Further Reductions of Normal Forms for Dynamical Systems

We propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields, with concrete and interesting applications. We use lower order nonlinear terms in the normal form for the simplifications of higher order terms. Our approach is applicable for both the non nilpotent and the nilpotent cases. For dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal (or unique) with respect to equivalence by formal near identity transformations. We can compute at the same time a formal diffeormorphism that realizes the normalization. Comparisons with other methods are given for several examples