检测和确定两地地图的保存度量和积分

IF 1 Q3 Engineering

Journal of Computational Dynamics Pub Date : 2022-01-01 DOI:10.3934/jcd.2022014

E. Celledoni, C. Evripidou, D. McLaren, B. Owren, G. Quispel, B. Tapley

{"title":"检测和确定两地地图的保存度量和积分","authors":"E. Celledoni, C. Evripidou, D. McLaren, B. Owren, G. Quispel, B. Tapley","doi":"10.3934/jcd.2022014","DOIUrl":null,"url":null,"abstract":"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 Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps, thus lending weight to a previous ansatz [2]. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.","PeriodicalId":37526,"journal":{"name":"Journal of Computational Dynamics","volume":"104 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Detecting and determining preserved measures and integrals of birational maps\",\"authors\":\"E. Celledoni, C. Evripidou, D. McLaren, B. Owren, G. Quispel, B. Tapley\",\"doi\":\"10.3934/jcd.2022014\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"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 Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps, thus lending weight to a previous ansatz [2]. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.\",\"PeriodicalId\":37526,\"journal\":{\"name\":\"Journal of Computational Dynamics\",\"volume\":\"104 1\",\"pages\":\"\"},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Computational Dynamics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3934/jcd.2022014\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"Engineering\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Computational Dynamics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/jcd.2022014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Engineering","Score":null,"Total":0}

引用次数: 5

摘要

本文利用离散达布多项式的方法计算有理图的保留测度和积分。该方法基于协因式和达布多项式的使用，并依赖于符号代数工具的使用。给定足够的计算能力，可以找到大多数(如果不是全部的话)保留的有理积分(甚至是一些非理性积分)。在一些例子中，我们展示了如何使用这种方法来确定和检测所考虑的有理映射的保留测度和积分，从而为之前的分析提供权重[2]。许多例子来自Kahan-Hirota-Kimura对完全可积常微分方程系统的离散化。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Detecting and determining preserved measures and integrals of birational maps

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 Darboux polynomials and relies on the use of symbolic algebra tools. Given sufficient computing power, most, if not all, rational preserved integrals can be found (and even some non-rational ones). We show, in a number of examples, how it is possible to use this method to both determine and detect preserved measures and integrals of the considered rational maps, thus lending weight to a previous ansatz [2]. Many of the examples arise from the Kahan-Hirota-Kimura discretization of completely integrable systems of ordinary differential equations.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Journal of Computational Dynamics Engineering-Computational Mechanics

CiteScore

2.30

自引率

10.00%

发文量

期刊介绍： JCD is focused on the intersection of computation with deterministic and stochastic dynamics. The mission of the journal is to publish papers that explore new computational methods for analyzing dynamic problems or use novel dynamical methods to improve computation. The subject matter of JCD includes both fundamental mathematical contributions and applications to problems from science and engineering. A non-exhaustive list of topics includes * Computation of phase-space structures and bifurcations * Multi-time-scale methods * Structure-preserving integration * Nonlinear and stochastic model reduction * Set-valued numerical techniques * Network and distributed dynamics JCD includes both original research and survey papers that give a detailed and illuminating treatment of an important area of current interest. The editorial board of JCD consists of world-leading researchers from mathematics, engineering, and science, all of whom are experts in both computational methods and the theory of dynamical systems.