\(QS8_{4}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(8\) | \(4422\) | \(22\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -x \, y+a \, (-1+2 \, x^{2}+y^{2}) \\ \dot{y} = -1+2 \, x^{2}+y^{2} \end{cases}\]
with parameters: \(a = 0.1\)
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 \(A V12\) 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 \(1\) 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 \(Chap 2 15\) in {B. Imane and B. Souad}, Global phase portraits of quadratic differential systems exhibiting an invariant algebraic curve or an algebraic cubic first integral, {Ph.D. Universite Mohamed el Bachir}, (2020).
- With name \(13\) in {A. Belfar and R. Benterki}, Qualitative dynamics of five quadratic polynomial differential systems exhibiting five classical cubic algebraic curves, Rend. Circ. Mat. Palermo (2) { bf 72} (2023), no.~1, 393--420; MR4543844
- With name \(15\) 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 \(A03\) in {C. A. Buzzi and D. J. Tonon}, Quadratic planar systems with two parallel invariant straight lines, Qual. Theory Dyn. Syst. { bf 7} (2009), no.~2, 295--316; MR2486677
- With name \(87\) 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 \(12\) 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 \(PP4\) 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 \(R2\) 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.; MR3441696Note (for name \(R2\)): 3 separatrices are orbits
- With name \(P17\) 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 \(P17\)): missed arrows
- With names \(Fig. 08\), \(Fig. 22\) and \(Fig. 25\) in {J. Llibre and C. Pantazi}, Global phase portraits of the quadratic systems having a singular and irreducible invariant curve of degree 3. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 33 (2023), no. 1, Paper No. 2350003, 54 pp.
- With name \(2\) 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 \(Ric. 29\) 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 \(S IV 4\) 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 \(S^2_{3,4}\) 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 \(V56\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic differential systems with a weak focus of second order, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}.
- With name \(Portrait 34\) 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 name \(P17\) 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; MR2745521
Neighbours of Codimension 1
- Through the border \(QS37_{6}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS23_{1}^{(0)}\).
- Through the border \(QS8_{3}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS8_{3}^{(0)}\).
- Through the border \(QS8_{4}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS11_{15}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{13}^{(0)}\).
- Through the border \(QS11_{16}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{13}^{(0)}\).
- Through the border \(QS11_{16}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{14}^{(0)}\).
- Through the border \(QS11_{18}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{14}^{(0)}\).
- Through the border \(QS11_{20}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS10_{16}^{(0)}\).
Comments
This phase portrait appears in J. C. Artés, J. Llibre and D. Schlomiuk (emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{16} (2006), {3127--3194}) featuring a weak focus of second order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS8_{4}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.