Affiliation:
1. Department of Mathematics, Mathematical Analysis and Applications Laboratory, University Mohamed El Bachir El Ibrahimi of Bordj Bou Arréridj 34030 , El Anasser , Algeria
Abstract
Abstract
The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in
R
2
{{\mathbb{R}}}^{2}
, 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.
Reference23 articles.
1. F. Dumortier, J. Llibre, and J. C. Artés, Qualitative theory of planar differential systems, Springer-Verlag, Universitext (UTX), 2006, DOI: https://doi.org/10.1007/978-3-540-32902-2.
2. J.C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, J. Differential Equations 107 (1994), no. 1, 80–95, DOI: https://doi.org/10.1006/jdeq.1994.1004.
3. J. C. Artés, J. Llibre, D. Schlomiuk, and N. Vulpe, Geometric Configurations of Singularities of Planar Polynomial Differential Systems, Birkhäuser Cham, Basel, 2021, DOI: https://doi.org/10.1007/978-3-030-50570-7.
4. A. Belfar and R. Benterki, Qualitative dynamics of quadratic systems exhibiting reducible invariant algebraic curve of degree 3, Palestine J. Math. 11 (2021), no. II, 1–12, https://pjm.ppu.edu.
5. A. Belfar and R. Benterki, Qualitative dynamics of quadratic polynomial differential system exhibiting an algebraic cubic first integral. Submitted (2022).