Poincare' normal forms and simple compact Lie groups

TL;DR

This paper classifies the behavior of Poincaré-Dulac normal forms for dynamical systems in real space that are symmetric under simple compact Lie groups, extending to systems with zero linear parts and discussing convergence issues.

Contribution

It provides a classification of normal forms for systems with Lie group symmetries, including zero linear parts, and discusses the convergence of normalizing transformations.

Findings

Classification of normal forms under simple compact Lie group symmetries.

Extension of normal form analysis to systems with zero linear part.

Discussion on the convergence of normalizing transformations.

Abstract

We classify the possible behaviour of Poincar\'e-Dulac normal forms for dynamical systems in $R^n$ with nonvanishing linear part and which are equivariant under (the fundamental representation of) all the simple compact Lie algebras and thus the corresponding simple compact Lie groups. The ``renormalized forms'' (in the sense of previous work by the author) of these systems is also discussed; in this way we are able to simplify the classification and moreover to analyze systems with zero linear part. We also briefly discuss the convergence of the normalizing transformations.