\(QS77_{2}^{(2)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(77\)	\(441\)	\(3310\)

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 name \(U^2_{BC,19}\) in Missing reference in BC1
With names \(A V9\) and \(A V11\) 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 \(A V11\)): The system has 1 limit cycle.
With name \(5S22\) 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 \(Fig11.5c (5)\) and \(Fig11.5c (6)\) 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 \(Fig11.5c (5)\)): The system has 1 limit cycle.
With names \(QS77_{2}^{(2)}\) and \(QS77_{2}^{(2)}\) 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 \(QS77_{2}^{(2)}\)): The system has 1 limit cycle.

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 \(QS77_{2}^{(2)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.