A dynamical systems approach to WKB-methods: The simple turning point

TL;DR

This paper introduces a dynamical systems approach to the classical WKB method for second order differential equations with turning points, bridging hyperbolic and elliptic regimes using geometric singular perturbation theory.

Contribution

It presents a novel dynamical systems framework for WKB analysis at turning points, requiring only finite smoothness and connecting hyperbolic and elliptic theories.

Findings

Provides an alternative WKB approach based on dynamical systems.

Bridges hyperbolic and elliptic theories in slow-fast systems.

Applicable to singular perturbation problems with turning points.

Abstract

In this paper, we revisit the classical linear turning point problem for the second order differential equation $\epsilon^2 x'' +\mu(t)x=0$ with $\mu(0)=0,\,\mu'(0)\ne 0$ for $0<\epsilon\ll 1$. Written as a first order system, $t=0$ therefore corresponds to a turning point connecting hyperbolic and elliptic regimes. Our main result is that we provide an alternative approach to WBK that is based upon dynamical systems theory, including GSPT and blowup, and we bridge -- perhaps for the first time -- hyperbolic and elliptic theories of slow-fast systems. As an advantage, we only require finite smoothness of $\mu$. The approach we develop will be useful in other singular perturbation problems with hyperbolic-to-elliptic turning points.