Integrable systems in the plane with center type linear part

• Autorzy: Javier Chavarriga
• Języki publikacji: EN

Abstrakty

EN

We study integrability of two-dimensional autonomous systems in the plane with center type linear part. For quadratic and homogeneous cubic systems we give a simple characterization for integrable cases, and we find explicitly all first integrals for these cases. Finally, two large integrable system classes are determined in the most general nonhomogeneous cases.

Słowa kluczowe

EN

integrable systems in the plane; center-focus problem

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Applicationes Mathematicae
• Rocznik: 1993-1995
• Tom: 22
• Numer: 2
• Strony: 285-309

Daty

wydano

1994

otrzymano

1993-05-05

Twórcy

autor

Javier Chavarriga

• Departamento de Matematicas, Escuela Técnica Superior de Ingeniería Agraria, Universidad de Lérida, C/ Rovira Roure, 177, 25006 Lérida, 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 type, Mat. Sb. 30 (72) (1952), 181-196 (in Russian); English transl. in Amer. Math. Soc. Transl. 100 (1954).
• [2] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304.
• [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: Ordinary Differential Equations and Operators, 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] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J., 1960.
• [7] S. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312.
• [8] S. Shi, On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields, ibid. 52 (1984), 52-57.
• [9] K. S. Sibirskiĭ, Introduction to the Algebraic Theory of Invariant Differential Equations, Manchester University Press, New York, 1988.
• [10] H. Żołądek, On certain generalization of Bautin's Theorem, preprint, Institute of Mathematics, University of Warsaw, 1991.