Global bifurcations in the two-sphere: a new perspective (Q1656354)

The authors consider families of vector fields on the two-sphere depending on \(k\) parameters which take values in a topological open ball \(B\in\mathbb{R}^k\). Some of the definitions make sense on any manifold \(M\) although the authors consider \(M=S^2\), the two-sphere. First they provide the definition that two vector fields on \(M\) are orbitally topologically equivalent, then the definition that two vector fields are weak topologically equivalent. They remark that the first definition is too strict because it distinguishes families with the same simple dynamics whereas the second is too lousy because does not distinguish some very different bifurcations. Therefore, they provide a new definition of vector fields which are \textit{moderately topologically equivalent}. With this definition in hand, the main result of the paper is an example of a non-empty open set of vector fields on the two-sphere such that each family of this set is moderately structurally stable. Moreover, the moderate topological classification of these families has a numerical invariant that may take any positive value. This is the statement of the main result as written in the paper. This result gives a negative answer to an Arnold's conjecture.