Continuous Dynamical Systems

提案编号：DMS-0204081 PI：Kuperbeg摘要研究将在三维和更高维流形上进行连续动力系统，重点是流最小的紧致不变子集。继续工作有关的真实的解析解塞弗特猜想的非周期性流动作用于三维领域，PI计划分类的最小集在这样的流量方面的同调和形状理论的属性。形状等效（在Borsuk的意义上）于多面体的最小集合经常作为吸引子出现在流中，并且特别有趣。非多面体形状的极小集通常用圆轨道或其他具有UV-性质的可动集来近似，它们具有非常复杂的动力学性质，这些性质可以通过近似不变集族的结构来研究。圆轨道在天体力学中是极其重要的，因为它们是运动的路径。更小的粒子，尘埃，可以沿着编织的轨道排列，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，非周期性的，但“圆形”，或“图形”，如著名的Denjoy连续。与不可压缩流体相关联的保体积动力系统对特殊不变集合之外的流动施加限制，并且也是研究的重要领域。 动力系统理论的发展是为了给真实的物理现象提供一种数学上的严格描述。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。 动力系统连接了数学的几个分支，如分析、拓扑学、几何学、代数学和组合学。

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.