# Global bifurcations on the two sphere: first steps of a new theory

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Differential equations deal with the same matters as children do: pictures in the plane. If a picture related to a differential 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 abrupt changes are the subject of the bifurcation theory. This talk gives a survey of the first three years of development of a new 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. Natalya Goncharuk and the speaker defined a set called large bifurcation support; bifurcations that occur in a small neighborhood of this set determine the global bifurcations on the two-sphere. This result is a starting point for systematic classification of global bifurcations in two-parameter families. New examples of structurally unstable three-parameter families will be demonstrated. These are joint results of the speaker and his collaborators: N. Goncharuk, D. Filimonov, Yu. Kudryashov, N. Solodovnikov, I. Schurov and others. The talk will be addressed to a broad audience.