\(QS5_{2}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(5\) | \(4431\) | \(110110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = y-x \, y/2 \\ \dot{y} = e^{2} \, x/5-e \, y+x^{2}-y^{2} \end{cases}\]
with parameters: \(e = 0.5\)
has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.
The phase portrait appears in the following papers
- With name \(6\) 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 names \(Chavarriga\) and \(CLS5\) in {M. Alberich-Carramiñana, A. Ferragut and J. Llibre}, Quadratic planar differential systems with algebraic limit cycles via quadratic plane, Cremona maps, Adv. Math. { bf 389} (2021), Paper No. 107924, 38 pp.; MR4290137Note (for name \(Chavarriga\)): The system has limit cycles with distribution \((0,0,1)\).Note (for name \(CLS5\)): The system has limit cycles with distribution \((0,0,1)\).
- With names \(Fig. 36\), \(Fig. 37\) and \(Fig. 38 c\) 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 \(Fig 5.139 S^2_{12,2}\) 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 \(Fig 5.139 S^2_{12,2}\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(S^2_{12,2}\) 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 \(V30\) 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 \(V14\) in {J. C. Artés, J. Llibre and D. Schlomiuk}, The geometry of quadratic polynomial differential systems with a weak focus and an invariant straight line, emph{ Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{20}, no. 11 (2010), 3627--3662.
Neighbours of Codimension 1
- Through the border \(QS7_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS74_{4}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{3}^{(0)}\).
- Through the border \(QS74_{3}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{10}^{(0)}\).
- Through the border \(QS31_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{3}^{(0)}\).
- Through the border \(QS31_{5}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{3}^{(0)}\).
- Through the border \(QS31_{4}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{3}^{(0)}\).
- Through the border \(QS5_{2}^{(1)}\), by means of a bifurcation of type \(D(b)\), we reach the neighbor \(QS5_{2}^{(0)}\).
- Through the border \(QS5_{6}^{(1)}\), by means of a bifurcation of type \(D(c)\), we reach the neighbor \(QS5_{3}^{(0)}\).
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 \(QS5_{2}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.