Turrittin's Theorem revisited. The real case

TL;DR

This paper extends Turrittin's theorem to real linear systems of ODEs, providing real-normal forms and transformations, and reviews the complex case to adapt the proofs.

Contribution

It establishes a real version of Turrittin's theorem, ensuring all normal forms and transformations have real coefficients.

Findings

Real polynomial and formal normal forms are constructed.

Transformations used are explicitly real-coefficient.

The complex case is reviewed to adapt proofs to the real setting.

Abstract

We establish a real version of Turrittin's result on polynomial and formal normal forms of linear systems of ODEs with meromorphic coefficients. Both the normal forms or the transformations used have only real coefficients. In order to adapt the proofs to the real case, we make a review of the result in the complex case.