Galois differential algebras and categorical discretization of dynamical systems

TL;DR

This paper introduces a categorical framework for discretizing diverse dynamical systems with variable coefficients using Galois differential algebras, preserving key solution properties.

Contribution

It develops a novel categorical approach linking Galois differential algebras to dynamical systems discretization, enabling solution preservation.

Findings

Integrable maps retain continuous solutions

Linear cases share Picard-Vessiot groups

Framework applies to a broad class of systems

Abstract

A categorical theory for the discretization of a large class of dynamical systems with variable coefficients is proposed. It is based on the existence of covariant functors between the Rota category of Galois differential algebras and suitable categories of abstract dynamical systems. The integrable maps obtained share with their continuous counterparts a large class of solutions and, in the linear case, the Picard-Vessiot group.