Analytic normalization of analytically integrable differential systems near a periodic orbit

TL;DR

This paper proves that an analytic differential system with a periodic orbit, which is integrable, can be transformed into a simplified normal form, extending previous results from singularities to periodic orbits.

Contribution

It establishes that analytically integrable systems near a periodic orbit are analytically equivalent to their Poincaré-Dulac normal form, extending existing theory from singularities to periodic orbits.

Findings

Analytic integrability implies equivalence to normal form near periodic orbits.

Extends integrability results from singularities to periodic orbits.

Provides a foundation for simplifying analysis of integrable systems around periodic orbits.

Abstract

For an analytic differential system in $\mathbb R^n$ with a periodic orbit, we will prove that if the system is analytically integrable around the periodic orbit, i.e. it has $n-1$ functionally independent analytic first integrals defined in a neighborhood of the periodic orbit, then the system is analytically equivalent to its Poincar\'e--Dulac type normal form. This result is an extension for analytic integrable differential systems around a singularity to the ones around a periodic orbit.