Abstract
The centers of the polynomial differential systems with a linear center perturbed by homogeneous polynomials have been studied for the degrees s = 2, 3, 4, 5. They are completely classified for s = 2, 3, and partially classified for s = 4, 5. In this paper we recall these results for s = 2, 3, 4, 5, and we give new centers for s = 6, 7
Similar content being viewed by others
References
Andronov, A. A., Leontovich, E. A., Gordon, I. I., Maier, A. G.: Theory of bifurcations of dynamic systems on a plane, John Wiley and Sons, New York-Toronto, 1967
Andreev, A.: Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular points. Translation of AMS, 8, 187–207 (1958)
Dumortier, F.: Singularities of vector fields on the plane. J. Differential Equations, 23, 53–106 (1977)
Poincaré, H.: Mémoire sur les courbes définies par les équations différentielles. Journal de Mathématiques, 37, 375–422 (1881); 8 , 251–296 (1882); Oeuvres de Henri Poincaré, vol. I, Gauthier-Villars, Paris, 3–84, 1951
Liapunov, M. A.: Problème général de la stabilité du mouvement, Annals of Mathematics Studies, 17, Pricenton University Press, New Jersey, 1947
Kapteyn, W.: On the midpoints of integral curves of differential equations of the first degree. Nederl. Akad. Wetensch. Verslag. Afd. Natuurk., 19, 1446–1457 (1911) (Dutch)
Kapteyn, W.: New investigations on the midpoints of integrals of differential equations of the first degree. Nederl. Akad. Wetensch. Verslag Afd. Natuurk., 20, 1354–1365; 21, 27–33 (1912) (Dutch)
Malkin, K. E.: Criteria for the center for a differential equation. Volz. Math. Sb., 2, 87–91 (1964)
Sibirskii, K. S.: On the number of limit cycles in the neighborhood of a singular point. Differential Equations, 1, 36–47 (1965)
Chavarriga, J., Giné, J.: Integrability of a linear center perturbed by fourth degree homogeneous polynomial. Publ. Mat., 40, 21–39 (1996)
Chavarriga, J., Giné, J.: Integrability of a linear center perturbed by fifth degree homogeneous polynomial. Publ. Mat., 41, 335–356 (1997)
Blows, T. R., Lloyd, N. G.: The number of limit cycles of certain polynomial differential equations. Proc. Roy. Soc. Edinburgh, 98A, 215–239 (1984)
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. Sb., 30(72), 181–196 (1952); Amer. Math. Soc. Transl., 100, 397–413 (1954)
Żołądek, H.: Eleven small limit cycles in a cubic vector field. Nonlinearity, 8, 843–860 (1995)
Shi, S. L.: On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields. J. Differential Equations, 52, 52–57 (1984)
Shi, S. L.: A method of constructing cycles without contact around a weak focus. J. Differential Equations, 41, 301–312 (1981)
Chavarriga, J.: Integrable systems in the plane with a center type linear part. Appl. Math. (Warsaw), 22(2), 285–309 (1994)
Schlomiuk, D.: Algebraic and geometric aspects of the theory of polynomials vector fields, Bifurcations and Periodic Orbits of Vector Fields (Montreal, PQ, 1992), 429–467, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 408, Kluwer Academic, Publ., Dordrecht, 1993
Chavarriga, J.: A class of integrable polynomial vector fields. Appl. Math. (Warsaw), 23(3), 339–350 (1995)
Cima, A., Gasull, A., Man̄osa, V., Man̄osas, F.: Algebraic properties of the Lyapunov and period constants. Rocky Mountain. J. Math., 27, 471–501 (1997)
Lukashevich, N. A.: Isochronicity of a center for certain systems of differential equations. Diff. Equations, 1, 220–226 (1965)
Dulac, H.: Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre. Bull. Sci. Math., 32, 230–252 (1908)
Frommer, M.: Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmheitsstellen. Math. Ann., 109, 395–424 (1934)
Lunkevich, V. A., Sibirskii, K. S.: Integrals of a general quadratic differential system in cases of a center. Diff. Equations, 18, 563–568 (1982)
Lunkevich, V. A., Sibirskii, K. S.: Integrals of a system with a homogeneous third–degree nonlinearity in the case of a center. Diff. Equations, 20, 563–568 (1984)
Nemytskii, V. V., Stepanov, V. V.: Qualitative Theory of Differential Equations, Dover Publ., New York, 1989
Rousseau, C., Schlomiuk, D.: Cubic vector fields symmetric with respect to a center. J. Differential Equations, 123, 388–436 (1995)
Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: On the integrability of two-dimensional flows. J. Differential Equations, 157(1), 163–182 (1999)
Cherkas, L. A.; Conditions for a center for a certain Liénard equation (Russian). Differencial’nye Uravnenija, 12(2), 292–298 (1976)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is partially supported by a DGICYT grant number BFM 2002-04236-C02-01 and by DURSI of Government of Catalonia "Distinció de la Generalitat de Catalunya per a la promoció de la recerca universitària".
Rights and permissions
About this article
Cite this article
Giné, J. The Center Problem for a Linear Center Perturbed by Homogeneous Polynomials. Acta Math Sinica 22, 1613–1620 (2006). https://doi.org/10.1007/s10114-005-0623-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-005-0623-4