If you have a good understanding of the Transversality Theorem of Rene Thom please write, I would like to correspond with you.
Here is a related question I am trying to answer:
If a dynamical system T is obtained from another, S, by a codimension n bifurcation (with n > 1), then is there a finite sequence of codimension 1 bifurcations starting at S and ending at T?
(With thanks to Morris Hirsch)
search terms: k-jet extension, c-infinity maps, submanifold, transversal