Rescaled Poincar\'e map via blowup of singularities

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.