Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields

Schlomiuk, Dana

doi:10.1007/978-94-015-8238-4_10

Cited by 106 publications

(84 citation statements)

References 26 publications

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“…The aim of this paper is to find the integrable cases of system (1.1) when s = 5 (see Theorem 1). The integrable cases for quadratic systems, s = 2, and cubic homogeneous systems, s = 3, have been studied by several authors Bautin [1], Chavarriga [2], Coppel [5], Lloyd [6], Lunkevich and Sibirskii [7], Schlomiuk [9] andŻoladek [13]. Some integrable cases of system (1.1) when s = 4 have been determinated by Chavarriga and Giné [4].…”

Section: Introductionmentioning

confidence: 99%

Integrability of linear center perturbed by a fifth degree homogeneous polynomial

Mesa¹,

Soriano²

1997

Publicacions Matemàtiques

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In this work we study the integrability of two-dimensional autonomous system in the plane with linear part of center type and non-linear part given by homogeneous polynomials of fifth degree. We give a simple characterisation for the integrable cases in polar coordinates. Finally we formulate a conjecture about the independence of the two classes of parameters which appear on the system; if this conjecture is true the integrable cases found will be the only possible ones.

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Section: Introductionmentioning

confidence: 99%

Integrability of linear center perturbed by a fifth degree homogeneous polynomial

Mesa¹,

Soriano²

1997

Publicacions Matemàtiques

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“…The characterization of the centers of the polynomial differential systems started with the classes of all the quadratic polynomial differential systems and the linear polynomial systems with homogeneous polynomial nonlinearities of degree 3, see for instance [1,27,28,29,30]. Unfortunately in the present we are very far from having the classification of all the centers of the cubic polynomial differential systems.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 97%

Centers for a class of generalized quintic polynomial differential systems

Giné

Llibre

Valls³

2014

Applied Mathematics and Computation

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“…This theory started with Darboux [10] in 1878. For more details and results on the Darbouxian theory of integrability for planar polynomial vector fields, see [1,3,4,6,12,[14][15][16][17][18][19]. Here, we study the inverse problem.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Polynomial differential systems having a given Darbouxian first integral*1

Llibre

2004

Bulletin des Sciences Math�matiques

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The Darbouxian theory of integrability allows to determine when a polynomial differential system in C 2 has a first integral of the kind f Here, we solve the inverse problem, i.e. we characterize the polynomial vector fields in C 2 having a given Darbouxian function as a first integral.On the other hand, using information about the degree of the invariant algebraic curves of a polynomial vector field, we improve the conditions for the existence of an integrating factor in the Darbouxian theory of integrability.

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Algebraic and Geometric Aspects of the Theory of Polynomial Vector Fields

Cited by 106 publications

References 26 publications

Integrability of linear center perturbed by a fifth degree homogeneous polynomial

Integrability of linear center perturbed by a fifth degree homogeneous polynomial

Centers for a class of generalized quintic polynomial differential systems

Polynomial differential systems having a given Darbouxian first integral*1

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