Darboux first integrals and linearizability of quadratic–quintic Duffing–van der Pol oscillators

Abstract

Nonlinear oscillators described by polynomial Liénard differential equations arise in a variety of mathematical and physical applications. For a family of generalized Duffing–van der Pol oscillators we classify Darboux integrable cases and explicitly construct the corresponding generalized Darboux first integrals. We demonstrate that Darboux integrability is in strong correlation with the linearizability via the generalized Sundman transformations. We establish that the general solutions can be written in a parametric form. We prove that there are no limit cycles in integrable cases with autonomous Darboux first integrals.