Further Normalization

Abstract

In this volume we have considered the standard normalization procedure of Poincaré, that depends on generating functions (for the changes of coordinates) h _m which are obtained as solutions of the relative homological equations. But these solutions are not unique whenever the homological operator, which will be denoted here as L ₀, has a nontrivial kernel. Functions h _m differing for an element δh _m ∈(L ₀) produce the same effect on F _m, but not on higher order terms F _k (k > m). Correspondingly, the normal form is not unique, and the unfolding of NFs - which can be described as the most general resonant system [i.e. a system with nonlinear part in Ker(L ₀)] - is exhaustive, but in general not minimal. That is, the Poincaré-Dulac theory provides a list of NFs such that any system can be (formally) reduced to one of the NFs of the list, but we are not guaranteed that the NFs are pairwise not conjugated.