On the convergence of WKB approximations of the damped Mathieu equation

Abstract

Consider the differential equation ${ m\ddot{x} +\gamma \dot{x} -x\epsilon \cos(\omega t) =0}$, $0 \leq t \leq T$. The form of the fundamental set of solutions are determined by Floquet theory. In the limit as $m \to 0$ we can apply WKB theory to get first order approximations of this fundamental set. WKB theory states that this approximation gets better as $m \to 0$ in the sense that the difference in sup norm is bounded as function of $m$ for a given $T$. However, convergence of the periodic parts and exponential parts are not addressed. We show that there is convergence to these components. The asymptotic error for the characteristic exponents are $O(m^2)$ and $O(m)$ for the periodic parts.