Holomorphic foliations in ${\Bbb C}{\Bbb P}(2)$ having an invariant algebraic curve

Cerveau, Dominique; Neto, Alcides Lins

doi:10.5802/aif.1278

Cited by 125 publications

(125 citation statements)

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“…To see the relation between the functional classes of the inverse integrating factors and their associated first integrals see Theorem 3 of [14], see also [7]. Many works deal with the integrability problem of a system of the form (2), see for instance [3,5,7,20,24] and references therein. The main idea of these works is to consider the singular points of the system and to give necessary conditions on the eigenvalues associated to the linear part of each singular point in order that the system has a first integral of a particular functional class.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

Giné¹,

Grau²,

Llibre³

2014

Publicacions Matemàtiques

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

Giné¹,

Grau²,

Llibre³

2014

Publicacions Matemàtiques

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“…A natural question which goes back to Poincaré [36] is: to give an effective procedure to find N . Partial answers to this question were given by Cerveau and Lins Neto [3], Carnicer [2], Campillo and Carnicer [1], and Walcher [42]. These results depend on either restricting the nature of the polynomial differential system, or more specifically on the singularities of its invariant algebraic curves.…”

Section: An Open Question For Planar Polynomial Vector Fieldsmentioning

confidence: 97%

On the Darboux Integrability of Polynomial Differential Systems

Llibre

Zhang

2011

Qual. Theory Dyn. Syst.

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Abstract. This is a survey on recent results on the Darboux integrability of polynomial vector fields in R n or C n with n ≥ 2. We also provide an open question and some applications based on the existence of such first integrals. IntroductionIn many branches of applied mathematics, physics and, in general, in applied sciences appear nonlinear ordinary differential equations. If a differential equation or vector field defined on a real or complex manifold has a first integral, then its study can be reduced by one dimension. Therefore a natural question is: Given a vector field on a manifold, how to recognize if this vector field has a first integral defined on an open and dense subset of the manifold? In general this question has no a good answer up to now.In this survey we provide sufficient conditions for the existence of a first integral for polynomial vector fields in R n or C n with n ≥ 2. An open question which essentially goes back to Poincaré is presented. Finally some applications of the existence of this kind of first integrals to physical problems, centers, foci, limit cycles and invariant hyperplanes are mentioned. Darboux theory of integrabilitySince any polynomial differential system in R n can be thought as a polynomial differential system inside C n we shall work only in C n . If our initial differential system is in R n , once we get a complex first integral of this system inside C n the real and the imaginary parts of it are real first integrals. Moreover if that complex first integral is rational, the same occur for its real and imaginary parts. In short in the rest of the paper we shall work in C n .In this section we study the existence of first integrals for polynomial vector fields in C n through the Darboux theory of integrability. The algebraic theory of integrability is a classical one, which is related with the first part of the Hilbert's 16th problem [19]. This kind of integrability is usually called Darboux integrability, and it provides a link between the integrability of polynomial vector fields and the number of invariant algebraic hypersurfaces that they have (see Darboux [13] and Poincaré [36]). This theory shows the fascinating relationships between integrability (a topological phenomenon) and the existence of exact algebraic invariant hypersurfaces formed by solutions for the polynomial vector field. This theory is now known as the Darboux theory of integrability, see for more details the Chapter 8 of [16].2010 Mathematics Subject Classification. 34A34, 34C05, 34C14.

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“…A natural question, going back to Poincaré [16], is to give an effective procedure to find N . Partial answers to this question were given by Cerveau and Lins Neto [4], Carnicer [2], and Campillo and Carnicer [1]. These results depend on either restricting the nature of the foliation given by (1) or the singularities of the algebraic curves.…”

Section: Introductionmentioning

confidence: 97%

A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

Christopher¹,

Llibre²

2001

Proc. Amer. Math. Soc.

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Holomorphic foliations in ${\Bbb C}{\Bbb P}(2)$ having an invariant algebraic curve

Cited by 125 publications

References 5 publications

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

Polynomial and rational first integrals for planar homogeneous polynomial differential systems

On the Darboux Integrability of Polynomial Differential Systems

A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals

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