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Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves

Benterki Rebiha, Belfar Ahlam

Demonstratio Mathematica

2023

Vol. 56, nr 1

art. no. 20220218

The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R2, particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.