Lectures on bifurcations in versal families

In these lectures we consider the ways in which the disposition of the phase curves of a vector field can alter in a neighbourhood of a singularity as the parameters on which the vector field depends vary. A technical convenience in the study of such changes are certain deformations having a special universality property – the so-called versal families. Our results are presented mainly in the form of explicit formulae for versal families and an analysis of the corresponding bifurcation diagrams. As an application of the general theory we give a classification of the singularities of the decrement of general two-parameter families of linear autonomous systems and a classification of the singularities of the neutral surface (stability boundary) of general three-parameter families of linear systems; we also treat the topologically versal deformations of singular points of non-linear systems of ordinary differential equations for all cases of degeneracy of codimension 1 and for some of codimension 2; we indicate applications to the theory of hydrodynamical stability.