Global bifurcations in the two-sphere: a new perspective

Abstract

We construct an open set of structurally unstable three parameter families whose weak and so called moderate topological classification 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 $({ℝ}_{+},a)\to ({ℝ}_{+},b)$. More generally, for any positive integers $d$ and $d^{\prime }$, we construct an open set of families whose topological classification has a germ of a smooth map $({ℝ}_{+}^d,a)\to ({ℝ}_{+}^{d^{\prime }},b)$ 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.