A note on Liouvillian first integrals and invariant algebraic curves

Giné, Jaume; Grau, Maite; Llibre, Jaume

doi:10.1016/j.aml.2012.09.008

Search citation statements

Order By: Relevance

Paper Sections

Select...

Combining Theorem 12 Of [38] We Can Get the Next Result1

Remark1

Introduction1

Citation Types

Supporting

Mentioning

Contrasting

Year Published

2013

2018

Publication Types

Select...

Article6

Relationship

Self Cite3

Independent3

Authors

Journals

Cited by 6 publications

(3 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We remark that Theorem 1 and Corollary 2 are extension of Theorem 1 and Corollary 2 of [12] from two dimensional polynomial differential systems to any finite dimensional polynomial differential systems.…”

Section: Combining Theorem 12 Of [38] We Can Get the Next Resultmentioning

confidence: 99%

Liouvillian Integrability Versus Darboux Polynomials

Llibre

Valls²,

Zhang

2016

Qual. Theory Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Combining Theorem 12 Of [38] We Can Get the Next Resultmentioning

confidence: 99%

Liouvillian Integrability Versus Darboux Polynomials

Llibre

Valls²,

Zhang

2016

Qual. Theory Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

show abstract

“…From Odani's result [28], it easily follows that system (5) does not have any finite invariant algebraic curve. The existence of finite invariant algebraic curves for an arbitrary planar differential system is shown in [15,17,18]. From Singer's results system (5) is Liouvillian integrable if it admits an integrating factor of the form (2).…”

Section: Remarkmentioning

confidence: 99%

On the extensions of the Darboux theory of integrability

2013

Self Cite

View full text Add to dashboard Cite

Recently some extensions of the classical Darboux integrability theory to autonomous and non-autonomous polynomial vector fields were completed. The classical Darboux integrability theory and its recent extensions are based on the existence of algebraic invariant hypersurfaces. However the algebraicity of the invariant hypersurfaces is not necessary and the unique necessary condition is the algebraicity of the cofactors associated to them. In this paper a more general extension of the classical Darboux integrability theory is established.

show abstract

“…A solution of system (1.3) has either empty intersection with the zero set of F or it is entirely contained in F = 0. Existence of invariant algebraic curves is a substantial measure of integrability, for more details see, for instance [11][12][13][14]. It is an important problem to classify all irreducible invariant algebraic curves of a dynamical system.…”

Section: Introductionmentioning

confidence: 99%

Invariant algebraic curves for Liénard dynamical systems revisited

Demina

2018

Applied Mathematics Letters

View full text Add to dashboard Cite

A novel algebraic method for finding invariant algebraic curves for a polynomial vector field in C 2 is introduced. The structure of irreducible invariant algebraic curves for Liénard dynamical systems x t = y, y t = −f (x)y − g(x) with deg g = deg f + 1 is obtained. It is shown that there exist Liénard systems that possess more complicated invariant algebraic curves than it was supposed before. As an example, all irreducible invariant algebraic curves for the Liénard differential system with deg f = 2 and deg g = 3 are obtained. All these results seem to be new.

show abstract

A note on Liouvillian first integrals and invariant algebraic curves

Cited by 6 publications

References 12 publications

Liouvillian Integrability Versus Darboux Polynomials

Liouvillian Integrability Versus Darboux Polynomials

On the extensions of the Darboux theory of integrability

Invariant algebraic curves for Liénard dynamical systems revisited

Contact Info

Product

Resources

About