\(QS5_{6}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(5\) | \(4221\) | \(211111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -100 \, x \, (1-x+6 \, y)+e \, y \\ \dot{y} = y \, (1+5 \, x/2-y) \end{cases}\]
with parameters: \(e = -2\)
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 names \(Filipstov\) and \(CLS6\) 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 \(Filipstov\)): The system has limit cycles with distribution \((0,0,1)\).Note (for name \(CLS6\)): The system has limit cycles with distribution \((0,0,1)\).
- With names \(Fig. 39 i\), \(Fig. 40 c\), \(Fig. 41\) and \(Fig. 42\) 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 \(Fig 5.162 S^2_{12,6}\) 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 \(Fig 5.162 S^2_{12,6}\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(S^2_{12,6}\) 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 names \(V31\) and \(V51\) 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}.Note (for name \(V51\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(V31\) in {J. Llibre and D. Schlomiuk}, Geometry of quadratic differential systems with a weak focus of third order, emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343.
- With name \(V15\) 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 \(QS7_{7}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS7_{6}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS74_{12}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{2}^{(0)}\).
- Through the border \(QS74_{13}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{14}^{(0)}\).
- Through the border \(QS74_{11}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS10_{15}^{(0)}\).
- Through the border \(QS31_{13}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{1}^{(0)}\).
- Through the border \(QS31_{12}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS19_{2}^{(0)}\).
- Through the border \(QS5_{4}^{(1)}\), by means of a bifurcation of type \(D(c)\), we reach the neighbor \(QS5_{3}^{(0)}\).
- Through the border \(QS5_{5}^{(1)}\), by means of a bifurcation of type \(D(d)\), we reach the neighbor \(QS5_{4}^{(0)}\).
- Through the border \(QS5_{10}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{6}^{(0)}\).
- Through the border \(QS5_{7}^{(1)}\), by means of a bifurcation of type \(D\), we reach the neighbor \(QS5_{7}^{(0)}\).
Comments
This phase portrait appears in J. Llibre and D. Schlomiuk (emph{Canad. J. of Math.}, textbf{56}, no. 2 (2004), 310--343) featuring a weak focus of third order. Since the portrait is of codimension 0, a configuration structurally equivalent to \(QS5_{6}^{(0)}\) could potentially exhibit up to three limit cycles (or compound double/triple arrangements) bifurcating from the focus.