\(QS22_{3}^{(2)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(22\) | \(10\) | \(2111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = (cm \, x^{2})/10 - y + 2 \, ch \, x \, y \\ \dot{y} = x + (-2 \, ch + cl) \, x^{2} + (-cg + cm/10) \, x \, y + 2 \, ch \, y^{2} \end{cases}\]
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 \(B4a=C4=B6a=B8=B10\) and \(B4b=C4=B6b=B8=b10\) in {J. C. Artés and J. Llibre}, Phase portraits for quadratic systems having a focus and one anti-saddle, emph{Rocky Mountain J. Math.}, textbf{24} (1994), 875--889.Note (for name \(B4a=C4=B6a=B8=B10\)): uncompleteNote (for name \(B4b=C4=B6b=B8=b10\)): uncomplete
- With name \(P19\) in {J. C. Artés and C. Trullàs}, Quadratic Differential Systems with a Weak Focus of First-Order and a Finite Saddle-Node, {International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}
- With name \(BD13\) in {J. C. Artés}, Systems of class BD, {Preprint} (2026).
- With name \(P03\) in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).
- With names \(7S02\), \(7S03\), \(7S13\) and \(7.10L1\) in Missing reference in QWF1SN02Note (for name \(7S02\)): The system has limit cycles with distribution \((1,0)\).Note (for name \(7S13\)): The system has limit cycles with distribution \((2,0)\).Note (for name \(7.10L1\)): The system has limit cycles with distribution \((d,0)\).
- With names \(5.7L2\) and \(5.7L3\) 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 \(5.7L2\)): missed orbit inside limit cycleNote (for name \(5.7L2\)): The system has limit cycles with distribution \((1,0)\).
- With name \(5.7L3\) 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 \(5.7L1\) in {J. C. Artés, A. C. Rezende and R. Oliveira}, Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node, emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, { bf 23}, no. 8 (2013), 1350140, 21 pp.
- With name \(2.7L20\) in {J. C. Artés, A. C. Rezende and R. D. S. Oliveira}, The geometry of quadratic polynomial differential systems with a finite and an infinite saddle-node (C), emph{Internat. J. Bifur. Chaos Appl. Sci. Engrg.}, textbf{25}, no. 3 (2015), 1530009, 111 pp.
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. Given that the portrait is of codimension 1, hyperbolic limit cycles can be generated without breaking its other unstable features. However, multiple or compound limit cycle configurations are not guaranteed, as they might be incompatible with the pre-existing unstable properties of the system.