Investigating Nonlinear Dynamics with Topological Methods

该项目从应用和理论两个角度研究了拓扑学和非线性动力学的相互作用。复杂的拓扑结构自然而然地出现在非线性动力系统中，经常“承载”系统的动态特性。理解这种拓扑结构对于理解系统的动态和长期行为是必不可少的。计划的内容是：(I)具有给定性质的“大多数”(即一般的)动力系统中存在的拓扑学的进一步理论研究，(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)保面积系统的理论和应用研究，(Iv)一维动力系统中拓扑与不变测度如何相关的调查，以及(V)拓扑马蹄形的进一步研究(与J.A.York ke和Maryland小组的其他人)。该项目建立在研究人员已经单独完成的工作基础上，并与其他数学家和科学家合作。这项工程需要并将使用拓扑学和动力学两方面的工具。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系统，如天气，或不同流体在搅拌时的运动，或太阳系，在运行时，随着集合相互映射，有趣的拓扑结构就会出现。理解这些集合对于理解非线性系统的复杂行为是至关重要的。有时复杂的拓扑可以在形成时观察到，它是某些类型的复杂动力学的标志。这能说得很准确吗？它能被“量化”吗？例如，想一想研究人员特别感兴趣的一种现象--搅拌。搅拌在每个厨房都会发生，而且似乎是一个简单的过程。但情况远非如此：在工业环境和自然界中，搅拌发生在许多不同的环境中，因为化学品或药物，或冷热空气，或冷热水，混合在一起。在工业中，最常见的目标是高效地实现混合物质的均质。化学工程师的实验和模拟结果表明，目前使用的这些过程并不能产生均质性。如何改进这些过程？这是这个项目的结果应该适用的问题之一。更广泛地说，计划的项目是研究复杂动态系统的拓扑，以及复杂拓扑的存在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.