# 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.

## Key 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.