303 citations · 399 across the 20 of their papers we have counts for
22 papers
Liouville integrable Lotka-Volterra systems
Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel
We present homogeneous -parameter families of Liouville integrable - and -dimensional Lotka-Volt…
On trilinear and quadrilinear equations associated with the lattice Gel'fand-Dikii hierarchy
P. H. van der Kamp, F. W. Nijhoff, D. I. McLaren +1
Introduced in 2012, by Zhang, Zhao, and Nijhoff, the trilinear Boussinesq equation is the natural form of the equation for the -function of the lattice Boussinesq system. In thi…
On a quadratic Poisson algebra and integrable Lotka-Volterra systems with solutions in terms of Lambert's W function
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel
We study a class of integrable nonhomogeneous Lotka-Volterra systems whose quadratic terms are defined by an antisymmetric matrix and whose linear terms consist of three blocks. We…
Trees and superintegrable Lotka-Volterra families
Peter H. van der Kamp, G. R. W. Quispel, D. I. McLaren
To any tree on vertices we associate an -dimensional Lotka-Volterra system with parameters and, for generic values of the parameters, prove it is superintegrable, i.e…
Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation
Peter H. van der Kamp, Robert I. McLachlan, David I. McLaren +1
We show that any Lotka--Volterra tree-system associated with an -vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also…
Birational maps from polarization and the preservation of measure and integrals
Robert I McLachlan, David I McLaren, G R W Quispel
The main result of this paper is the discretization of Hamiltonian systems of the form , where is a constant symmetric matrix and $W\colon\mathbb{R}^n…
Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families
G. R. W. Quispel, Benjamin K. Tapley, D. I. McLaren +1
We present a method to construct superintegrable -component Lotka-Volterra systems with parameters. We apply the method to Lotka-Volterra systems with components for…
Deducing properties of ODEs from their discretization
G. R. W. Quispel, D. I. McLaren, C. Evripidou
We show that some hard to detect properties of quadratic ODEs (eg certain preserved integrals and measures) can be deduced more or less algorithmically from their Kahan discretizat…
A novel 8-parameter integrable map in
G R W Quispel, D I McLaren, P H van der Kamp
We present a novel 8-parameter integrable map in . The map is measure-preserving and possesses two functionally independent 2-integrals, as well as a measure-preservi…
Homogeneous Darboux polynomials and generalising integrable ODE systems
Peter H. van der Kamp, D. I. McLaren, G. R. W. Quispel
We show that any system of ODEs can be modified whilst preserving its homogeneous Darboux polynomials. We employ the result to generalise a hierarchy of integrable Lotka-Volterra s…
Using discrete Darboux polynomials to detect and determine preserved measures and integrals of rational maps
E Celledoni, C Evripidou, D I McLaren +4
In this Letter we propose a systematic approach for detecting and calculating preserved measures and integrals of a rational map. The approach is based on the use of cofactors and…
Detecting and determining preserved measures and integrals of birational maps
Elena Celledoni, Charalambos Evripidou, David McLaren +3
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 Da…
Three classes of quadratic vector fields for which the Kahan discretization is the root of a generalised Manin transformation
Peter H. van der Kamp, Elena Celledoni, Robert I. McLachlan +3
We apply Kahan's discretisation method to three classes of 2-dimensional quadratic vector fields with quadratic, resp cubic, resp quartic Hamiltonians. We show that the maps obtain…
Generalised Manin transformations and QRT maps
Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel
Manin transformations are maps of the plane that preserve a pencil of cubic curves. They are the composition of two involutions. Each involution is constructed in terms of an invol…
Geometric and integrability properties of Kahan's method
Elena Celledoni, David McLaren, Brynjulf Owren +1
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 discre…
Two classes of quadratic vector fields for which the Kahan discretization is integrable
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
Applying Kahan's discretization to the reduced Nahm equations, we obtain two classes of integrable mappings.
The use of discrete gradient methods for total variation type regularization problems in image processing
V Grimm, R I McLachlan, D McLaren +2
Discrete gradient methods are well-known methods of Geometric Numerical Integration, which preserve the dissipation of gradient systems. The preservation of the dissipation of a sy…
Volume Preservation by Runge-Kutta Methods
Philipp Bader, David I McLaren, G. R. W. Quispel +1
It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but i…
Discretization of polynomial vector fields by polarization
Elena Celledoni, Robert I. McLachlan, David I. McLaren +2
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 energy whe…
Integrability properties of Kahan's method
Elena Celledoni, Robert I McLachlan, David I McLaren +2
We present several novel examples of integrable quadratic vector fields for which Kahan's discretization method preserves integrability. Our examples include generalized Suslov and…
Projection methods and discrete gradient methods for preserving first integrals of ODEs
R. A. Norton, D. I. McLaren, G. R. W. Quispel +2
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show…
Preserving energy resp. dissipation in numerical PDEs using the "Average Vector Field" method
E. Celledoni, V. Grimm, R. I. McLachlan +4
We give a systematic method for discretizing Hamiltonian partial differential equations (PDEs) with constant symplectic structure, while preserving their energy exactly. The same m…