While generic vector fields on S² are structurally stable, locally
generic three-parameter families of vector fields on S² can be
structurally unstable, i.e. locally generic close families experience
topologically different bifurcations. This effect was recently
discovered by Yu. Ilyashenko, I. Schurov, and the speaker.
It seems natural to ask how far can finite-parameter families fall from
being structurally stable: can the classification have multiple
numerical invariants? functional invariants? do these invariants survive
if we add more parameters?
I will discuss some examples showing that finite-parameter families can
be *very* far from being structurally stable. The talk is based on a
joint work with Yu. Ilyashenko, I. Schurov, N. Goncharuk, and
N. Solodovnikov.