\(QS38_{19}^{(1)}\)

Description

Topological configuration of singularities: \(s,a,sn;S,N,N\)

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(38\)	\(441\)	\(321111\)

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: \(h = \frac{-211}{2000}, \quad c = \frac{9549}{50000}, \quad n = \frac{4}{5}, \quad e = 0.000001\)

has the following phase portrait done with P4. If you want, you may download the P4 file here. Since the image is not clear enough, we have added a ZOOM of it.

The phase portrait appears in the following papers

With names \(V16\) and \(V19\) 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)}Note (for name \(V16\)): The system has 1 limit cycle.
With name \(U^1_{A40}\) 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 \(2S4\) 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}.

Bifurcations in codimension 0

Bifurcation interface between the codimension 0 portrait \(QS25_{2}^{(0)}\) and \(QS10_{9}^{(0)}\).

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.