On the computation of extended rescaled Poincare maps for singular vector fields

摘要

In this paper, we aim to compute extended rescaled Poincare maps for singular vector fields. For explicit 2-dimensional linear vector fields, we are able to compute the extended rescaled Poincare maps upto second order derivatives. For singular vector fields, we show that the extended rescaled Poincare maps over the non-degenerate singularity are equal to the extended rescaled Poincare maps of the linearized vector field. As an application, we study the central model over certain aperiodic singular chain recurrent class for d-dimensional (d ≥ 4) flows and proves that it must have a chain recurrent central segment over the singularity, which obstructs the creation of horseshoes.