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.

Keywords: hyperbolic polycycles, гиперболические полициклы, структурная устойчивость, structural stability, бифуркации, bifurcations, separatrix connections, сепаратрисные связки