This monograph concerns the question of how the solution of a system of ODE's varies when the differential equation varies. The goal is to give non-zero asymptotic expansions for the solution in terms of a parameter expressing how some coefficients go to infinity. A particular class of families of equations is considered, where the answer exhibits a new kind of behaviour not seen in most work known until now. The techniques include Laplace transform and the method of stationary phase, and combinatorial techniques for estimating the contributions of terms in an infinite series expansion for the solution. Addressed primarily to researchers in algebraic geometry, ordinary differential equations and complex analysis, the book will also be of interest to applied mathematicians working on asymptotics of singular perturbations and the numerical solution of ODE's.