Stabilization Methods for Dynamical Systems

动力系统的稳定方法

• 批准号: 2247553
• 负责人: Scott Schmieding
• 金额: $ 33.2万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-04-15 至 2026-03-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247553&HistoricalAwards=false
• 关键词: Stabilization; Methods; Dynamical; Systems

The study of dynamical systems concerns the structure of systems which evolve in time. Their study connects with several areas within mathematics, as well as physics and other fields of science. This project is focused on groups of symmetries of dynamical systems, exploring how properties of the underlying system manifest in, and constrain, the structure of these symmetry groups. A key class of systems are the hyperbolic homeomorphisms, and the core of the project concerns a new stabilized framework for studying the collection of automorphisms of these systems. Symbolic systems, including shifts of finite type, also play a key role, and have important connections to other areas such as ergodic theory and information theory, and their symmetry groups exhibit a wide range of complexity. This project will advance the field by investigating new techniques and invariants for classifying symmetries of systems through the use of stabilization. The project also provides opportunities for graduate students to conduct research. The project is aimed at a notion of reconstruction: the extent to which the structure of symmetry groups, and their stabilized counterparts, can be used to reconstruct various dynamical properties of the underlying system. The PI intends to further develop and apply various stabilized invariants, notable local entropy, to the study of such automorphism groups with the goal of recovering dynamical invariants, such as topological entropy and zeta functions. Such invariants have been used successfully in the symbolic setting, and one aim of the project is to extend this to certain smooth settings. One goal of the project is to study the question of whether the stabilized automorphism group of a shift of finite type is a complete invariant up to topological conjugacy. In addition, the project explores the extent to which stabilized results, and invariants derived from there, might be used to analyze the classical automorphism groups.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对动态系统的研究涉及随时间发展的系统的结构。他们的研究与数学以及物理和其他科学领域的几个领域联系在一起。该项目的重点是动态系统的对称组，探讨了基础系统的属性如何在这些对称组的结构中表现出来并约束。一类关键的系统是双曲线同构的，该项目的核心涉及一个新的稳定框架，用于研究这些系统的自动形态的收集。符号系统，包括有限类型的转移，也起着关键作用，并与其他领域（例如ergodic理论和信息理论）及其对称群体具有重要的联系。该项目将通过研究新技术和不变式来通过使用稳定化来对系统对称进行分类，从而推进该领域。该项目还为研究生提供了研究的机会。该项目针对重建的概念：对称组的结构及其稳定对应物的结构可用于重建基础系统的各种动力学特性。 PI打算进一步开发和应用各种稳定的不变性，值得注意的局部熵，以研究此类自动形态群体，目的是恢复动态不变性，例如拓扑熵和ZETA功能。此类不变的人已在符号环境中成功使用，该项目的一个目的是将其扩展到某些平滑的设置。该项目的一个目标是研究有限类型转变的稳定自动构成群体是否是完全不变的拓扑结合性的问题。此外，该项目探讨了稳定结果的程度以及从那里得出的不变性，可用于分析经典的自身形态群体。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的智力优点和更广泛的影响审查标准通过评估来进行评估的。