Robust and generic mechanisms in smooth dynamics

平稳动力学中稳健且通用的机制

• 批准号: 1402852
• 负责人: Anne Wilkinson
• 金额: $ 60万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2020-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1402852&HistoricalAwards=false
• 关键词: Robust; generic; mechanisms; smooth; dynamics

The PI is a leading expert in the field of partially hyperbolic dynamics; dynamics is the study of systems, physical or mathematical, that evolve over time according to a deterministic set of rules. Hyperbolic dynamical systems are those which display chaotic, unpredictable features at every point; they are both naturally occurring and well-studied. There are other dynamical systems called KAM systems (named after Kolomogorov, Arnold, and Moser), which have stable regions of regular motion. Partially hyperbolic systems provide a more general class of dynamical systems than either, and include systems that combine hyperbolicity in some directions with KAM behavior in other directions. Partially hyperbolic systems occur widely in dynamical systems arising in physics; for example planetary motion usually contains partially hyperbolic subdynamics, and the effective construction of particle accelerators (used in biological imaging, as well as theoretical physics) requires a detailed understanding of both KAM and partially hyperbolic dynamics. The PI has a well-developed research plan of over 15 years studying partially hyperbolic systems and is poised to raise the theory of these systems to a new level of generality and applicability. The impacts of this research will be seen in future applications to systems in biology, physics and engineering. The PI is currently collaborating with the particle accelerator group at Fermilab to explore some of these potential applications.The research supported by this grant is guided by the far-reaching goal of developing a general theory of partially hyperbolic systems along the lines of the hyperbolic theory developed in the past 40 years. In particular the PI proposes to study: ergodic properties of conservative partially hyperbolic diffeomorphisms; physical measures for (dissipative) partially hyperbolic diffeomorphisms; rigidity phenomena connected to partially hyperbolic group actions; and ergodicty of singular partially hyperbolic systems. These research goals will be carried out through a variety of modalities, including published papers in peer-reviewed journals, supervising Ph.D. students, and public speaking, both at research conferences and to the general public.

PI是部分双曲动力学领域的领先专家。动力学是对物理或数学的系统的研究，它们会根据确定性的规则集随时间而发展。 双曲线动力学系统是每个点都显示混乱，不可预测的功能的系统。 它们都是自然发生的，并且是经过充分研究的。 还有其他称为KAM系统的动力系统（以Kolomogorov，Arnold和Moser的名字命名），它们具有常规运动的稳定区域。 部分双曲系统提供的动态系统比任何一个都提供了更一般的动力系统，并包括将某些方向上的双曲线与其他方向上的KAM行为相结合的系统。 部分双曲系统广泛存在于物理中引起的动力系统中。例如，行星运动通常包含部分双曲线亚动力学，并且有效地构造了粒子加速器（用于生物成像以及理论物理学）需要详细了解KAM和部分双曲动力学。 PI拥有一项完善的研究计划，该计划已超过15年，研究部分双曲线系统，并准备将这些系统的理论提升到新的一般性和适用性水平。这项研究的影响将在未来对生物学，物理和工程系统中的应用中进行。 PI目前正在Fermilab的粒子加速器组合作，以探索其中一些潜在的应用。该赠款支持的研究是由过去40年来开发的双曲线理论沿途发展的一般双曲机系统的一般理论的指导。特别是PI提出的研究：保守的部分双曲线差异； （耗散）部分双曲线差异的物理措施；刚度与部分双曲组作用有关的僵化现象；和奇异的部分双曲系统。 这些研究目标将通过多种方式实现，包括在同行评审期刊中发表的论文，监督博士学位。学生和公开演讲，包括研究会议和公众。