Using aromas to search for preserved measures and integrals in Kahan's method

TL;DR

This paper introduces an aromatic series-based algorithm to identify preserved measures and integrals in Kahan's method for quadratic differential equations, significantly simplifying the analysis compared to traditional approaches.

Contribution

The paper develops a novel aromatic series approach to detect preserved measures and integrals in Kahan's method, reducing complexity in parameter-rich systems.

Findings

The aromatic series method simplifies the identification of invariants.

The algorithm demonstrates effectiveness on various examples.

It offers a more interpretable alternative to discrete Darboux polynomial methods.

Abstract

The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyze. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic funtions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.