Investigating Nonlinear Dynamics with Topological Methods

该项目从应用和理论两个角度研究拓扑学和非线性动力学的相互作用。复杂拓扑在非线性动力系统中自然产生，并且经常“携带”系统的动力学。理解这种拓扑结构对于理解这种结构的动力学和长期行为是至关重要的。计划是（i）对“大多数”拓扑结构的进一步理论研究（即，（ii）搅拌和湍流的应用研究（与罗格斯大学的化学工程师合作），（iii）面积保持系统的理论和应用研究，（iv）一维动力系统中拓扑和不变测度之间关系的研究，（v）拓扑马蹄铁的进一步研究（与J.A. Yorke和其他来自马里兰州的人）。该项目的基础上已经完成的工作由调查bothalone和与其他数学家和科学家合作。拓扑学和动力系统的工具都是需要的，并将用于执行该项目。有关搅拌的理论结果将得到实验室实际实验和观察的支持。 每当一个非线性动力学系统，如天气，或不同流体的运动，因为他们被搅拌，或太阳系，是“操作”，有趣的拓扑结构出现作为集映射回本身。理解这些集合对于理解非线性系统的复杂行为至关重要。有时，复杂的拓扑结构可以在它形成时观察到，并且是某些复杂动力学的标志。这可以精确吗？可以“量化”吗？例如，考虑搅拌，这是研究者特别感兴趣的现象。搅拌发生在每个厨房，似乎是一个简单的过程。但事实远非如此：搅拌发生在工业环境中，也发生在自然界中许多不同的环境中，如化学品或药物，或冷热空气，或冷热水混合。在工业中，最常见的目标是有效地实现混合物质的均匀性。化学工程师的实验和模拟结果表明，目前使用的那些工艺不能得到均匀的结果。如何改进这些工艺？这是本项目的结果应该适用的问题之一。更一般地说，计划的项目是复杂动态系统的拓扑结构的研究，以及复杂拓扑结构对非线性系统的影响，反之亦然。

The project is an investigation of the interplay of topology and nonlinear dynamics from both applied and theoretical viewpoints. Complicatedtopology arises naturally in nonlinear dynamical systems and frequently"carries" the dynamics of the system. Understanding this topology isessential to understanding the dynamics and the long-term behavior of thesystem. Planned are (i) further theoretical study of the topology presentin "most" (i.e., generic) dynamical systems with a given property, (ii)applied study of stirring and turbulence (with chemical engineers atRutgers), (iii) theoretical and applied study of area-preserving systems,(iv) an investigation of how topology and invariant measures arerelated in one-dimesional dynamical systems, and (v) further study oftopological horseshoes (with J. A. Yorke and others from the Marylandgroup). The project builds on work already done by the investigator bothalone and in collaboration with other mathematicians and scientists. Thetools of both topology and dynamical systems are needed and will be usedto carry out the project. Theoretical results concerning stirring will bebacked up by actual experiment and observation in the lab. Whenever a nonlinear dynamical system, such as the weather, or the motion of different fluids as they are stirred, or the solar system, is"operating", interesting topology arises as sets are mapped back acrossthemselves. Understanding these sets is crucial to understanding thecomplicated behavior of nonlinear systems. Sometimes the complicatedtopology can be observed as it forms, and is a marker for certain kindsof complicated dynamics. Can this be made precise? Can it be "quantified"?Consider stirring, a phenomenon of special interest to the investigator,for example. Stirring occurs in every kitchen, and seems to be a simpleprocess. But this is far from the case: Stirring occurs in industrialsettings and in nature in many different contexts as chemicals ormedicines, or hot and cold air, or hot and cold water, are mixed. Mostoften in industry achieving homogeneity of the mixed substance efficientlyis the goal. Experimental and simulated results by chemical engineersdemonstrate that those processes used today do not result in homogeneity.How can those processes be improved? That is one of the problems to whichthe results of this project should be applicable. More generally, theproject planned is a study of the topology of complicated dynamicalsystems, and both the implications of the presence of complicated topologyfor nonlinear systems and vice versa.