Diagonalisation method in the asymptotical theory for solutions of systems of ODE

TL;DR

This paper presents a diagonalisation method to analyze the asymptotic behavior of solutions for parameter-dependent systems of ODEs, simplifying complex problems to solvable diagonal cases through matrix approximation.

Contribution

It introduces an approach to reduce a broad class of parameter-dependent ODE systems to diagonal form using smooth matrix approximations, facilitating asymptotic analysis.

Findings

The method effectively simplifies asymptotic analysis of ODE systems.

It provides a systematic way to approximate matrix functions for large parameters.

The approach enhances solvability of complex ODE systems with parameter dependence.

Abstract

The question about asymptotical behaviour of solutions for the system $\dot x=A_\nu x+f$ for big values of the parameter $\nu\in\frak A$ is considered. An approach to the reduction of a large class of problems to easily solvable problem with diagonal coefficient matrix is described. This approach is based on the approximation of summable matrix function $A_\nu$ by specially constructed smooth matrix functions.