Dimension increase and splitting for Poincare'-Dulac normal forms

TL;DR

This paper explores how increasing the dimension and splitting in Poincare'-Dulac normal forms can aid in analyzing nonlinear dynamical systems, extending ideas from integrable to non-integrable cases.

Contribution

It introduces a novel approach of dimension increase and splitting for normal forms, inspired by methods used in integrable systems, applied to non-integrable systems.

Findings

Dimension increase facilitates analysis of nonlinear systems.

Splitting methods reveal new structural insights.

Applicable to both integrable and non-integrable systems.

Abstract

Integration of nonlinear dynamical systems is usually seen as associated to a symmetry reduction, e.g. via momentum map. In Lax integrable systems, as pointed out by Kazhdan, Kostant and Sternberg in discussing the Calogero system, one proceeds in the opposite way, enlarging the nonlinear system to a system of greater dimension. We discuss how this approach is also fruitful in studying non integrable systems, focusing on systems in normal form.