Global bifurcations in the two-sphere: a new perspective

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.