Continuous Dynamical Systems

建议编号：DMS-0204081 PI：Kuperbeg摘要将对三维及更高维流形上的连续动力系统进行研究，重点是流动最小的紧致不变子集。继续关于作用在三维球面上的非周期流动的Seifert猜想的实解析解的工作，PI计划根据它们的同调和形状理论性质来分类这种流动中的极小集。极小集形状等价(在Borsuk意义下)多面体经常作为吸引子出现在流中，并且具有特殊的意义。非多面体形状的极小集通常被圆轨道或其他具有UV性质的可移动集逼近，它们具有非常复杂的动力学特性，可以通过逼近不变集族的结构来研究。圆形轨道在天体力学中是极其重要的，因为它们看起来像是运动的路径。更小的粒子，尘埃，可以沿着辫子轨道排列，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“圆形”，或“图形”，如众所周知的Denavy连续体。与不可压缩流体有关的体积守恒动力系统对特殊不变量集合之外的流动施加了限制，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的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、物理学的各个领域、生物学和经济学。动力系统连接了几个数学分支，如分析、拓扑、几何、代数和组合学。

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.