Regularisation for Planar vector fields

This paper serves as a first foray on regularisation for planar vector fields. Motivated by singularities in celestial mechanics, the block regularisation of a generic class of degenerate singularities is studied. The paper is concerned with asymptotic properties of the transition map between a section before and after the singularity. Block regularisation is reviewed before topological and explicit conditions for the C-0-regularity of the map are given. Computation of the C-1-regularisation is reduced to summing residues of a rational function. It is shown that the transition map is in general only finitely differentiable and a method of computing the map is conveyed. In particular, a perturbation of an example derived from the four-body problem is shown to be C-4/3. The regularisation of all homogeneous quadratic vector fields is computed.