Rescaled Poincar\'e map via blowup of singularities

TL;DR

This paper extends the rescaled Poincaré map to nondegenerate singularities using blowup techniques, showing it aligns with the linearization and analyzing its nonlinear behavior in specific cases.

Contribution

It proves the equivalence of the extended rescaled Poincaré map with the linearization at nondegenerate singularities and computes second-order approximations for certain 2D linear vector fields.

Findings

Extended rescaled Poincaré map equals linearization at nondegenerate singularities.

Second order of the map generally does not vanish except in the focus case.

Rescaled Poincaré map is typically nonlinear near singularities.

Abstract

Through blowup \cite{T}, the rescaled Poicar\'e map is continued to nondegenerate singularities in \cite{GY,CY}. For a nondegenerate singularity of a $C^1$ vector field, we prove that the extended rescaled Poincar\'e map over the singurity equals the counterpart of the vector field's linearization. For singularities of two dimensional linear vector fields of three types, we compute the rescaled Poincar\'e maps upto the second order. We show that except for the focus case, the second order generally does not vanish, and the rescaled Poincar\'e map is generally nonlinear.