  We construct an open set of structurally unstable three parameter families
whose weak and so called moderate topological classification defined below has
a numerical invariant that may take an arbitrary positive value. Here and below
"families" are "families of vector fields in the two-sphere". This result
disproves an Arnold's conjecture of 1985. Then we construct an open set of six
parameter families whose moderate topological classification has a functional
invariant. This invariant is an arbitrary germ of a smooth map $(\mathbb R_+,
a)\to(\mathbb R_+, b)$. More generally, for any positive integers $d$ and $d'$,
we construct an open set of families whose topological classification has a
germ of a smooth map $\left(\mathbb R_+^d, a\right)\to\left(\mathbb R_+^{d'},
b\right)$ as an invariant. Any smooth germ of this kind may be realized as such
an invariant. These results open a new perspective of the global bifurcation
theory in the two sphere. This perspective is discussed at the end of the
paper.
