Movable Singularities of Equations of Liénard Type

Filipuk, Galina; Halburd, R. G.

doi:10.1007/bf03321744

Cited by 9 publications

(20 citation statements)

References 7 publications

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“…The class of ODEs considered here arises from the fact that the function W which remains bounded as the singularity is approached is taken to be linear in y . This generalises the result for equations of Liénard type in [2].…”

Section: Resultssupporting

confidence: 87%

“…In this article we consider a class of equations of the form (1) with the quasi-Painlevé property where it suffices to take W to be linear in y . Although this class of equations does not include any of the Painlevé equations, it generalises the result in [2] for equations of Liénard type.…”

Section: Introductionmentioning

confidence: 82%

“…[5], [7], [9], and subsequently for classes of equations with the quasi-Painlevé property, e.g. [10], [1], [2] and [3]. Although the local structure of a solution of (1) is simple in the way that it is finitely branched about the movable singularities described here, its global structure can be complicated.…”

Section: Introductionmentioning

confidence: 99%

“…, where the functions in the numerators are evaluated at (ẑ, α(ẑ)). Equating the coefficient of (z −ẑ) −3/2 to zero gives c 2 1 = −f (ẑ, α(ẑ)), where the choice of branch for c 1 can be absorbed into the choice of branch for (z −ẑ) 1/2 . Equating the coefficient of (z −ẑ) −1 to zero and using (34) one obtains…”

mentioning

confidence: 99%

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A Class of Non-Linear ODEs with Movable Algebraic Singularities

Kecker

2012

Comput. Methods Funct. Theory

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A class of nonlinear second-order rational ODEs is studied for which it is shown that any movable singularity of a solution that can be reached along a finite length curve is an algebraic branch point. Some conditions need to be imposed on the equations including the existence of certain formal algebraic series solutions. An example is discussed demonstrating the degree of restriction for the parameters of the equation.

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Section: Resultssupporting

confidence: 87%

Section: Introductionmentioning

confidence: 82%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A Class of Non-Linear ODEs with Movable Algebraic Singularities

Kecker

2012

Comput. Methods Funct. Theory

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“…Of central importance in all of these works is the introduction of new variables in a neighborhood of a movable singularity, so that the original singularity can be characterized by an explicit regular initial value problem in the new variables. The proof of Theorem 1 presented here follows the same philosophy, only, as in earlier work by the authors [10,11], the existence of appropriate regular initial value problems is proved using recurrence relations for certain coefficient functions, without the need to write down the initial value problems explicitly. Other work on the Painlevé property for the Painlevé equations that does not explicitly use the isomonodromy problems for these equations (which would explicitly use their integrability) includes Joshi and Kruskal [12] and Hu and Yan [13].…”

Section: Introductionmentioning

confidence: 90%

Rational ODEs with Movable Algebraic Singularities

Filipuk

Halburd

2009

Stud Appl Math

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A class of second-order rational ordinary differential equations, admitting certain families of formal algebraic series solutions, is considered. For all solutions of these equations, it is shown that any movable singularity that can be reached by analytic continuation along a finite-length curve is an algebraic branch point. The existence of these formal series expansions is straightforward to determine for any given equation in the class considered. We apply the theorem to a family of equations, admitting different kinds of algebraic singularities. As a further application we recover the known fact for generic values of parameters that the only movable singularities of solutions of the Painlevé equations P II -P VI are poles.

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On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

Filipuk

Kecker

2021

Results Math

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The method of blowing up points of indeterminacy of certain systems of two ordinary differential equations is applied to obtain information about the singularity structure of the solutions of the corresponding non-linear differential equations. We first deal with the so-called Painlevé example, which passes the Painlevé test, but the solutions have more complicated singularities. Resolving base points in the equivalent system of equations we can explain the complicated structure of singularities of the original equation. The Smith example has a solution with non-isolated singularity, which is an accumulation point of algebraic singularities. Smith’s equation can be written as a system in two ways. We show that the sequence of blow-ups for both systems can be infinite. Another example that we consider is the Painlevé-Ince equation. When the usual Painlevé analysis is applied, it possesses both positive and negative resonances. We show that for three equivalent systems there is an infinite sequence of blow-ups and another one that terminates, which further gives a Laurent expansion of the solution around a movable pole. Moreover, for one system it is even possible to obtain the general solution after a sequence of blow-ups.

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Movable Singularities of Equations of Liénard Type

Cited by 9 publications

References 7 publications

A Class of Non-Linear ODEs with Movable Algebraic Singularities

A Class of Non-Linear ODEs with Movable Algebraic Singularities

Rational ODEs with Movable Algebraic Singularities

On Singularities of Certain Non-linear Second-Order Ordinary Differential Equations

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