Kahan Discretizations of Skew-Symmetric Lotka-Volterra Systems and Poisson Maps

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2021-07-30 DOI:10.1007/s11040-021-09399-x

C. A. Evripidou, P. Kassotakis, P. Vanhaecke

引用次数: 1

Abstract

The Kahan discretization of the Lotka-Volterra system, associated with any skew-symmetric graph Γ, leads to a family of rational maps, parametrized by the step size. When these maps are Poisson maps with respect to the quadratic Poisson structure of the Lotka-Volterra system, we say that the graph Γ has the Kahan-Poisson property. We show that if Γ is connected, it has the Kahan-Poisson property if and only if it is a cloning of a graph with vertices \(1,2,\dots ,n\), with an arc i → j precisely when i < j, and with all arcs having the same value. We also prove a similar result for augmented graphs, which correspond with deformed Lotka-Volterra systems and show that the obtained Lotka-Volterra systems and their Kahan discretizations are superintegrable as well as Liouville integrable.

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斜对称Lotka-Volterra系统和泊松映射的Kahan离散化

Lotka-Volterra系统的Kahan离散化，与任何偏对称图Γ相关联，导致一组有理映射，由步长参数化。当这些映射是Lotka-Volterra系统的二次泊松结构的泊松映射时，我们说图Γ具有Kahan-Poisson性质。我们证明了如果Γ是连通的，当且仅当它是一个顶点为\(1,2,\dots ,n\)的图的克隆时，它具有Kahan-Poisson性质，当i ＆lt;J，所有的弧都有相同的值。我们还证明了与变形Lotka-Volterra系统对应的增广图的类似结果，并证明了所得到的Lotka-Volterra系统及其Kahan离散化是超可积的，也是Liouville可积的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.