Analytic linearization of nonlinear perturbations of Fuchsian systems

TL;DR

This paper demonstrates that nonlinear perturbations of Fuchsian systems with commuting monodromy and positive eigenvalues can be analytically linearized through a unique correction function, even when not initially linearizable.

Contribution

It introduces a method to achieve analytic linearization of certain nonlinear Fuchsian systems by constructing a unique correction function under specific conditions.

Findings

Existence of a unique correction function for linearization.

Linearization is possible when the linear part has commuting monodromy.

Applicable to systems with eigenvalues having positive real parts.

Abstract

Nonlinear perturbation of Fuchsian systems are studied in regions including two singularities. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable). Nevertheless, it is shown that in the case when the linear part has commuting monodromy, and the eigenvalues have positive real parts, there exists a unique correction function of the nonlinear part so that the corrected system becomes analytically linearizable.