Nonlinear perturbations of Fuchsian systems: corrections and linearization, normal forms

TL;DR

This paper investigates nonlinear perturbations of Fuchsian systems near singularities, demonstrating non-linearizability in general, identifying obstructions, and establishing conditions for formal linearization and classification through normal forms.

Contribution

It introduces a constructive method to find obstructions to linearization and shows the existence of a unique formal correction for polynomial nonlinear parts.

Findings

Nonlinear perturbations are generally not analytically linearizable.

Obstructions to linearization are characterized as a countable set of numbers.

Existence and uniqueness of a formal correction for polynomial nonlinear parts are proven.

Abstract

Nonlinear perturbation of Fuchsian systems are studied in a region including two singularities. It is proved that such systems are generally not analytically equivalent to their linear part (they are not linearizable) and the obstructions are found constructively, as a countable set of numbers. Furthermore, assuming a polynomial character of the nonlinear part, it is shown that there exists a unique formal "correction" of the nonlinear part so that the "corrected" system is formally linearizable. Normal forms of these systems are found, providing also their classification.