A method to tackle first-order ordinary differential equations with Liouvillian functions in the solution

Duarte, L.G.S.; Duarte, Sérgio; Mota, L.A.C.P. da

doi:10.1088/0305-4470/35/17/306

Cited by 22 publications

(37 citation statements)

References 11 publications

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“…In [23,21,22] the authors give an algorithm to compute such integrating factors. The key point is the computation of exponential factors.…”

Section: Liouvillian First Integralsmentioning

confidence: 99%

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Chèze¹

2011

Journal of Complexity

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International audienceIn this paper we study planar polynomial differential systems of this form: dX/dt=A(X, Y ), dY/dt= B(X, Y ), where A,B belongs to Z[X, Y ], degA ≤ d, degB ≤ d, and the height of A and B is smaller than H. A lot of properties of planar polynomial differential systems are related to irreducible Darboux polynomials of the corresponding derivation: D =A(X, Y )dX + B(X, Y )dY . Darboux polynomials are usually computed with the method of undetermined coefficients. With this method we have to solve a polynomial system. We show that this approach can give rise to the computation of an exponential number of reducible Darboux polynomials. Here we show that the Lagutinskii-Pereira's algorithm computes irreducible Darboux polynomials with degree smaller than N, with a polynomial number, relatively to d, log(H) and N, binary operations. We also give a polynomial-time method to compute, if it exists, a rational first integral with bounded degree

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“…In [23,21,22] the authors give an algorithm to compute such integrating factors. The key point is the computation of exponential factors.…”

Section: Liouvillian First Integralsmentioning

confidence: 99%

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Chèze¹

2011

Journal of Complexity

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“…The classical result of the equivalence of the two representations of a Darbouxian first integral is proved in [30,13,17], [33,Satz 2], and in [34,Lemma 2 p. 205]. Singer [36] proves that a vector field admitting a first integral built by successive integrations, exponentiations and algebraic extensions of K(x, y) (so a Liouvillian function), also admits a Liouvillian first integral of the form given in Definition 1.…”

Section: (Eq)mentioning

confidence: 99%

“…With this approach if D 0 has k Darboux polynomials and the bound on the degree of i f ei i is bigger than k then we have to study at least 2 k situations. In [2,16,17], in order to find a Liouvillian first integral, the authors compute a Darbouxian integrating factor R = e P/Q i f ci i . With our approach the integrating factor R is related to the equation (L) in the following way:…”

Section: (Eq)mentioning

confidence: 99%

Symbolic Computations of First Integrals for Polynomial Vector Fields

Chèze

Combot

2019

Found Comput Math

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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

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“…In our previous two works (Chandrasekar et al 2005(Chandrasekar et al , 2006, we have studied in some detail the extended modified Prelle-Singer (PS) procedure (Prelle & Singer 1983;Duarte et al 2002), so as to apply it to a class of second-and thirdorder nonlinear ordinary differential equations (ODEs) and have solved several physically interesting nonlinear systems and identified a number of important linearization procedures. We now wish to extend the procedure to coupled ODEs.…”

Section: Introductionmentioning

confidence: 99%

On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

2008

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Continuing our study on the complete integrability of nonlinear ordinary differential equations (ODEs), in this paper we consider the integrability of a system of coupled first-order nonlinear ODEs of both autonomous and non-autonomous types. For this purpose, we modify the original Prelle-Singer (PS) procedure so as to apply it to both autonomous and non-autonomous systems of coupled first-order ODEs. We briefly explain the method of finding integrals of motion (time-independent as well as timedependent integrals) for two and three coupled first-order ODEs by extending the PS method. From this we try to answer some of the open questions in the original PS method. We also identify integrable cases for the two-dimensional Lotka-Volterra system and three-dimensional Rössler system as well as other examples including nonautonomous systems in a straightforward way using this procedure. Finally, we develop a linearization procedure for coupled first-order ODEs.

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A method to tackle first-order ordinary differential equations with Liouvillian functions in the solution

Cited by 22 publications

References 11 publications

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Computation of Darboux polynomials and rational first integrals with bounded degree in polynomial time

Symbolic Computations of First Integrals for Polynomial Vector Fields

On the complete integrability and linearization of nonlinear ordinary differential equations. III. Coupled first-order equations

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