Nilpotent singularities in generic 4-parameter families of 3-dimensional vector fields

Abstract

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 generic 3-parameter families. In codimension 3 all such singularities are mutually C-D equivalent. We give a proof of this, provide a good normal form for 3-parameter unfoldings, and show that all non-wandering behaviour in such an unfolding is of small amplitude. We also show that for codimension 4 there are exactly 5 types of singularities for C-D equivalence. The proof relies on normal form theory, blowing-up, and estimation of Abelian integrals. (C) 1996 Academic Press, Inc.