ANALYTIC SINGULARITY AND SINGULAR PERTURBATION IN C(2)

Given a family of ordinary differential equations (epsilon(alpha)) : epsilondu/dz = f(z, u, alpha) in C2 with a small parameter epsilon and a control parameter alpha, we are interested in the local existence of a pair (alpha*, u*) which have an asymptotic expansion in power of epsilon and such that u* is a solution of (epsilon(alpha*)). The main result of our study establishes a main connection between the existence of such a pair and a property of an unfolding of the singularity of the first approximation of the function f.