\(QS15_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(15\) | \(44\) | \(211211\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = x^{2}+2 \, x \, y \\ \dot{y} = -1-2 \, x \, y-y^{2} \end{cases}\]
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 \(AA^s_2\) 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 name \(80\) 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 \(10\) 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 \(11\) 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 name \(P11\) 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 \(11\) 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 names \(Fig 5 P2B\) and \(Fig 7 P2C\) in {A. M. Travaglini}, Integrability and geometryof quadratic differential systems with invariant hyperbolas, {Ph. D., Uni. de Sao Paulo} (2026).
- With name \(V1\) in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Topological classification of quadratic polynomial differential systems with a finite semi-elemental triple saddle, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 26} (2016), no.~11, 1650188, 26 pp.; MR3566296
- With name \(S^2_{6,1}\) 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 \(25\) in {J. C. Artés and J. Llibre}, Quadratic Hamiltonian vector fields, emph{J. Differential Equations}, { bf 107} (1994), 80--95.
- With name \(Ham 25\) 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 names \(Fig10 3\) and \(Fig10 5\) 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.
- With names \(Fig 1.1 a\), \(Fig 1.1 c\) and \(Fig 1.28\) 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; MR1660222Note (for name \(Fig 1.1 a\)): the quality of the pictures of this paper is very low and it is possible to misinterpret some orbits
Neighbours of Codimension 1
- Through the border \(QS15_{1}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS15_{1}^{(0)}\).
- Through the border \(QS28_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS1_{1}^{(0)}\).
- Through the border \(QS28_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS1_{2}^{(0)}\).
- Through the border \(QS28_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS1_{2}^{(0)}\).
- Through the border \(QS28_{4}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS1_{3}^{(0)}\).
- Through the border \(QS28_{5}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS1_{4}^{(0)}\).
- Through the border \(QS87_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS25_{1}^{(0)}\).