Rescaled Poincaré map via blowup of singularities
Through blowup \cite{T}, the rescaled Poicaré 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 Poincaré 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 Poincaré maps upto the second order. We show that except for the focus case, the second order generally does not vanish, and the rescaled Poincaré map is generally nonlinear.
math.DS↗