Integrable systems in the plane with center type linear part

Javier Chavarriga

Applicationes Mathematicae (1994)

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

Abstract

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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.

How to cite

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Chavarriga, 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

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  1. [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. [2] W. A. Coppel, A survey of quadratic systems, J. Differential Equations 2 (1966), 293-304. Zbl0143.11903
  3. [3] C. Li, Two problems of planar quadratic systems, Sci. Sinica Ser. A 26 (1983), 471-481. Zbl0534.34033
  4. [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. [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. [6] V. V. Nemytskiĭ and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, N.J., 1960. Zbl0089.29502
  7. [7] S. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312. Zbl0442.34029
  8. [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. [9] K. S. Sibirskiĭ, Introduction to the Algebraic Theory of Invariant Differential Equations, Manchester University Press, New York, 1988. 
  10. [10] H. Żołądek, On certain generalization of Bautin's Theorem, preprint, Institute of Mathematics, University of Warsaw, 1991. Zbl0838.34035

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