Exponential asymptotics and higher-order Stokes phenomena in singularly perturbed ODEs

Outside the area of exponential asymptotics, the concept of the higher-order Stokes phenomenon remains somewhat esoteric. The intention of this work is to provide several examples of relatively simple ordinary differential equations where the phenomenon arises, and to develop additional practical methodologies for its study.

In particular, we show how the higher-order Stokes phenomenon may be derived through the hyperterminant representation of the late-term divergence of an asymptotic expansion developed by Olde Daalhuis (J. Comp. Appl. Math. vol. 76, 1996, pp. 255-264). Lower-order components of the factorial-over-power divergence are considered, for which late-late-term divergence arises. Borel resummation of these lower-order components reveals the higher-order Stokes phenomenon, in which new components of the late-terms of the expansion are smoothly switched on with an error function dependence. The techniques are firstly demonstrated with a second-order linear inhomogeneous ODE that exemplifies the simplest example of higher-order Stokes phenomena. Further examples studied include higher-order equations and eigenvalue problems.