\(QS19_{3}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(19\) | \(22\) | \(110110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y-x \, y/2 \\ \dot{y} = 1+x^{2}-y^{2} \end{cases}\]
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 \(AA^n_6\) in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes, J. Dynam. Differential Equations { bf 33} (2021), no.~4, 1779--1821; MR4333383
- With names \(17\) and \(22\) in {R. Benterki and J. Llibre}, Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 4, Electron. J. Differential Equations { bf 2019}, Paper No. 15, 25 pp.; MR3919655
- With name \(8\) in {R. Benterki and A. Belfar}, Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves, Demonstr. Math. { bf 56} (2023), no.~1, Paper No. 20220218, 16 pp.; MR4592893
- With name \(106\) 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 \(35\) and \(54\) 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 \(P10\) in {J. Llibre and R. D. S. Oliveira}, Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3, Nonlinear Anal. { bf 70} (2009), no. 12, 6378--6379.Note (for name \(P10\)): missed arrows
- With name \(6\) 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 \(Fig 5.32 S^2_{11,3}\) 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.Note (for name \(Fig 5.32 S^2_{11,3}\)): The system has limit cycles with distribution \((1,0)\).
- With name \(S^2_{11,3}\) in {J. C. Artés, R. E. Kooij and J. Llibre}, Structurally stable quadratic vector fields, Mem. Amer. Math. Soc. { bf 134} (1998), no.~639, viii+108 pp.; MR1432139
- With name \(Fig 1.24 k\) 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 \(P10\) in {J. C. Artés, J. Llibre and N. Vulpe}, Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space $ Bbb R^{12$}, Rend. Circ. Mat. Palermo (2) { bf 59} (2010), no.~3, 419--449; MR2745521Note (for name \(P10\)): missing arrows
- With name \(V3\) 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.
Neighbours of Codimension 1
- Through the border \(QS22_{4}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS16_{1}^{(0)}\).
- Through the border \(QS85_{3}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS25_{1}^{(0)}\).
- Through the border \(QS19_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS31_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{2}^{(0)}\).
- Through the border \(QS31_{5}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{2}^{(0)}\).
- Through the border \(QS31_{6}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS5_{3}^{(0)}\).