Local Darboux first integrals of analytic differential systems

Llibre, Jaume; Walcher, Sebastian; Zhang, Xiang

doi:10.1016/j.bulsci.2013.09.007

Cited by 10 publications

(10 citation statements)

References 23 publications

(32 reference statements)

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“…Darboux theory of integrability was started in 1878 by Darboux [7,8], who provided a method to construct first integral using Darboux polynomials. This theory has been greatly developed in different aspects, see for instance [2,5,6,10,11,13,22,28,25,26,27,39] and the references therein. The Darboux theory of integrability has also been broadly applied to study dynamics of differential systems, for instance the center problem, limit cycles and global dynamical analysis, see for example [5,9,17,18,31,35,36,37] and the references quoted there.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Liouvillian Integrability Versus Darboux Polynomials

Llibre

Valls²,

Zhang

2016

Qual. Theory Dyn. Syst.

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

Liouvillian Integrability Versus Darboux Polynomials

Llibre

Valls²,

Zhang

2016

Qual. Theory Dyn. Syst.

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“…Non-algebraic invariant curves with polynomial cofactor can also be used in order to find a first integral for a system. This observation permits the generalization of the Liouvillian integrability theory given in [5][6][7]10], where a new kind of first integrals, not only the Liouvillian ones, is described.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

On Liouvillian integrability of the first–order polynomial ordinary differential equations

Giné¹,

Llibre²

2012

Journal of Mathematical Analysis and Applications

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a b s t r a c tRecently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Giné and Llibre (2012) [8]. In this note, we prove that, if a complex differential equation of the form y ′ = a 0 (x) + a 1 (x)y + · · · + a n (x)y n , with a i (x) polynomials for i = 0, 1, . . . , n, a n (x) ̸ = 0, and n ≥ 2, has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1, i.e., for linear polynomial differential equations.

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“…But such a first integral would also be a first integral of ẋ = Bx (see e.g. [22], Theorem 1), hence a quotient of power series in γ . But then γ is a first integral.…”

Section: Degree Bounds For Jacobi Multipliersmentioning

confidence: 99%

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

et al. 2021

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We discuss criteria for the nonexistence, existence and computation of invariant algebraic surfaces for three-dimensional complex polynomial vector fields, thus transferring a classical problem of Poincaré from dimension two to dimension three. Such surfaces are zero sets of certain polynomials which we call semi-invariants of the vector fields. The main part of the work deals with finding degree bounds for irreducible semi-invariants of a given polynomial vector field that satisfies certain properties for its stationary points at infinity. As a related topic, we investigate existence criteria and properties for algebraic Jacobi multipliers. Some results are stated and proved for polynomial vector fields in arbitrary dimension and their invariant hypersurfaces. In dimension three we obtain detailed results on possible degree bounds. Moreover by an explicit construction we show for quadratic vector fields that the conditions involving the stationary points at infinity are generic but they do not a priori preclude the existence of invariant algebraic surfaces. In an appendix we prove a result on invariant lines of homogeneous polynomial vector fields.

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Local Darboux first integrals of analytic differential systems

Cited by 10 publications

References 23 publications

Liouvillian Integrability Versus Darboux Polynomials

Liouvillian Integrability Versus Darboux Polynomials

On Liouvillian integrability of the first–order polynomial ordinary differential equations

Invariant Algebraic Surfaces of Polynomial Vector Fields in Dimension Three

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