In this paper we study the germ of a family of real analytic diffeomorphisms unfolding a period-doubling bifurcation of codimension k. Such families of diffeomorphisms are called codimension-k flip families. We prove that any such a family can be realized naturally as the Poincaré monodromy of an analytic family of elliptic vector fields of order k.

period-doubling, holomorphic maps, normal form, monodromy, Stokes phenomena, bifurcation, analytic foliation.