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

Yu. Ilyashenko,
Yu. Kudryashov,
I. Schurov

March, 2018

### 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 $(ℝ_+, a)→(ℝ_+, b)$.
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 $\left(ℝ_+^d, a\right)→\left(ℝ_+^{d’}, b\right)$ 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.

Publication

*Inventiones mathematicae*, **213** (2), 461–506