Exponentially Accurate Solution Tracking For Nonlinear Odes, The Higher Order Stokes Phenomenon And Double Transseries Resummation

We demonstrate the conjunction of new exponential-asymptotic effects in the context of a second order nonlinear ordinary differential equation (ODE) with a small parameter. First, we show how to use a hyperasymptotic, beyond-all-orders approach to seed a numerical solver of a nonlinear ODE with sufficiently accurate initial data so as to track a specific solution in the presence of an attractor. Second, we demonstrate the necessary role of a higher order Stokes phenomenon in analytically tracking the transition between asymptotic behaviours in a heteroclinic solution. Third, we carry out a double resummation involving both subdominant and sub-subdominant transseries to achieve the two-dimensional (in terms of the arbitrary constants) uniform approximation that allows the exploration of the behaviour of a two-parameter set of solutions across wide regions of the independent variable. This is the first time all three effects have been studied jointly in the context of an asymptotic treatment of a nonlinear ODE with a parameter. This paper provides an exponential-asymptotic algorithm for attacking such problems when they occur. The availability of explicit results would depend on the individual equation under study.