Differential systems with Fuchsian linear part: correction and linearization, normal forms and multiple orthogonal polynomials

TL;DR

This paper investigates differential systems with Fuchsian linear parts, identifying nonlinear corrections and normal forms using multiple orthogonal polynomials, thus advancing the understanding of their linearization and formal equivalence.

Contribution

It introduces a constructive method to find nonlinear corrections and normal forms for Fuchsian systems using multiple orthogonal polynomials, extending the theory of linearization.

Findings

Nonlinear obstructions to linearization are characterized.

Constructive procedures for corrections and normal forms are developed.

Multiple orthogonal polynomials are instrumental in the analysis.

Abstract

Differential systems with a Fuchsian linear part are studied in regions including all the singularities in the complex plane of these equations. Such systems are not necessarily analytically equivalent to their linear part (they are not linearizable) and obstructions are found as a unique nonlinear correction after which the system becomes formally linearizable. More generally, normal forms are found. The corrections and the normal forms are found constructively. Expansions in multiple orthogonal polynomials and their generalization to matrix-valued polynomials are instrumental to these constructions.