Normal Form for Module of Equivariants

Abstract

The subject of dynamical systems is concerned with:

1. i.          Given an matrix  describe the behavior , in a neighborhood of the origin the solutions of all systems of differential equations having a rest point at the origin with the linear part Ax
2. ii.          Describe the behavior (near origin) of all systems close to a system of the type described in i above.

The  normal form is intended to be the ‘’simplest ‘’form in which any system of intended type can be transformed by changing the coordinates in a prescribed manner. But if a normal form is thought as the ‘’simplest’’ form into which a system can be placed there might be  disagreement as to what is considered simplest. A systematic policy for deciding what accounts for simplest is called normal form style. The important normal form styles are Semisimple, Innerproduct and SL(2) Or Triad styles.

The unfolding of the normal form is intended to be the simplest form in which all systems close to the original system can be transformed

I will present the procedure for obtaining the normal form using the inner product style and explain the structure of normal form using the language of equivariants over a ring of invariants and give an algorithm suitable for use in symbolic computation systems procedure