Integrability and solvability of polynomial Liénard differential systems

Abstract

We provide the necessary and sufficient conditions of Liouvillian integrability for Liénard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Liénard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of polynomials arising in the systems. It is demonstrated that if the degree of a polynomial responsible for the restoring force is higher than the degree of a polynomial producing the damping, then a generic Liénard differential system is not Liouvillian integrable with the exception for a linear Liénard system. However, for any fixed degrees of the polynomials describing the damping and the restoring force there always exist subfamilies possessing Liouvillian first integrals. A number of novel Liouvillian integrable subfamilies parameterized by an arbitrary polynomial are presented. In addition, we study the existence of non-autonomous Darboux first integrals and non-autonomous Jacobi last multipliers with a time-dependent exponential factor.