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

In this paper, we revisit the classical linear turning point problem for the second order differential equation ${\epsilon }^2{x}^{\mathrm{\prime }\mathrm{\prime }}+\mu (t)x=0$ with $\mu (0)=0,{\mu }^{\mathrm{\prime }}(0)\mathrm{\ne }0$ 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.