\(QS74_{11}^{(1)}\)
Description
Phase Portrait
Topological Invariants
| TCSP | Fin Sep | Inf Sep |
|---|---|---|
| \(74\) | \(421\) | \(211111\) |
Example
The quadratic differential system
\[\begin{cases} \dot{x} = (1-e) \, x^{2}+2 \, h \, x \, y+(n-1-2 \, h) \, y^{2} \\ \dot{y} = y+c \, (1-e) \, x^{2}+(2+2 \, h-n) \, (1-e) \, x \, y-2 \, c \, x \, y+(c-2-2 \, h+2 \, n) \, y^{2} \end{cases}\]
with parameters: \(h = \frac{-21}{5}, \quad c = 18, \quad n = 10, \quad e = 0.1\)
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 names \(V10\) and \(V60\) in {J. C. Artés and L. Cairó}, Phase portraits of quadratic differential systems with a weak focus and a (1,1) SN, {Preprint} (2026).Note (for name \(V60\)): The system has limit cycles with distribution \((0,1)\).
- With name \(U^1_{C28}\) 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 \(an27 Fig 2.43\) in {X. Huang}, Qualitative analysis or certain nonlinear differential equations, {Ph.D. U. Delft}, (1996).
- With name \(1S12\) 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 \(an23 Fig. 15\) in {J. W. Reyn and X. H. Huang}, Separatrix configurations of quadratic systems with finite multiplicity three and a $M^0_{1,1$ type of critical point at infinity}, Report U. Delft (1997?).
Bifurcations in codimension 0
- Bifurcation interface between the codimension 0 portrait \(QS10_{15}^{(0)}\) and \(QS5_{6}^{(0)}\).
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. Given that the portrait is of codimension 1, hyperbolic limit cycles can be generated without breaking its other unstable features. However, multiple limit cycle configurations are not guaranteed, as they might be incompatible with the pre-existing unstable properties of the system.