Nonlocal deformations of autonomous invariant curves for Liénard equations with quadratic damping

Abstract

We consider a family of nonlinear oscillators with quadratic damping, that generalizes the Liénard equation. We show that certain nonlocal transformations preserve autonomous invariant curves of equations from this family. Thus, nonlocal transformations can be used for extending known classification of invariant curves to the whole equivalence class of the corresponding equation, which includes non-polynomial equations. Moreover, we demonstrate that an autonomous first integral for one of two non-locally related equations can be constructed in the parametric form from the general solution of the other equation. In order to illustrate our results, we construct two integrable subfamilies of the considered family of equations, that are non-locally equivalent to two equations from the Painlevé–Gambier classification. We also discuss several particular members of these subfamilies, including a traveling wave reduction of a nonlinear diffusion equation, and construct their invariant curves and first integrals.