Regularising transformations for the \((n,n+1)\)-Liénard equations
type: Item
instance of: scholarly article
title: Regularising transformations for the \((n,n+1)\)-Liénard equations
author: Galina Filipuk
published in: Letters in Mathematical Physics
publication date: 2022-11-21T00:00:00Z
review text: In this article, the author is interested in regularizing transformations for the Liénard equations of the form \[ y''=F(z,y)y'+G(z,y), \tag{L} \] where \(F\) and \(G\) are polynomials in \(y\) of degrees \(n\) and \(n+1\) respectively. It is well-known that such equations have movable singularities, the locations of which depend on initial conditions. In [\textit{G. Filipuk} and \textit{R. G. Halburd}, Comput. Methods Funct. Theory 9, No. 2, 551--563 (2009; Zbl 1194.34165)], these ones were studied under certain additional resonance condition. In particular, it was proved that any movable singularity of \(y\) that can be reached by analytic continuation along a finite length curve is algebraic. In the present paper, the author proposes to revisit this result by using a geometric approach. In particular, she constructs for \(n=1,2,3\) explicit transformations for regularizing Eq. (L) and, as an application, she displays a transformation for the Duffing-van der Pol equation which leads to its regularization.
reviewed by: Pascal Remy
Mathematics Subject Classification ID: 34M35
Mathematics Subject Classification ID: 34B30
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 34M04
Mathematics Subject Classification ID: 34M25
zbMATH DE Number: 7621076
zbMATH Keywords: algebraic singularities
zbMATH Keywords: movable singularities
zbMATH Keywords: Liénard equations
MaRDI profile type: Publication
full work available at URL: s11005-022-01606-2
OpenAlex ID: W4308398099
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DOI: 10.1007/S11005-022-01606-2
description: scientific article; zbMATH DE number 7621076