(PP2\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant ellipse, J. Fixed Point Theory Appl. { bf 27} (2025), no.~4, Paper No. 89, 18 pp.; MR4957938
- With name \(PP02\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant parabola, Electron. J. Qual. Theory Differ. Equ. { bf 2025}, Paper No. 66, 54 pp.; MR5018064
- With name \(PP33\) in {J. Llibre and H. X. Ou}, Quadratic systems with two invariant real straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(P1\) in {J. Llibre and J. Yu}, Global phase portraits of quadratic systems with an ellipse and a straight line as invariant algebraic curves, Electron. J. Differential Equations { bf 2015}, No. 314, 14 pp.; MR3441696
- With name \(Fig 39 P3W\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(C_2 4\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Abel quadratic differential systems of second kind, Electron. J. Differential Equations { bf 2024}, Paper No. 50, 38 pp.; MR4793966
- With name \(QS122_{1}^{(6)}\) in {J. C. Artés and N. Vulpe}, The codimension of the phase portraits for degenerate quadratic differential systems, Bul. Acad. c Stiin c te Repub. Mold. Mat. { bf 2024}, no.~3(106), 29--53; MR4967334
- With name \(D3\) in {B. Coll, A. Gasull and J. Llibre}, Quadratic systems with a unique finite rest point, emph{Publ. Mat.}, textbf{32} (1988), 199--259.
- With name \(Fig6 7\) in {P. de Jager}, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, emph{J. Differential Equations}, textbf{87} (1990), 169--204.