The Smooth Classifications Of Poincaré Type Maps On R~3

This thesis is devoted to studying the smooth normal forms and smooth classifcations ofPoincarétype maps at their fxed points on R3. There are two parts in this paper, the frst partgives the smooth polynomial normal forms of Poincarétype maps on R3. The second part givesthe classifcations of Poincarétype maps on R3.In the frst part of this thesis, the smooth polynomial normal forms of Poincarétype mapsnear their fxed points on three-dimensisonal space are considered. Based on the classical normalform theory such as Poincaré-Dulac theorem, Chen theorem and Samovol theorem, we considertheir resonant normal forms under smooth conjugating equivalence in order to get their smoothpolynomial normal forms by the resonant transformations.In the second part, we study the classifcations of Poincarétype maps on R3. We provethat except for two cases any two Poincarétype maps with the similar linearized matrixes andgeneric nonlinear parts, are at least C3conjugated.