Deducing properties of ODEs from their discretization

TL;DR

This paper demonstrates that properties of quadratic ODEs, such as preserved integrals and measures, can be inferred from their discretizations using Darboux Polynomials, with extensions to higher degree ODEs.

Contribution

It introduces a method to deduce properties of ODEs from their discretizations, leveraging Darboux Polynomials, including for higher degree and order equations.

Findings

Properties like preserved integrals can be algorithmically deduced from discretizations.

Darboux Polynomials are effective tools for analyzing discretized ODEs.

Extensions to higher degree/order ODEs are possible with other birational discretizations.

Abstract

We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretization, using Darboux Polynomials (DPs). Somewhat similar results hold for ODEs of higher degree/order using certain other birational discretization methods.