Hypergeometric First Integrals of the Duffing and van der Pol Oscillators
Mathematical Physics
2019-03-08 v3 math.MP
Exactly Solvable and Integrable Systems
Abstract
The autonomous Duffing oscillator, and its van der Pol modification, are known to admit time-dependent first integrals for specific values of parameters. This corresponds to the existence of Darboux polynomials, and in fact more can be shown: that there exist Liouvillian first integrals which do not depend on time. They can be expressed in terms of the Gauss and Kummer hypergeometric functions, and are neither analytic, algebraic nor meromorphic. A criterion for this to happen in a general dynamical system is formulated as well.
Keywords
Cite
@article{arxiv.1706.02506,
title = {Hypergeometric First Integrals of the Duffing and van der Pol Oscillators},
author = {Tomasz Stachowiak},
journal= {arXiv preprint arXiv:1706.02506},
year = {2019}
}
Comments
20 pages, 4 color figures; 3rd version revised and accepted in JDE