A New Algorithm for Finding Rational First Integrals of Polynomial Vector Fields

Ferragut, Antoni; Giacomini, Héctor

doi:10.1007/s12346-010-0021-x

Cited by 24 publications

(25 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…All the Puiseux series that solve equation (2.1) can be obtained with the help of the Painlevé methods, see, for example [12,[18][19][20][21][22][23]. Some methods and algorithms related to first integrals, algebraic functions, and Puiseux series are described in [24][25][26][27].…”

Section: Proof Of Main Resultsmentioning

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

Section: Proof Of Main Resultsmentioning

confidence: 99%

Invariant algebraic curves for Liénard dynamical systems revisited

Demina

2018

Applied Mathematics Letters

View full text Add to dashboard Cite

show abstract

“…Thus our bound on the degree of F corresponds to a bound on the degree of the polynomials P, Q, f i . In [19], Ferragut and Giacomini have proposed a method to compute rational first integrals with bounded degree. This approach does not follow the previous strategy.…”

Section: (Eq)mentioning

confidence: 99%

“…The algorithm proposed in this article is based on a generalization of the extactic curve and follows the idea used in [19] and [5]. We give then a uniform strategy with a uniform complexity to compute rational, Darbouxian, Liouvillian, and Riccati first integrals.…”

Section: (Eq)mentioning

confidence: 99%

Symbolic Computations of First Integrals for Polynomial Vector Fields

Chèze

Combot

2019

Found Comput Math

View full text Add to dashboard Cite

In this article we show how to generalize to the Darbouxian, Liouvillian and Riccati case the extactic curve introduced by J. Pereira. With this approach, we get new algorithms for computing, if it exists, a rational, Darbouxian, Liouvillian or Riccati first integral with bounded degree of a polynomial planar vector field. We give probabilistic and deterministic algorithms. The arithmetic complexity of our probabilistic algorithm is inÕ(N ω+1 ), where N is the bound on the degree of a representation of the first integral and ω ∈ [2; 3] is the exponent of linear algebra. This result improves previous algorithms. Our algorithms have been implemented in Maple and are available on authors' websites. In the last section, we give some examples showing the efficiency of these algorithms. First integrals and Symbolic computations and Complexity analysis 1

show abstract

“…The remarkable values and remarkable curves of rational first integrals of planar differential systems were first introduced by Poincaré in [23], and afterwards studied by several authors, see [8,10,11]. It has been shown in the literature that the remarkable curves play an important role in the phase portrait as they are strongly related to its separatrices.…”

Section: Remarkable Values Of Rational First Integralsmentioning

confidence: 99%

Detection of Special Curves Via the Double Resultant

Ferragut

García-Saldaña

Gasull

2015

Qual. Theory Dyn. Syst.

Self Cite

View full text Add to dashboard Cite

Abstract. We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.

show abstract

A New Algorithm for Finding Rational First Integrals of Polynomial Vector Fields

Abstract: Abstract. We present a new method to compute rational first integrals of a planar polynomial vector field. The algorithm is in general much faster than the usual methods and also allows to compute the remarkable curves associated to the rational first integral of the system.

Cited by 24 publications

References 20 publications

Invariant algebraic curves for Liénard dynamical systems revisited

Invariant algebraic curves for Liénard dynamical systems revisited

Symbolic Computations of First Integrals for Polynomial Vector Fields

Detection of Special Curves Via the Double Resultant

Contact Info

Product

Resources

About