On the flat remainder in normal forms of families of analytic planar   saddles
  We give an explicit expression for the (finitely) flat remainder after analytic normal form reduction of a family of planar saddles of diffeomorphisms or vector fields. We distinguish between a rational or irrational ratio of the moduli of the eigenvalues at the saddle for a certain value of the parameter. 
Patrick Bonckaert and Freek Verstringe