Asymptotic Poincar\'e compactification and finite-time singularities

TL;DR

This paper extends the asymptotic decomposition method for vector fields with finite-time singularities using Poincaré's central extension, providing a new way to analyze singularity behavior in dynamical systems.

Contribution

It introduces a novel extension of asymptotic decomposition via Poincaré's central extension, enabling detailed analysis of finite-time singularities in dynamical systems.

Findings

Describes a bundle of asymptotic systems governing singularity behavior

Applies method to model a universe at maximum expansion

Discusses implications for structural stability and phase space

Abstract

We provide an extension of the method of asymptotic decompositions of vector fields with finite-time singularities by applying the central extension technique of Poincar\'e to the dominant part of the vector field on approach to the singularity. This leads to a bundle of fan-out asymptotic systems whose equilibria at infinity govern the dynamics of the asymptotic solutions of the original system. We show how this method can be useful to describe a single-fluid isotropic universe at the time of maximum expansion, and discuss possible relations of our results to structural stability and non-compact phase spaces.