Theory of global bifurcations of vector fields on the two-sphere

The talk presents the first steps of the new theory named in the title. Global bifurcations in one-parameter families will be classified (joint work with N. Solodovnikov). The question: Who bifurcates? will be discussed. The answer allows us to distinguish the part of the phase portrait of a degenerate vector field that actually bifurcates when the field is perturbed (joint result with N. Goncharuk.)
This result opens a possibility of classification of the families of vector fields with the higher number of parameters. The obstacles for this classification are numeric and functional moduli that occur when the number of the parameters is too large (joint work with Yu. Kudryashov and I. Schurov).