Normal Form for Local Dynamical Systems

Abstract

Normal Form is a theory that applies in the neighbourhood of an orbit of a vector field map. The theory provides an algorithmic way to generate a sequence of non-linear coordinate changes that eliminate as much non-linearity as possible at each order (where order refers to terms in Taylors series about an orbit).The normal form is intended to be the simplest form into which any system of the intended type can be transformed by changing the coordinates in a prescribed manner. Interestingly the form of non-linear that cannot be eliminated by such coordinate changes is determined by the structure of the linear part of the vector field map.

This section consists of some background knowledge, theorems and definitions necessary for understanding the concept of normal form for local dynamical systems. We briefly discuss the concept of ring of invariants and module of equivariants, and use the Groebner basis methods to compute a Groebner basis for the ideal of relations among the basic invariants.