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