New Directions in Partially Hyperbolic Dynamics

部分双曲动力学的新方向

• 批准号: 1823150
• 负责人: Andriy Gogolyev
• 金额: $ 11.74万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-09-18 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1823150&HistoricalAwards=false
• 关键词: New; Directions; Partially; Hyperbolic; Dynamics; New; Directions; Partially; Hyperbolic; Dynamics

This project seeks to gain a deeper understanding of chaotic dynamical systems. Chaotic dynamics is abundant in the real world and appears throughout sciences and engineering. It can appear in simple mechanical mechanisms such as a double pendulum or in complicated natural phenomena such as convection currents in the atmosphere. The study of such naturally arising examples is extremely challenging and we are very far from a good understanding of them. The partially hyperbolic dynamical systems that will be studied in this project are simplified mathematical models of chaotic dynamical systems that arise in nature. The emphasis of the project will be on a wide range of problems which are new in the field or have received only marginal attention in the past. Potentially, this research may have impacts in physics, biology, chemistry, engineering and other areas where chaotic dynamical systems arise.Anosov and partially hyperbolic diffeomorphisms are the prime examples of chaotic dynamical systems. In the past decades the study of ergodic and chaotic properties in partially hyperbolic dynamics has flourished, while geometric aspects haven't received as much attention. This will be a wide-ranging program to develop geometric structure theory for partially hyperbolic dynamics. The program includes: (1) further development of the Anosov bundle theory; (2) development of topological and global structural stability for partially hyperbolic diffeomorphisms; (3) initiation of the smooth conjugacy program for 3-dimensional partially hyperbolic diffeomorphisms. Further, the project will to bring partially hyperbolic vision beyond the dominated splitting paradigm and take first steps into the realm of homological and coarse hyperbolicity. Finally, the project will expand work on new examples in partially hyperbolic dynamics. New examples have intrinsic value and can be used as additional ground for the structure theory to be applied and tested. The project will use a diverse blend of dynamics and three and higher dimensional topology.

该项目旨在深入了解混乱的动力系统。 混乱的动力学在现实世界中很丰富，并且在整个科学和工程中出现。 它可以以简单的机械机制出现，例如双摆或复杂的自然现象，例如大气中的对流电流。 对这种自然产生的例子的研究极具挑战性，我们对它们的了解很远。 该项目将在本项目中研究的部分双曲动力系统是自然界中出现的混乱动力学系统的简化数学模型。 该项目的重点将是多种问题，这些问题在该领域是新的，或者过去仅受到边际关注。 这项研究可能会影响物理，生物学，化学，工程和混乱动力学系统出现的其他领域。Anosov和部分双曲线差异性是混乱动力学系统的主要例子。在过去的几十年中，部分双曲线动力学中对奇异和混乱的性能的研究蓬勃发展，而几何方面并没有得到太多关注。 这将是一个开发部分双曲动力学的几何结构理论的广泛程序。 该计划包括：（1）进一步发展Anosov捆绑理论； （2）发展部分双曲线差异的拓扑和全球结构稳定性； （3）启动平滑共轭程序，用于三维部分双曲线差异。 此外，该项目将使部分双曲线视觉超出主导的分裂范式，并迈入同源和粗糙的双曲线领域的第一步。最后，该项目将在部分双曲动力学中扩展有关新示例的工作。新示例具有内在价值，可以用作应用和测试结构理论的附加基础。该项目将使用各种动力学和三维拓扑结合。