\(QS13_{1}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Inf Sep |
|---|---|
| \(13\) | \(100100\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y+2 \, x \, y \\ \dot{y} = x^{2}+y^{2}+e2+e1 \, y \end{cases}\]
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 \(2\) 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 \(PP07\) in {J. C. Artés, J. Llibre and Huaxin Ou}, Quadratic systems with two invariant straight lines and an invariant hyperbola, {Preprint} (2026).
- With name \(H4\) 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 \(42\) 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 names \(4.13b\) and \(4.34a\) 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 \(6.2\) in {D. Schlomiuk and N. Vulpe}, Integrals and phase portraits of planar quadratic differential systems with invariant lines of at least five total multiplicity, emph{Rocky Mountain J. Math.}, textbf{38}, no. 6 (2008), 2015--2075.
- With name \(51\) 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; MR3941199
- With name \(IV\) in {T. Li and J. Llibre}, Phase portraits of separable quadratic systems and a bibliographical survey on quadratic systems, Expo. Math. { bf 39} (2021), no.~4, 540--565; MR4340762
- With name \(Fig. 4 a\) in {J. Llibre, W. F. Pereira and C. G. Pessoa}, Phase portraits of Bernoulli quadratic polynomial differential systems, Electron. J. Differential Equations { bf 2020}, Paper No. 48, 19 pp.; MR4102990
- With name \(P08\) in {J. Llibre and C. Valls}, Global phase portraits for the Abel quadratic polynomial differential equations of second kind with $Z_2$-symmetries, Canad. Math. Bull. { bf 61} (2018), no.~1, 149--165; MR3746481
- With name \(4L3\) in {M. C. Mota, R. D. S. Oliveira and A. M. Travaglini}, The interplay among the topological bifurcation diagram, integrability and geometry for the family { bf QSH(D)}, Geom. Dedicata { bf 217} (2023), no.~6, Paper No. 95, 42 pp.; MR4631488
- With name \(17\) 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 \(Fig2 E^1M^-1_02,1M^1_02,1(b)\) in {J. W. Reyn}, Phase portraits of quadratic systems without finite critical points, Nonlinear Anal. { bf 27} (1996), no.~2, 207--222; MR1389478
- With name \(Ric. 1\) 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 \(U^1_{D21}\) 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 \(Fig 11 p3F\), \(Fig 13 P2G\) and \(Fig 25 P2M\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(2\) in {A. Gasull, S. Li Ren and J. Llibre}, Chordal quadratic systems, emph{Rocky Mountain J. Math.}, textbf{16}, no. 4 (1986), 751--782.
- With names \(Fig 1.21 h\) and \(Fig 1.29 e\) 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
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS13_{1}^{(0)}\) and \(QS13_{1}^{(0)}\).