\(QS79_{1}^{(2)}\)

Description

Topological configuration of singularities: \(s,a,a;(1,2)E-H,S\)

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(79\)	\(421\)	\(3101\)

The quadratic differential system

\[\begin{cases} \dot{x} = P_x(x,y) \\ \dot{y} = P_y(x,y) \end{cases}\]

has the following phase portrait done with P4.

With names \(A V89\) and \(A V91\) in {J. C. Artés, M. C. Mota and A. C. Rezende}, Quadratic systems possessing an infinite elliptic-saddle or an infinite nilpotent saddle, Internat. J. Bifur. Chaos Appl. Sci. Engrg. { bf 34} (2024), no.~11, Paper No. 2430023, 43 pp.; MR4801966Note (for name \(A V91\)): The system has limit cycles with distribution \((0,1)\).
With name \(5S26\) 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 \(Fig10.2 (2)\) and \(Fig10.2 (3)\) in {J. W. Reyn and X. H. Huang}, Phase portraits of quadratic systems with finite multiplicity three and a degenerate critical point at infinity, Rocky Mountain J. Math. { bf 27} (1997), no.~3, 929--978; MR1490285Note (for name \(Fig10.2 (3)\)): The system has limit cycles with distribution \((0,1)\).
With names \(QS79_{1}^{(2)}\) and \(QS79_{1}^{(2)}\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Phase portraits of a family of real quadratic differential systemspossessing a nilpotent or intricate singularity at infinity, {Preprint} (2026).Note (for name \(QS79_{1}^{(2)}\)): The system has limit cycles with distribution \((0,1)\).

This phase portrait appears in J. C. Artés and L. Cairó ({Preprint} (2026)) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS79_{1}^{(2)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.