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## Global bifurcations in the two-sphere: a new perspective

Inventiones Mathematicae. 2018. Vol. 213. No. 2. P. 461-506.
Yu. Ilyashenko, Kudryashov Y., I. Schurov.

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 (R_+, a) → (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 (R_+^d, a) → (R_+^{d'}, b) 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.