arXiv subjects

David McLaren

Publications and source records attributed to David McLaren.

3 recordsLinked to original sources

arXiv2024

An Elementary Construction of Modified Hamiltonians and Modified Measures of 2D Kahan Maps

We show how to construct in an elementary way the invariant of the KHK discretisation of a cubic Hamiltonian system in two dimensions. That is, we show that this invariant is expressible as the product of the ratios of affine polynomials defining the prolongation of the three parallel sides of a hexagon. On the vertices of such a hexagon lie the indeterminacy points of the KHK map. This result is obtained analysing the structure of the singular fibres of the known invariant. We apply this construction to several examples, and we prove that a similar result holds true for a case outside the hypotheses of the main theorem, leading us to conjecture that further extensions are possible.

math.NA↗

arXiv2022

Detecting and determining preserved measures and integrals of birational maps

In this paper we use the method of discrete Darboux polynomials to calculate preserved measures and integrals of rational maps. The approach is based on the use of cofactors and Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.

math.NA↗

arXiv2019

Geometric and integrability properties of Kahan's method

Given a quadratic vector field on \mathbb{R}^n possessing a quadratic first integral depending on two of the independent variables, we give a constructive proof that Kahan's discretization method exactly preserves a nearby modifed integral. Building on this result, we present a family of integrable quadratic vector fields (including the Euler top) whose Kahan discretization is a novel 10-parameter family of integrable maps.

math.NA↗