# Integrable systems in the plane with center type linear part

Applicationes Mathematicae (1994)

- Volume: 22, Issue: 2, page 285-309
- ISSN: 1233-7234

## Access Full Article

top## Abstract

top## How to cite

topChavarriga, Javier. "Integrable systems in the plane with center type linear part." Applicationes Mathematicae 22.2 (1994): 285-309. <http://eudml.org/doc/219096>.

@article{Chavarriga1994,

abstract = {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.},

author = {Chavarriga, Javier},

journal = {Applicationes Mathematicae},

keywords = {integrable systems in the plane; center-focus problem; integrability; two-dimensional autonomous systems; center type linear part; first integrals},

language = {eng},

number = {2},

pages = {285-309},

title = {Integrable systems in the plane with center type linear part},

url = {http://eudml.org/doc/219096},

volume = {22},

year = {1994},

}

TY - JOUR

AU - Chavarriga, Javier

TI - Integrable systems in the plane with center type linear part

JO - Applicationes Mathematicae

PY - 1994

VL - 22

IS - 2

SP - 285

EP - 309

AB - 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.

LA - eng

KW - integrable systems in the plane; center-focus problem; integrability; two-dimensional autonomous systems; center type linear part; first integrals

UR - http://eudml.org/doc/219096

ER -

## References

top- [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). Zbl0059.08201
- [2] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304. Zbl0143.11903
- [3] C. Li, Two problems of planar quadratic systems, Sci. Sinica Ser. A 26 (1983), 471-481. Zbl0534.34033
- [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. Zbl0499.34017
- [6] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J., 1960. Zbl0089.29502
- [7] S. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312. Zbl0442.34029
- [8] S. Shi, On the structure of Poincaré-Lyapunov constants for the weak focus of polynomial vector fields, ibid. 52 (1984), 52-57. Zbl0534.34059
- [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. Zbl0838.34035

## Citations in EuDML Documents

top## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.