[图书][B] Nonlinear Systems and Their Remarkable Mathematical Structures: Volume 1

N Euler - 2018 - taylorfrancis.com
Nonlinear Systems and Their Remarkable Mathematical Structures, Volume 1 aims to
describe the recent progress in nonlinear differential equations and nonlinear dynamical …

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 …

A three-dimensional generalization of QRT maps

J Alonso, YB Suris, K Wei - Journal of Nonlinear Science, 2023 - Springer
We propose a geometric construction of three-dimensional birational maps that preserve two
pencils of quadrics. The maps act as compositions of involutions, which, in turn, act along …

Linearly implicit local and global energy-preserving methods for PDEs with a cubic Hamiltonian

S Eidnes, L Li - SIAM Journal on Scientific Computing, 2020 - SIAM
We present linearly implicit methods that preserve discrete approximations to local and
global energy conservation laws for multisymplectic partial differential equations with cubic …

Discretization of polynomial vector fields by polarization

E Celledoni, RI McLachlan… - … of the Royal …, 2015 - royalsocietypublishing.org
A novel integration method for quadratic vector fields was introduced by Kahan in 1993.
Subsequently, it was shown that Kahan's method preserves a (modified) measure 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 …

Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps

E Celledoni, C Evripidou, DI McLaren… - Journal of Physics A …, 2019 - iopscience.iop.org
In this letter we propose a systematic approach for detecting and calculating integrals and
preserved measures of rational maps. The approach is based on the use of cofactors and …

Geometric and integrability properties of Kahan's method: the preservation of certain quadratic integrals

E Celledoni, DI McLaren, B Owren… - Journal of Physics A …, 2019 - iopscience.iop.org
Given a quadratic vector field on possessing a quadratic first integral depending on two of
the independent variables, we give a constructive proof that Kahan's discretization method …

Growth and integrability of some birational maps in dimension three

M Graffeo, G Gubbiotti - Annales Henri Poincaré, 2024 - Springer
Motivated by the study of the Kahan–Hirota–Kimura discretisation of the Euler top, we
characterise the growth and integrability properties of a collection of elements in the …