\(QS3_{1}^{(0)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(3\) | \(4321\) | \(11\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -x+2 \, x \, y+m \, y \\ \dot{y} = y-x^{2}+y^{2} \end{cases}\]
with parameters: \(m = 1\)
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 \(A V05\) 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 names \(1\), \(5\) and \(10\) in {R. Benterki and J. Llibre}, Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 4, Electron. J. Differential Equations { bf 2019}, Paper No. 15, 25 pp.; MR3919655
- With names \(Fig 5.61 S^2_{5,1}\), \(Fig 5.64 S^2_{5,1}\), \(Fig 5.66 S^2_{5,1}\) and \(Fig 5.105 S^2_{5,1}\) 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 \(Fig 5.61 S^2_{5,1}\)): The system has limit cycles with distribution \((1,0,0)\).Note (for name \(Fig 5.64 S^2_{5,1}\)): The system has limit cycles with distribution \((0,1,0)\).Note (for name \(Fig 5.66 S^2_{5,1}\)): The system has limit cycles with distribution \((0,1,0)\).Note (for name \(Fig 5.105 S^2_{5,1}\)): The system has limit cycles with distribution \((0,0,1)\).
- With name \(S IV 2\) 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 \(S^2_{5,1}\) 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 \(V46\) 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 \(V28\) 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 \(QS73_{1}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS8_{1}^{(0)}\).
- Through the border \(QS73_{2}^{(1)}\), by means of a bifurcation of type \(C\), we reach the neighbor \(QS8_{5}^{(0)}\).
- Through the border \(QS30_{1}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS16_{1}^{(0)}\).
- Through the border \(QS30_{2}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS16_{1}^{(0)}\).
- Through the border \(QS30_{3}^{(1)}\), by means of a bifurcation of type \(A\), we reach the neighbor \(QS16_{1}^{(0)}\).
- Through the border \(QS3_{2}^{(1)}\), by means of a bifurcation of type \(D(c)\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS3_{1}^{(1)}\), by means of a bifurcation of type \(D(b)\), we reach the neighbor \(QS3_{1}^{(0)}\).
- Through the border \(QS7_{2}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS5_{2}^{(0)}\).
- Through the border \(QS7_{3}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS5_{3}^{(0)}\).
- Through the border \(QS7_{5}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS5_{5}^{(0)}\).
- Through the border \(QS7_{7}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS5_{6}^{(0)}\).
- Through the border \(QS7_{8}^{(1)}\), by means of a bifurcation of type \(B\), we reach the neighbor \(QS5_{7}^{(0)}\).
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 \(QS3_{1}^{(0)}\) could potentially exhibit up to two limit cycles (or a compound double limit cycle) bifurcating from the focus.