Regularising transformations for the $$(n,\,n+1)$$-Liénard equations

Abstract

We discuss regularising transformations for the Liénard equations of the form $$y''=F(z,y)y'+G(z,y)$$ y ′ ′ = F ( z , y ) y ′ + G ( z , y ) , where F and G are polynomials of degrees n and $$n+1$$ n + 1 respectively using the geometric approach. As a particular case we find a transformation for the Duffing–van der Pol equation which leads to the regularisation.