Regularising transformations for the \((n,n+1)\)-Liénard equations
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.
scientific article; zbMATH DE number 7621076