Normalization flow and Poincar\'e-Dulac theory

TL;DR

This paper introduces a novel normalization flow approach for Poincaré-Dulac theory, proving full formal normalization and providing convergence bounds, along with a new proof of the Siegel-Brjuno theorem.

Contribution

It develops a continuous averaging method to construct normalization flows and proves convergence properties, offering new insights into normal form theory.

Findings

Achieves full formal normalization in the algebra of formal vector fields.

Establishes a lower bound on the radius of convergence for normalization.

Provides a new proof of the Siegel-Brjuno theorem.

Abstract

In this article, we develop a new approach to the Poincar\'e--Dulac normal form theory for a system of differential equations near a singular point. Using the continuous averaging method, we construct a normalization flow that moves a vector field to its normal form. We prove that, in the algebra of formal vector fields (given by power series), the normalization procedure achieves full normalization. When convergence is taken into account, we show that the radius of convergence admits a lower bound of order $1/(1+A\delta)$, with $A>0$, as $\delta \to +\infty$. Based on the methods of this work and on the approaches of \cite{Tres2}, we provide a new proof of the Siegel--Brjuno theorem on the convergence of the normalizing transformation.