A class of integrable polynomial vector fields

Javier Chavarriga

Applicationes Mathematicae (1995)

  • Volume: 23, Issue: 3, page 339-350
  • ISSN: 1233-7234

Abstract

top
We study the integrability of two-dimensional autonomous systems in the plane of the form = - y + X s ( x , y ) , = x + Y s ( x , y ) , where Xs(x,y) and Ys(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).

How to cite

top

Chavarriga, Javier. "A class of integrable polynomial vector fields." Applicationes Mathematicae 23.3 (1995): 339-350. <http://eudml.org/doc/219136>.

@article{Chavarriga1995,
abstract = {We study the integrability of two-dimensional autonomous systems in the plane of the form $=-y+X_s(x,y)$, $=x+Y_s(x,y)$, where Xs(x,y) and Ys(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).},
author = {Chavarriga, Javier},
journal = {Applicationes Mathematicae},
keywords = {integrable systems in the plane; center-focus problem; particular solutions; integrability; two-dimensional autonomous systems; homogeneous polynomials},
language = {eng},
number = {3},
pages = {339-350},
title = {A class of integrable polynomial vector fields},
url = {http://eudml.org/doc/219136},
volume = {23},
year = {1995},
}

TY - JOUR
AU - Chavarriga, Javier
TI - A class of integrable polynomial vector fields
JO - Applicationes Mathematicae
PY - 1995
VL - 23
IS - 3
SP - 339
EP - 350
AB - We study the integrability of two-dimensional autonomous systems in the plane of the form $=-y+X_s(x,y)$, $=x+Y_s(x,y)$, where Xs(x,y) and Ys(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).
LA - eng
KW - integrable systems in the plane; center-focus problem; particular solutions; integrability; two-dimensional autonomous systems; homogeneous polynomials
UR - http://eudml.org/doc/219136
ER -

References

top
  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 of type (R), Mat. Sb. 30 (72) (1952), 181-196 (in Russian); English transl.: Amer. Math. Soc. Transl. 100 (1954), 397-413. Zbl0059.08201
  2. [2] J. Chavarriga, Integrable systems in the plane with a center type linear part, Appl. Math. (Warsaw) 22 (1994), 285-309. Zbl0809.34002
  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: 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] 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. Zbl0790.34031
  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] H. Żołądek, On a certain generalization of Bautin's Theorem, preprint, Institute of Mathematics, University of Warsaw, 1991. Zbl0838.34035

NotesEmbed ?

top

You must be logged in to post comments.