Various Equivalence Relations in Global Bifurcation Theory

We discuss various definitions of equivalence for bifurcations of vector fields on the sphere and give a large number of examples (both known and new) that illustrate the advantages and disadvantages of different definitions. In addition to the classical definitions of strong and weak equivalence, we consider new notions of Sing-equivalence and moderate equivalence. These definitions seem to be more relevant to and consistent with the intuitive notion of equivalent bifurcations. They were introduced and used to describe the structural instability of some finite-parameter families of vector fields on the sphere and to study invariants of their classification.