A class of integrable polynomial vector fields

• Autorzy: Javier Chavarriga
• 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).

Słowa kluczowe

EN

integrable systems in the plane; center-focus problem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1995-1996
• Tom: 23
• Numer: 3
• Strony: 339-350

Daty

wydano

1995

otrzymano

1994-12-20

Twórcy

autor

Javier Chavarriga

• 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.