Regularisation for Planar vector fields

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.

On the C8/3-regularisation of simultaneous binary collisions in the collinear 4-body problem

“…How can non-smoothness arise in a smooth (or even polynomial) planar dynamical system? The answer to this question as described in [10] is: Through a block map past a degenerate equilibrium point. A degenerate equilibrium point can, e.g., be such that the negative x-axis is the stable manifold and the positive x-axis is the unstable manifold.…”

“…In this case, a Poincaré map between transversal sections to the x-axis can be defined. In [10] it was shown that this map generically cannot be extended to y = 0 in a smooth way. The reason is that orbits passing the equilibrium with y > 0 may be sufficiently different from those passing the equilibrium with y < 0.…”

“…Before describing in more detail the strategy of the proof we would like to explain the essence of the mechanism in a toy model first introduced in [10]. How can non-smoothness arise in a smooth (or even polynomial) planar dynamical system?…”

“…10 gives the asymptotic expansion of D − 2,γα , D + 1,γα whilst Lemma 4.11 gives the asymptotic expansion of Tγ α . Composing the maps and keeping track of the known error, we have that,…”

On the C 8/3-regularisation of simultaneous binary collisions in the planar four-body problem

“…10 gives the asymptotic expansion of D − 2,γ α , D + 1,γ α whilst lemma 4.11 gives the asymptotic expansion of T γα . Composing the maps and keeping track of the known error, we have that,π + γ = D − 2,γ α • T γα • D + 1,γ α (εβ α0 , εγ α0 , εδ α0 , u 0 ) = D − 2,γ α • T γα ε 1/3 γ O(ε 3 ln ε) , u 0 + O(ε 3 ln ε) O(ε 3 ln ε) + O(ε 8/3 ) , ε 2/3 γ −1/3 0 β 0 1 + O(ε 2 ln ε) + O(ε 8/3 ) , ε 2/3 γ −1/3 0 δ 0 1 + O(ε 3 ln ε) + O(ε 7/3 ) , u 0 + O(ε 3 ln ε) + O(ε 8/3 )= εβ α0 + O(ε 3 ), εγ α0 + O(ε 3 ), εδ α0 + O(ε 3 ), u 0 + O(ε 8/3 ) .…”