On the convergence of WKB approximations of the damped Mathieu equation
- 1 June 2021
- journal article
- research article
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (6), 062702
- https://doi.org/10.1063/1.5145267
Abstract
The form of the fundamental set of solutions of the damped Mathieu equation is determined by Floquet theory. In the limit as m → 0, we can apply WKB theory to get first order approximations of the fundamental set. WKB theory states that this approximation gets better as m → 0 and T is fixed. However, convergence of the periodic part and characteristic exponent is not addressed. We show that they converge to those predicted by WKB theory. We also provide a rate of convergence that is not dependent on T.Keywords
Funding Information
- National Science Foundation (DMS 1615045)
This publication has 6 references indexed in Scilit:
- Overdamped dynamics of a Brownian particle levitated in a Paul trapPhysical Review A, 2020
- The stochastic Mathieu's equationProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2008
- Stability of the Damped Mathieu Equation With Time DelayJournal of Dynamic Systems, Measurement, and Control, 2003
- The positivity of the Lyapunov exponent and the absence of the absolutely continuous spectrum for the almost-Mathieu equationJournal of Mathematical Physics, 1984
- Resonant excitation of motion perpendicular to galactic planesMonthly Notices of the Royal Astronomical Society, 1981
- Error bounds for the Liouville–Green (or WKB) approximationMathematical Proceedings of the Cambridge Philosophical Society, 1961