Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations

A technique for calculating exponentially small terms beyond all orders in singularly perturbed ordinary differential equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line-smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples.