\(QS19_{2}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(19\) | \(10\) | \(111110\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = -y-(v-b) \, x^{2}/3+2 \, a \, x \, y+n \, y^{2} \\ \dot{y} = x+a \, x^{2}+(2 \, v+b)/3 \, x \, y-a \, y^{2} \end{cases}\]
with parameters: \(a = 1, \quad b = -9, \quad v = -.7967, \quad n = 1\)
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 \(7.11L5\) 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 \(1.7L3\) 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 name \(U^1_{D28}\) 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.
- With name \(7S13\) 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 \(7S13\) 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 \(7S1\) 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.
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS19_{1}^{(0)}\) and \(QS19_{2}^{(0)}\).
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. 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.