This book concerns the question of how the solution of asystem of ODE's varies when the differential equationvaries. The goal is to give nonzero asymptotic expansionsfor the solution in terms of a parameter expressing how somecoefficients go to infinity. A particular classof familiesof equations is considered, where the answer exhibits a newkind of behavior not seen in most work known until now. Thetechniques include Laplace transform and the method ofstationary phase, and a combinatorial technique forestimating the contributions of terms in an infinite seriesexpansion for the solution. Addressed primarily toresearchers inalgebraic geometry, ordinary differentialequations and complex analysis, the book will also be ofinterest to applied mathematicians working on asymptotics ofsingular perturbations and numerical solution of ODE's.