On the integrability of some forced nonlinear oscillators

Abstract

We consider integrability properties of a family of forced nonlinear oscillators, which generalizes the Liénard equation. We demonstrate that some forced oscillators with previously known first integrals can be linearized via certain nonlocal transformations. Furthermore, we show that the whole family of Liénard (n,n+1) equations with arbitrary external forcing admits a first integral. We study in detail the case of the Liénard (3,4) equation due to its value for applications. We prove that despite the fact that this equation possesses one Darboux first integral and can be linearized, it does not have an additional Darboux integral and, hence, is not Darboux integrable. Therefore, we demonstrate that certain nonlocal transformations do not preserve the property of Darboux integrability.