The topological analysis of the electron localization function (ELF) provides a convenient theoretical framework to characterize chemical bonds. This method does not rely on the particular approximations that are made in actual quantum chemical calculations of the electronic structure. In principle, it can be applied to exact wave functions as well as to experimental electron densities. Introduction of a control space, such as a set of reaction pathways, allows extension of the analysis to chemical reactions. The study of the bifurcations occurring during such processes is of particular interest for their classification and their qualitative description. This is achieved with the help of Rene Thom's catastrophe theory. The following examples are discussed: the ammonia inversion, the breaking of the ethane C-C bond, and the breaking of the dative bond in NH3BH3. The types of catastrophe and their unfolding have been determined for each of these processes. As by-products, nonempirical definitions of covalent and dative bonds are proposed.