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1995-1996 | 23 | 3 | 339-350
Tytuł artykułu

A class of integrable polynomial vector fields

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We study the integrability of two-dimensional autonomous systems in the plane of the form $\dotx=-y+X_s(x,y)$, $\doty=x+Y_s(x,y)$, where X_s(x,y) and Y_s(x,y) are homogeneous polynomials of degree s with s≥2. First, we give a method for finding polynomial particular solutions and next we characterize a class of integrable systems which have a null divergence factor given by a quadratic polynomial in the variable $(x^2+y^2)^{s/2-1}$ with coefficients being functions of tan^{−1}(y/x).
Rocznik
Tom
23
Numer
3
Strony
339-350
Opis fizyczny
Daty
wydano
1995
otrzymano
1994-12-20
Twórcy
  • Departament de Matemàtica, Escola Técnica Superior D'Enginyeria Agrària, Universitat de Lleida, Avda. Alcalde Rovira Roure, 177, 25006 Lleida, Spain
Bibliografia
  • [1] N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center of type (R), Mat. Sb. 30 (72) (1952), 181-196 (in Russian); English transl.: Amer. Math. Soc. Transl. 100 (1954), 397-413.
  • [2] J. Chavarriga, Integrable systems in the plane with a center type linear part, Appl. Math. (Warsaw) 22 (1994), 285-309.
  • [3] C. Li, Two problems of planar quadratic systems, Sci. Sinica Ser. A 26 (1983), 471-481.
  • [4] N. G. Lloyd, Small amplitude limit cycles of polynomial differential equations, in: Lecture Notes in Math. 1032, Springer, 1983, 346-356.
  • [5] V. A. Lunkevich and K. S. Sibirskiĭ , Integrals of a general quadratic differential system in cases of a center, Differential Equations 18 (1982), 563-568.
  • [6] D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in: Bifurcations and Periodic Orbits of Vector Fields, Kluwer Acad. Publ., 1993, 429-467.
  • [7] S. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312.
  • [8] H. Żołądek, On a certain generalization of Bautin's Theorem, preprint, Institute of Mathematics, University of Warsaw, 1991.
Typ dokumentu
Bibliografia
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Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv23i3p339bwm
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