Yulij Ilyashenko (National Research University - Higher School of Economy, Moscou)

First steps of the global bifurcation theory on the two sphere

Diferential equations deal with the same matters as children do: pictures in the plane. If a picture related to a diferential equation remains (topologically) the same after the equation is slightly
perturbed, this equation is structurally stable. If it is not, abrupt changes of the corresponding picture may occur under a small perturbation. These abriupt changes are the subject of the bifurcation theory. This talk manifests the first steps of a new born branch of the bifurcation theory: global bifurcations on the two sphere. Bifurcations in generic one-parameter families were classified; the answer appeared to be quite unexpected. An important and non-trivial question "who bifurcates?" was answered. 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. These are joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov, Yu. Kudryashov, N. Solodovnikov, I. Schurov and others. The development of the bifurcation theory will be outlined from the very beginning. Some open problems will be stated.