Integrability and solvability of polynomial Liénard differential systems

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 wi...

Full description

Saved in:
Bibliographic Details
Published in:Studies in applied mathematics (Cambridge) 2023-04, Vol.150 (3), p.755-817
Main Author: Demina, Maria V.
Format: Article
Language:English
Subjects:
Damping
Darboux integrability
Integral calculus
Integrals
invariant algebraic curves
Liouvillian integrability
Liénard differential systems
Nonlinear systems
Polynomials
Puiseux series
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 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 Liénard differential system is not Liouvillian integrable with the exception of linear Liénard 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 nonautonomous Darboux first integrals and nonautonomous Jacobi last multipliers with a time‐dependent exponential factor.
ISSN:0022-2526
1467-9590
DOI:10.1111/sapm.12556