Global bifurcations on the two sphere

This talk manifests the first steps of a new born branch of the bifurcation theory: global bifurcations on the two sphere. It appeared that there exist structurally unstable generic families of vector fields in the plane. In all the previous works on the planar bifurcations, the result was described by a finite number of phase portraits that may occur under the perturbations of degenerate vector fields. In the global theory, this is no more the case. Even three-parameter families of vector fields on the two sphere may have numeric invariants, and six–parameter families may have functional invariants. No versal families exist any more. A continual set of germs of generic bifurcation diagrams may occur even in the four–parameter families. A natural question arises: given a degenerate vector field, how to determine, what part of its phase portrait actually bifurcates? How to classify bifurcations in the low (one and two)-parameter families, where numeric invariants are not expected? All these questions, except for the classification of the two-parameter families, are answered by the speaker and his collaborators: Nataliya Goncharuk, Dmitry Filimonov, Yury Kudryashov, Nikita Solodovnikov, Ilya Schurov and others. Some open problems will be stated.
All the necessary definitions will be given during the talk.