Birational maps from polarization and the preservation of measure and integrals

RI McLachlan, DI McLaren… - Journal of Physics A …, 2023 - iopscience.iop.org
The main result of this paper is the discretization of second-order Hamiltonian systems of the
form $\ddot x=-K\nabla W (x) $, where K is a constant symmetric matrix and …

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

G Bogfjellmo, E Celledoni, R McLachlan… - Mathematics of …, 2024 - ams.org
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 …

The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

A Laurent, RI McLachlan, HZ Munthe-Kaas… - Forum of Mathematics …, 2023 - cambridge.org
Aromatic B-series were introduced as an extension of standard Butcher-series for the study
of volume-preserving integrators. It was proven with their help that the only volume …

Analogues of Kahan's method for higher order equations of higher degree

ANW Hone, GRW Quispel - … Algebraic and Geometric Aspects of Integrable …, 2020 - Springer
Kahan introduced an explicit method of discretization for systems of first order differential
equations with nonlinearities of degree at most two (quadratic vector fields). Kahan's method …

On the preservation of second integrals by Runge-Kutta methods

BK Tapley - arXiv preprint arXiv:2105.10929, 2021 - arxiv.org
One can elucidate integrability properties of ordinary differential equations (ODEs) by
knowing the existence of second integrals (also known as weak integrals or Darboux …

The Lie derivative and Noether's theorem on the aromatic bicomplex

A Laurent - arXiv preprint arXiv:2307.07984, 2023 - arxiv.org
The aromatic bicomplex is an algebraic tool based on aromatic Butcher-trees and used in
particular for the explicit description of volume-preserving affine-equivariant numerical …

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

A Laurent - Journal of Computational Dynamics, 2023 - hal.science
The aromatic bicomplex is an algebraic tool based on aromatic Butcher trees and used in
particular for the explicit description of volume-preserving affine-equivariant numerical …

Measure preservation and integrals for Lotka--Volterra -systems and their Kahan discretisation

PH van der Kamp, RI McLachlan, DI McLaren… - arXiv preprint arXiv …, 2023 - arxiv.org
We show that any Lotka--Volterra $ T $-system associated with an $ n $-vertex tree $ T $ as
introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We …

Dynamical degrees of birational maps from indices of polynomials with respect to blow-ups II. 3D examples

J Alonso, YB Suris, K Wei - arXiv preprint arXiv:2307.09939, 2023 - arxiv.org
The goal of this paper is the exact computation of the degrees $\text {deg}(f^ n) $ of the
iterates of a birational map $ f:\mathbb {P}^ N\dashrightarrow\mathbb {P}^ N $. In the …

A novel 8-parameter integrable map in

GRW Quispel, DI McLaren… - arXiv preprint arXiv …, 2020 - arxiv.org
arXiv:2003.05588v2 [nlin.SI] 9 Jul 2020 Page 1 arXiv:2003.05588v2 [nlin.SI] 9 Jul 2020 A
novel 8-parameter integrable map in R4 GRW Quispel, DI McLaren and PH van der Kamp …