\(QS172_1\)

Description

Topological configuration of singularities: \(c,[ | , ∅];[ | , ∅]\)

Phase Portrait

Topological Invariants

TCSP
\(172\)

Example

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.

The phase portrait appears in the following papers

With name \(C 1.2L3\) in {J. C. Artés, C. Bujac, D. Schlomiuk and N. Vulpe}, Phase portraits of real quadratic differential systems possessing an invariant ellipse, {Preprint} (2026).
With name \(14\) in {J. C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe}, Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point, Rend. Circ. Mat. Palermo (2) { bf 70} (2021), no.~2, 923--945; MR4286006
With name \(1.8L1\) 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 name \(6\) in {A. Belfar and R. Benterki}, Qualitative dynamics of quadratic systems exhibiting reducible invariant algebraic curve of degree 3, Palest. J. Math. { bf 11} (2022), Special Issue II, 1--12; MR4447008
With name \(Chap 2 12\) in {B. Imane and B. Souad}, Global phase portraits of quadratic differential systems exhibiting an invariant algebraic curve or an algebraic cubic first integral, {Ph.D. Universite Mohamed el Bachir}, (2020).
With name \(P1\) in {D. Schlomiuk and X. Zhang}, Quadratic differential systems with complex conjugate invariant lines meeting at a finite point, emph{J. Differential Equations}, { bf 265}, no. 8 (2018), 3650--3684.Note (for name \(P1\)): In this article, some separatrices are drawn wider, but not all
With name \(r>0\) in {J. Llibre and J. Yu}, Global phase portraits of quadratic systems with an ellipse and a straight line as invariant algebraic curves, Electron. J. Differential Equations { bf 2015}, No. 314, 14 pp.; MR3441696
With name \(B\) in {J. Llibre, and C. Valls}, Global phase portraits of quadratic systems with a complex ellipse as invariant algebraic curve. Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 5, 801–811.
With name \(P16\) 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 name \(QS172_1\) in {J. C. Artés and N. Vulpe}, The codimension of the phase portraits for degenerate quadratic differential systems, Bul. Acad. c Stiin c te Repub. Mold. Mat. { bf 2024}, no.~3(106), 29--53; MR4967334
With name \(Vul29\) in {N. I. Vulpe}, Affine--invariant conditions for the topological distinction of quadratic systems with a center (in Russian), emph{Differentsial'nye Uravneniya}, textbf{19}, no. 3 (1983), 371--379. (Translation in emph{Differential Equations}, textbf{19} (1983), {273--280}.)