  We study bifurcations in finite-parameter families of vector fields on $S^2$.
Recent papers by Yu. Ilyashenko, N. Goncharuk, Yu. Kudryashov, I. Schurov, and
N. Solodovnikov provide examples of (locally generic) structurally unstable
families of 3-parameter vector fields: generic close 3-parameter families
experience different bifurcations. In this paper, we use these results to
construct new examples of few-parameter generic families of planar vector
fields such that their classification has many invariants. In particular, we
construct (a) 3-parameter families with infinitely many numerical invariants;
(b) 4-parameter families with arbitrarily many "robust" numerical invariants;
(c) 5-parameter families with functional invariants.
