\(QS40_{1}^{(2)}\)

Description

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

Phase Portrait

Topological Invariants

TCSP	Fin Sep	Inf Sep
\(40\)	\(422\)	\(31\)

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 \(U^2_AA 2\) in {J. C. Artés, R. D. S. Oliveira and A. C. Rezende}, Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes, J. Dynam. Differential Equations { bf 33} (2021), no.~4, 1779--1821; MR4333383
With name \(9S3\) 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 names \(20\) and \(23\) in {R. Benterki and A. Belfar}, Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves, Demonstr. Math. { bf 56} (2023), no.~1, Paper No. 20220218, 16 pp.; MR4592893Note (for name \(23\)): This porttrait cannot be right. Either there is a simple error in one arrow and we get something equivalent to 20, or there is a bigger error. We assume the first
With names \(FIg18 1\), \(FIg18 5\), \(Fig18 6a\), \(Fig18 12a\) and \(Fig20 15\) in {P. de Jager}, Phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues, emph{J. Differential Equations}, textbf{87} (1990), 169--204.Note (for name \(Fig18 12a\)): The system has 1 limit cycle.

This phase portrait appears in J. C. Artés and C. Trullàs ({International Journal of Bifurcation and Chaos, Vol. 36, No. 6 (2026) 2630013 (99 pages)}) featuring a weak focus of first order. Consequently, a configuration structurally equivalent to \(QS40_{1}^{(2)}\) could potentially exhibit an additional limit cycle bifurcating from the focus.