Regularisation for Planar vector fields. (19th November 2019)

• Record Type: Journal Article
• Authors: Duignan, Nathan; Dullin, Holger R
• Is Part Of: Nonlinearity. Volume 33:Number 1(2020)
• Page Start: 106
• Page End: 138
• Publication Date: 2019-11-19
• Subjects: dynamical systems -- celestial mechanics -- regularisation -- isolating blocks -- planar vector fields; 37C10 -- 37C15 -- 37C25 -- 70F10 -- 70F15; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab34d9
• Languages: English
• ISSNs: 0951-7715

Abstract

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 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.