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 …
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
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 …
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
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 …
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 …
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 …
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 …
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 …
particular for the explicit description of volume-preserving affine-equivariant numerical …
Measure preservation and integrals for Lotka--Volterra -systems and their Kahan discretisation
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 …
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 …
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 …
novel 8-parameter integrable map in R4 GRW Quispel, DI McLaren and PH van der Kamp …