Geometric Solutions of a Quadratic Polynomial Differential System Having the First Integral

Belfar, Ahlam; Benterki, Rebiha; Sellami, Halla

doi:10.2478/tmmp-2025-0024

Abstract

In this paper, we classify the geometric solutions of a class of integrable quadratic polynomial differential systems. We provide a new class of quadratic polynomial differential system that exhibits the so-called cubic egg curve as a first integral, we analyze the geometric solutions of this class in the Poincaré disc.

References

ARTÉS, J. C.—LLIBRE, J.—SCHLOMIUK, D.—VULPE, N.: Geometric Configurations of Singularities of Planar Polynomial Differential Systems. Springer-Verlag, Berlin, 2021.
Search in Google Scholar Back to article
BAUTIN, N. N.: On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik, 30 (1952), 181–196, Amer. Math. Soc. Transl, 100 (1954), 1–19.
Search in Google Scholar Back to article
BELFAR, A.— BENTERKI, R.: Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree six, Tatra Mt. Math. Publ. 15 (2022), 129–144.
Search in Google Scholar Back to article
BENTERKI, R.—LLIBRE, J.: Phase portraits of quadratic polynomial differential systems having as solution some classical planar algebraic curves of degree 4, Electronic J. Differential Equ., 15) (2019), 1–25.
Search in Google Scholar Back to article
DULAC, H.: Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre, Bull. Sci. Math. Sér. (2), 32 (1908), 230–252.
Search in Google Scholar Back to article
DUMORTIER, F.—LLIBRE, J.—ARTÉS, J. C.: Qualitative Theory of Planar Differential Systems. Universitext, Spring-Verlag, Berlin, 2006.
Search in Google Scholar Back to article
KAPTEYN, W.: On the midpoints of integral curves of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederlands, 19 (1911), 1446–1457.
Search in Google Scholar Back to article
KAPTEYN, W.: New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354–1365.
Search in Google Scholar Back to article
LUNKEVICH, V. A.—SIBIRSKII, K. S.: Integrals of a general quadratic differential system in cases of a center, Differential Equations, 18 (1982), 563–568.
Search in Google Scholar Back to article
MARKUS, L.: Global structure of ordinary differential equations in the plane, Trans. Amer. Math Soc. 76 (1954), 127–148.
Search in Google Scholar Back to article
NEUMANN, D. A.: Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81.
Search in Google Scholar Back to article
PEIXOTO, M. M.: Dynamical systems, in: Proceedings of a Symposium held at the University of Bahia, Salvdor, Brasil, July 26–August14 1971, Acad. Press, New York, 1973. pp. 389–420.
Search in Google Scholar Back to article
REYN, J. W.: Phase Portraits of Planar Quadratic Systems. In:Mathematics and its Applications Vol. 583, Springer-New York, 2007.
Search in Google Scholar Back to article
SCHLOMIUK. D.: Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 99–841.
Search in Google Scholar Back to article
WEI YIN.—YANQIAN, YE.: On the conditions of a center and general integrals of quadratic differential systems, Acta Math. Sin. (Engl. Ser.) 17 (2001), 229–236.
Search in Google Scholar Back to article
YE, Y. Q ET AL.: Theory of Limit Cycles. 2nd ed. (Translated from the Chinese by Chi Y. Lo.). In: Translations of Mathematical Monographs, Vol. 66, American Math. Soc., Providence, RI, 1986.
Search in Google Scholar Back to article

Geometric Solutions of a Quadratic Polynomial Differential System Having the First Integral

Abstract

Paradigm

My account