Global bifurcations in the two-sphere: a new perspective

TL;DR

This paper introduces new classes of structurally unstable vector field families on the two-sphere with arbitrary invariants, disproving a longstanding conjecture and expanding the understanding of global bifurcations.

Contribution

It constructs open sets of vector field families with arbitrary numerical and functional invariants, providing a new perspective on global bifurcation theory in the two-sphere.

Findings

Disproves Arnold's 1985 conjecture.

Constructs families with arbitrary smooth map invariants.

Expands the classification framework for bifurcations.

Abstract

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.