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Regularisation for Planar Vector Fields
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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.
Forward citations
Cited by 2 Pith papers
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Normal Forms for Manifolds of Normally Hyperbolic Singularities and Asymptotic Properties of Nearby Transitions
The Dulac map near a normally hyperbolic manifold of saddle singularities is shown to admit an asymptotic expansion built from the Ecalle-Roussarie compensator, generalizing the planar saddle result.
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On the $ C^{8/3} $-Regularisation of Simultaneous Binary Collisions in the Collinear 4-Body Problem
A new geometric proof shows that simultaneous binary collisions in the collinear four-body problem are regularisable only up to C^{8/3}, with the obstruction caused by the first coupling term between the two binaries.
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