Computational Tools for Dynamical Systems

动力系统的计算工具

• 批准号: 1602300
• 负责人: Benjamin Hutz
• 金额: $ 8.33万
• 依托单位: Saint Louis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1602300&HistoricalAwards=false
• 关键词: Computational; Tools; Dynamical; Systems

Over the last several decades, the use of computers in the science, engineering, and medicine has expanded enormously. With the increasing complexity of problems to be solved and the increasing capacity of hardware to solve these problems, the field of computational mathematics became more essential. The state of current computational tools is still behind the state of current research. This project aims to narrow that gap in the area of dynamical systems by the resolution of open computational problems and their implementation in the computer algebra system: Sage. Dynamical systems have a myriad of applications throughout the sciences. They include time-reversing symmetries in physics, chaotic systems in weather prediction, cellular and genetic processes in biology and medicine, and cryptography and information security. With the increasing number of researchers in dynamical systems, there is a distinct need for a comprehensive set of computational tools. These tools will allow researchers to quickly and efficiently test and refine theories and to compute complicated examples. In addition, students can use these tools to quickly become involved in current research problems.In addition to expanding the general set of computational tools for dynamical systems, this project specifically addresses problems in the arithmetic of dynamical systems, and number theoretic questions arising from iterating functions. Of particular interest are points and subvarieties with finite forward orbits, called preperiodic. This project aims to study the computational problems associated with rational preperiodic points and canonical heights over number fields and iteration and canonical heights for subvarieties. Both aspects of this project have direct applications to working with post-critically finite maps.

在过去的几十年里，计算机在科学、工程和医学领域的应用得到了极大的扩展。 随着要解决的问题日益复杂以及解决这些问题的硬件能力不断增强，计算数学领域变得更加重要。当前计算工具的状态仍然落后于当前研究的状态。该项目旨在通过解决开放计算问题及其在计算机代数系统：Sage 中的实现来缩小动力系统领域的差距。动力系统在整个科学领域有着无数的应用。它们包括物理学中的时间反转对称性、天气预报中的混沌系统、生物学和医学中的细胞和遗传过程，以及密码学和信息安全。随着动力系统研究人员数量的不断增加，对一套全面的计算工具的需求明显增加。这些工具将使研究人员能够快速有效地测试和完善理论并计算复杂的示例。此外，学生可以使用这些工具快速参与当前的研究问题。除了扩展动力系统的通用计算工具集之外，该项目还专门解决动力系统的算术问题以及迭代函数引起的数论问题。特别令人感兴趣的是具有有限前向轨道的点和子类，称为前周期。该项目旨在研究与数域上的有理前周期点和规范高度以及子品种的迭代和规范高度相关的计算问题。该项目的两个方面都可以直接应用于处理后临界有限地图。