Continuous Dynamical Systems

连续动力系统

• 批准号: 0204081
• 负责人: Krystyna Kuperberg
• 金额: $ 16.47万
• 依托单位: Auburn University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-15 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0204081&HistoricalAwards=false
• 关键词: Continuous; Dynamical; Systems

Proposal Number: DMS-0204081 PI: Kuperbeg ABSTRACTResearch will be conducted on continuous dynamical systems onmanifolds of dimension three and higher, with emphasis on compact invariant subsets on which the flow is minimal. Continuing the work related to the real analytic solution to the Seifert conjecture on aperiodic flows acting on the three-dimensional sphere, the PI plans to classify the minimal sets in such flows with respect to their homological and shape theory properties. Minimal sets shape-equivalent (in the senseof Borsuk) to a polyhedron often appear in flows asattractors, and are of special interest. Minimal sets of non-polyhedral shape are usually approximated by circular orbits or by other movable sets with the so called UV-property.They have very complicated dynamics around them, which can be investigated through the structure of the family of the approximating invariant sets. Circular orbits are extremelyimportant in celestial mechanics as they appear as paths of movement. Smaller particles, dust, can align along a braided orbit, and in the continuous case, in a mathematical model, this "orbit" could be an invariant set, aperiodic but "circle-like", or "graph-like" such as the well-known Denjoy continua. A volume-preserving dynamical system, associated with incompressible fluids, imposes restrictions on the flow outside the special invariant collection and is an important area of study as well. The theory of dynamical systems was developed in an effort toprovide a mathematically rigorous description of real physical phenomena. It is therefore closely connected to many domains of science: mechanics, various areas of physics, biology, and economics. Dynamical systems connect several branches of mathematics such as analysis, topology, geometry, algebra, and combinatorics.

提案编号：DMS-0204081 PI：Kuperbeg 摘要 研究将针对三维及更高维流形上的连续动力系统进行，重点是流动最小的紧凑不变子集。继续与作用于三维球体的非周期流的 Seifert 猜想的实解析解相关的工作，PI 计划根据此类流中的最小集的同调和形状理论属性对它们进行分类。与多面体形状等价（在 Borsuk 意义上）的最小集合经常作为吸引子出现在流中，并且受到特别关注。非多面体形状的最小集通常由圆形轨道或具有所谓UV性质的其他可移动集来近似。它们周围具有非常复杂的动力学，可以通过近似不变集族的结构来研究它们。圆形轨道在天体力学中极其重要，因为它们表现为运动路径。较小的颗粒，灰尘，可以沿着编织轨道排列，在连续的情况下，在数学模型中，这个“轨道”可以是一个不变的集合，非周期但“类圆形”，或“类图形”，例如著名的 Denjoy 连续体。与不可压缩流体相关的体积保持动力系统对特殊不变集合之外的流动施加限制，也是一个重要的研究领域。 动力系统理论的发展是为了对真实物理现象提供严格的数学描述。因此，它与许多科学领域密切相关：力学、物理学、生物学和经济学的各个领域。 动力系统连接了多个数学分支，例如分析、拓扑、几何、代数和组合数学。