Integrability and solvability of polynomial Li\'{e}nard differential systems
Abstract
We provide the necessary and sufficient conditions of Liouvillian integrability for Li\'{e}nard differential systems describing nonlinear oscillators with a polynomial damping and a polynomial restoring force. We prove that Li\'{e}nard differential systems are not Darboux integrable excluding subfamilies with certain restrictions on the degrees of the polynomials arising in the systems. We demonstrate that if the degree of a polynomial responsible for the restoring force is greater than the degree of a polynomial producing the damping, then a generic Li\'{e}nard differential system is not Liouvillian integrable with the exception of linear Li\'{e}nard systems. However, for any fixed degrees of the polynomials describing the damping and the restoring force we present subfamilies possessing Liouvillian first integrals. As a by-product of our results, we find a number of novel Liouvillian integrable subfamilies. In addition, we study the existence of non-autonomous Darboux first integrals and non-autonomous Jacobi last multipliers with a time-dependent exponential factor.
Cite
@article{arxiv.2110.14306,
title = {Integrability and solvability of polynomial Li\'{e}nard differential systems},
author = {Maria V. Demina},
journal= {arXiv preprint arXiv:2110.14306},
year = {2022}
}