Geometry of Normal Forms for Dynamical Systems

TL;DR

This paper explores the geometric structure of vector fields in Poincare'-Dulac normal form, using Michel theory to facilitate analysis and address systems with inherent symmetries, enhancing understanding of dynamical systems.

Contribution

It introduces a constructive approach to analyze normal forms through group actions, especially for symmetric systems, advancing the geometric understanding of dynamical systems.

Findings

Develops a geometric framework for normal forms

Uses Michel theory for constructive analysis

Addresses symmetric systems in physics

Abstract

We discuss several aspects of the geometry of vector fields in (Poincare'-Dulac) normal form. Our discussion relies substantially on Michel theory and aims at a constructive approach to simplify the analysis of normal forms via a splitting based on the action of certain groups. The case, common in Physics, of systems enjoying an a priori symmetry is also discussed in some detail.