Document Zbl 1124.34326

The center problem for a linear center perturbed by homogeneous polynomials. (English) Zbl 1124.34326

Acta Math. Sin., Engl. Ser. 22, No. 6, 1613-1620 (2006).

The author studies the center problem for a special system of differential equations of the form \[ \dot x=-y+X_{s}(x,y),\qquad \dot y=x+Y_{s}(x,y), \] where \(X_{s}(x,y)\) and \(Y_{s}(x,y)\) are homogeneous polynomials of degree \(s,\) with \(s\geq2.\) It is known that 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 \(s=4,5.\) The author recalls these results and gives new centers for \(s=6,7.\)

Reviewer: Valery A. Gaiko (Minsk)

Cited in 26 Documents

MSC:

34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34A05	Explicit solutions, first integrals of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations

Keywords:

system of polynomial differential equations; center problem

Cite Review PDF

Full Text: DOI

References:

[1]	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
[2]	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) · Zbl 0079.11301
[3]	Dumortier, F.: Singularities of vector fields on the plane. J. Differential Equations, 23, 53–106 (1977) · Zbl 0346.58002 · doi:10.1016/0022-0396(77)90136-X
[4]	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 · JFM 13.0591.01
[5]	Liapunov, M. A.: Problème général de la stabilité du mouvement, Annals of Mathematics Studies, 17, Pricenton University Press, New Jersey, 1947
[6]	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)
[7]	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)
[8]	Malkin, K. E.: Criteria for the center for a differential equation. Volz. Math. Sb., 2, 87–91 (1964)
[9]	Sibirskii, K. S.: On the number of limit cycles in the neighborhood of a singular point. Differential Equations, 1, 36–47 (1965)
[10]	Chavarriga, J., Giné, J.: Integrability of a linear center perturbed by fourth degree homogeneous polynomial. Publ. Mat., 40, 21–39 (1996) · Zbl 0851.34001
[11]	Chavarriga, J., Giné, J.: Integrability of a linear center perturbed by fifth degree homogeneous polynomial. Publ. Mat., 41, 335–356 (1997) · Zbl 0897.34030
[12]	Blows, T. R., Lloyd, N. G.: The number of limit cycles of certain polynomial differential equations. Proc. Roy. Soc. Edinburgh, 98A, 215–239 (1984) · Zbl 0603.34020
[13]	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) · Zbl 0059.08201
[14]	\.Zołądek, H.: Eleven small limit cycles in a cubic vector field. Nonlinearity, 8, 843–860 (1995) · Zbl 0837.34042 · doi:10.1088/0951-7715/8/5/011
[15]	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) · Zbl 0534.34059 · doi:10.1016/0022-0396(84)90133-5
[16]	Shi, S. L.: A method of constructing cycles without contact around a weak focus. J. Differential Equations, 41, 301–312 (1981) · doi:10.1016/0022-0396(81)90039-5
[17]	Chavarriga, J.: Integrable systems in the plane with a center type linear part. Appl. Math. (Warsaw), 22(2), 285–309 (1994) · Zbl 0809.34002
[18]	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 · Zbl 0790.34031
[19]	Chavarriga, J.: A class of integrable polynomial vector fields. Appl. Math. (Warsaw), 23(3), 339–350 (1995) · Zbl 0839.34001
[20]	Cima, A., Gasull, A., Manosa, V., Manosas, F.: Algebraic properties of the Lyapunov and period constants. Rocky Mountain. J. Math., 27, 471–501 (1997) · Zbl 0911.34025 · doi:10.1216/rmjm/1181071923
[21]	Lukashevich, N. A.: Isochronicity of a center for certain systems of differential equations. Diff. Equations, 1, 220–226 (1965) · Zbl 0178.43301
[22]	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) · JFM 39.0374.01
[23]	Frommer, M.: Über das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung rationaler Unbestimmheitsstellen. Math. Ann., 109, 395–424 (1934) · JFM 60.1094.01 · doi:10.1007/BF01449147
[24]	Lunkevich, V. A., Sibirskii, K. S.: Integrals of a general quadratic differential system in cases of a center. Diff. Equations, 18, 563–568 (1982) · Zbl 0499.34017
[25]	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)
[26]	Nemytskii, V. V., Stepanov, V. V.: Qualitative Theory of Differential Equations, Dover Publ., New York, 1989 · Zbl 0089.29502
[27]	Rousseau, C., Schlomiuk, D.: Cubic vector fields symmetric with respect to a center. J. Differential Equations, 123, 388–436 (1995) · Zbl 0839.34035 · doi:10.1006/jdeq.1995.1168
[28]	Chavarriga, J., Giacomini, H., Giné, J., Llibre, J.: On the integrability of two-dimensional flows. J. Differential Equations, 157(1), 163–182 (1999) · Zbl 0940.37005 · doi:10.1006/jdeq.1998.3621
[29]	Cherkas, L. A.; Conditions for a center for a certain Liénard equation (Russian). Differencial’nye Uravnenija, 12(2), 292–298 (1976)

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.