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We construct an open set of structurally unstable three parameter familieswhose weak and so called moderate topological classification defined below hasa numerical invariant that may take an arbitrary positive value. Here and below"families" are "families of vector fields in the two-sphere". This resultdisproves an Arnold's conjecture of 1985. Then we construct an open set of sixparameter families whose moderate topological classification has a functionalinvariant. 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 agerm 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 suchan invariant. These results open a new perspective of the global bifurcationtheory in the two sphere. This perspective is discussed at the end of thepaper.

Global bifurcations in the two-sphere: a new perspective