Nonlinear Dynamics with Applications to Physical Systems

非线性动力学及其在物理系统中的应用

• 批准号: 0205128
• 负责人: Mark Levi
• 金额: $ 24万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205128&HistoricalAwards=false
• 关键词: Nonlinear; Dynamics; Applications; Physical; Systems

Proposal #0205128PI: Mark LeviInstitution: Penn State UniversityTitle: Nonlinear Dynamics with Applications to Physical SystemsABSTRACTThis project consists of three parts. The first part deals with a fundamental theoretical question of dynamics: To understand the nature of physically observable chaos. Despite the great amount of work done over the past five decades, the fundamental question of demonstrating genuine chaotic behavior in a physically realistic differential equation has not been answered. The answer seems finally within grasp, due to incremental progress in understanding two-dimensional mappings on the one hand and some new insights into differential equations on the other. The second part of the project deals with rapidly vibrating systems. High-frequency oscillations lead to fascinating phenomena, which have found their use in such areas as particle confinement, particle accelerators, and laser tweezers. A recent observation of the author shed a new light on these phenomena and opened connections with other fields, which will be further explored in this project, along with possible new applications. The third part of the project deals with the mathematical analysis of a geometrical object (a class of area-preserving cylinder maps) which is of independent mathematical interest on the one hand and gives insight into physical systems on the other. These systems include (i) charged particles in magnetic fields, (ii) the Frenkel-Kontorova model of an equilibrium configuration of electrons in a crystal lattice, and (iii) arrays of Josephson junctions.This project involves three different research directions. The first direction addresses a major gap in our understanding of how chaos really manifests itself in physical systems. Surprising as it may seem, despite the large amount of sometimes deep work in the field of "chaos," chaotic behavior has not been proven (understood in the mathematical sense) in realistic physical systems. The nature of chaos has really been understood only in mathematical models that are too simplified to include some key features of real systems (although even those simplified models can be quite complex). The theory, however, has reached a level where such understanding seems finally within grasp. A mathematical proof of the existence of chaos for a realistic physical system would be the culmination of a long development over the past five decades or more. The second direction of the project deals with the study of physical systems with rapid oscillations, where phenomena occur that are both utterly fascinating and useful. For example, subjecting the end of an open bicycle chain to high-frequency vibrations will, in theory, enable the chain to stand up stably (!) on its end. The theory which underlies this curiosity is also responsible for the workings of such seemingly unrelated devices as particle traps, particle accelerators, laser tweezers (which allow one to move a particle inside a cell by a laser beam without breaking the cell wall), and acoustic levitators. Over the past few years, the author has found an explanation of these effects; prior theory predicted the result but did not answer the "why." The explanation opened new directions, which we plan to explore further. We plan, in particular, to better understand the concept of vibrational control and explore the use of vibration for filtering (using acoustic waves). The third direction is of basic mathematical interest but has significance in several physical situations. The mathematical theory will give insight, otherwise inaccessible, into at least three different phenomena: (i) the behavior of particles in magnetic fields of certain patterns; (ii) the electric conductivity of certain types of crystals, and (iii) the electric behavior of some arrays of Josephson junctions known as superconducting quantum interference devices (the so-called SQUIDS).

提案#0205128PI: Mark LeviInstitution：宾夕法尼亚州立大学标题：非线性动力学及其在物理系统中的应用摘要本项目由三部分组成。第一部分涉及动力学的一个基本理论问题：理解物理上可观察到的混沌的本质。尽管在过去的五十年里做了大量的工作，但在物理上真实的微分方程中证明真正的混沌行为的基本问题还没有得到回答。答案似乎终于触手可及，一方面是由于对二维映射的理解取得了渐进式的进展，另一方面是由于对微分方程的一些新见解。项目的第二部分涉及快速振动系统。高频振荡导致了令人着迷的现象，这些现象已经在粒子限制、粒子加速器和激光镊子等领域得到了应用。作者最近的一次观察为这些现象提供了新的视角，并打开了与其他领域的联系，这将在本项目中进一步探索，以及可能的新应用。该项目的第三部分涉及几何对象（一类保持面积的柱面图）的数学分析，这一方面具有独立的数学兴趣，另一方面提供了对物理系统的洞察。这些系统包括(i)磁场中的带电粒子，（ii）晶格中电子平衡构型的Frenkel-Kontorova模型，以及（iii） Josephson结阵列。这个项目涉及三个不同的研究方向。第一个方向解决了我们对混沌如何在物理系统中真正表现出来的理解中的一个主要空白。令人惊讶的是，尽管在“混沌”领域进行了大量有时深入的工作，但混沌行为尚未在现实的物理系统中得到证明（在数学意义上被理解）。混沌的本质只有在数学模型中才能真正被理解，这些模型过于简化，无法包含真实系统的一些关键特征（尽管即使是那些简化的模型也可能相当复杂）。然而，这个理论已经达到了这样一个水平，似乎终于可以理解了。对现实物理系统中混沌存在的数学证明将是过去50多年来长期发展的高潮。该项目的第二个方向是研究具有快速振荡的物理系统，其中发生的现象既迷人又有用。例如，对一根打开的自行车链条的一端进行高频振动，从理论上讲，可以使链条稳定地站起来。这种好奇心背后的理论也解释了一些看似无关的设备的工作原理，如粒子陷阱、粒子加速器、激光镊子（允许人们在不破坏细胞壁的情况下通过激光束在细胞内移动粒子）和声波悬浮器。在过去的几年里，作者已经找到了对这些影响的解释；先验理论预测了结果，但没有回答“为什么”。这一解释开辟了新的方向，我们计划进一步探索。特别是，我们计划更好地理解振动控制的概念，并探索使用振动进行滤波（使用声波）。第三个方向具有基本的数学意义，但在几种物理情况下具有重要意义。数学理论将使我们深入了解至少三种不同的现象，否则我们是无法了解的：(i)粒子在特定模式的磁场中的行为；（ii）某些类型晶体的导电性，以及（iii）一些被称为超导量子干涉装置（所谓的squid）的约瑟夫森结阵列的电学行为。