\(QS67_{1}^{(4)}\)
Description
Phase Portrait
Topological Invariants
| TCSP |
|---|
| \(67\) |
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 \(B 2S7\) 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 \(6\) 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 \(11\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point, Rend. Circ. Mat. Palermo (2) { bf 70} (2021), no.~2, 923--945; MR4286006
- With name \(11\) 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; MR4447008Note (for name \(11\)): wrong arrows
- With names \(3\) and \(4\) in {R. Benterki and A. Belfar}, Global phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 6, Tatra Mt. Math. Publ. { bf 81} (2022), 129--144; MR4519365
- With name \(X1\) in {C. A. Buzzi, J. Llibre and P. H. R. Santana}, Phase portraits of $(2;0)$ reversible vector fields with symmetrical singularities, J. Math. Anal. Appl. { bf 503} (2021), no.~2, Paper No. 125324, 21 pp.; MR42612202
- With name \(E3\) 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 \(85\) 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 \(4.8b\) and \(4.42b\) 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 \(6.6\) 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 \(P14\) in {D. Schlomiuk and X. Zhang}, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, emph{J. Differential Equations}, { bf 265}, no. 8 (2018), 3650--3684.
- With name \(39\) 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 \(39\)): This needs 2 elliptic sectors
- With name \(P2\) 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 \(2\) in {J. Llibre, M. Messias and A. C. Reinol}, Normal forms and global phase portraits of quadratic and cubic integrable vector fields having two nonconcentric circles as invariant algebraic curves, Dynamical Systems, DOI: 10.1080/14689367.2016.1263600
- With name \(24\) 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 \(Fig1.21 1\) in {K. S. Sibirskii}, Introduction to the algebraic theory of invariants of differential equations (Russian), ``Shtiintsa'', Kishinev, 1982; MR0716501
- With name \(S IV 21\) 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 \(fig 1.1(d)\) in {B. Coll, A. Gasull and J. Llibre}, Quadratic systems with a unique finite rest point, emph{Publ. Mat.}, textbf{32} (1988), 199--259.