NILPOTENT SINGULARITIES IN GENERIC 4-PARAMETER FAMILIES OF 3-DIMENSIONAL VECTOR-FIELDS

This paper deals with singularities of vector fields in R(3) having a 1-jet linear conjugate to y(partial derivative/partial derivative x) z(partial derivative/partial derivative y). They first occur in gener ic 3-parameter families. In codimension 3 all such singularities are m utually C-D equivalent. We give a proof of this, provide a good normal form for 3-parameter unfoldings, and show that all non-wandering beha viour in such an unfolding is of small amplitude. We also show that fo r codimension 4 there are exactly 5 types of singularities for C-D equ ivalence. The proof relies on normal form theory, blowing-up, and esti mation of Abelian integrals. (C) 1996 Academic Press, Inc.