# A class of integrable polynomial vector fields

Applicationes Mathematicae (1995)

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

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topChavarriga, 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

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- [2] J. Chavarriga, Integrable systems in the plane with a center type linear part, Appl. Math. (Warsaw) 22 (1994), 285-309. Zbl0809.34002
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- [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] 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] S. Shi, A method of constructing cycles without contact around a weak focus, J. Differential Equations 41 (1981), 301-312. Zbl0442.34029
- [8] H. Żołądek, On a certain generalization of Bautin's Theorem, preprint, Institute of Mathematics, University of Warsaw, 1991. Zbl0838.34035

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