Research On Polynomial Normal Forms For Non-strongly 1-resonant Diffeomorphisms On R~3

The normal form theory is widely used to study nonlinear problems. Both its theory and application have particularly important significances. In recent years, the normal form theory has wide application on the Hilbert 16 th problem, bifurcation, classification of dynamical systems, embedding flow of diffeomorphism and so on, which increasingly arouses people’s concern.This present thesis primarily consists of two parts: In the first part, we discuss the finite determinacy and polynomial normal forms of diffeomorphisms near their nonstrongly 1-resonant fixed points on R3 under generic conditions. In the second part, we discuss the finite determinacy and polynomial normal forms of diffeomorphisms near their non-strongly 1-resonant fixed points on R3 under degenerated conditions.In the first part, we mainly discuss the non-strongly 1-resonant diffeomorphisms in which the linear part has multiple eigenvalues on R3 under generic conditions. Based on classical Poincare-Dulac normal form theory, we can get the classical resonant normal form through considering the resonant relations of eigenvalues. On the basis of the Ichikawa’s finite determinacy theory, take the resonant transformations. According to the Belistkii theory we get their finite determinacy and polynomial normal forms.In the second part, we mainly discuss the non-strongly 1-resonant diffeomorphisms in which the linear part has no multiple eigenvalues on R3 under degenerated conditions.Similarly, under taking the way of resonant transformations and combined with the Ichikawa and Belistkii theory the finite determinacy and polynomial normal forms of diffeomorphisms are obtained under smooth conjugating equivalence.