Regularisation for Planar Vector Fields

TL;DR

This paper investigates regularisation techniques for planar vector fields, focusing on singularities relevant to celestial mechanics, and provides conditions for different levels of regularity of the transition map.

Contribution

It introduces new conditions for C^0 and C^1 regularisation of transition maps in degenerate singularities, including explicit computations for quadratic vector fields.

Findings

Transition map is generally finitely differentiable.

C^1 regularisation reduces to summing residues of a rational function.

A perturbation example from the 4-body problem is C^{4/3}.

Abstract

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 a toy example derived from the 4-body problem is shown to be $ C^{4/3} $. The regularisation of all homogeneous quadratic vector fields is computed.