Nonlinear Dynamics with Applications to Physical Systems

提案#0205128PI：Mark LeviInstitution：宾夕法尼亚州立大学题目：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50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根本问题还没有得到回答。答案似乎终于在掌握之中了，一方面是因为对二维映射的理解不断进步，另一方面是因为对微分方程的一些新的见解。该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线索，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(一类保面积柱面映射)的数学分析，它一方面具有独立的数学兴趣，另一方面提供对物理系统的洞察。这些系统包括(I)磁场中的带电粒子，(Ii)晶格中电子平衡组态的Frenkel-Kontorova模型，以及(Iii)约瑟夫森结阵列。本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上的一个重大差距。令人惊讶的是，尽管在“混沌”领域做了大量有时很深入的工作，但混沌行为还没有在现实的物理系统中得到证明(从数学意义上理解)。混沌的本质只有在过于简化的数学模型中才能真正被理解，这些模型过于简化，无法包括真实系统的一些关键特征(尽管即使是这些简化的模型也可能相当复杂)。然而，这一理论已经达到了这样一种理解似乎终于可以理解的程度。一个现实的物理系统存在混沌的数学证明，将是过去50年或更长时间里长期发展的顶峰。该项目的第二个方向是研究具有快速振荡的物理系统，在这些系统中发生的现象既非常吸引人，又非常有用。例如，使开放的自行车链条的末端受到高频振动，理论上将使链条能够稳定地站立起来(！)在它的尽头。支持这种好奇心的理论也对粒子捕捉器、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在细胞内移动粒子)和声学悬浮设备的工作原理负责。在过去的几年里，作者找到了对这些影响的解释；先前的理论预测了结果，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。我们尤其计划更好地理解振动控制的概念，并探索利用振动进行过滤(使用声波)。第三个方向是基本的数学兴趣，但在几种物理情况下具有重要意义。这一数学理论将使人们能够洞察至少三种不同的现象：(I)粒子在某些模式的磁场中的行为；(Ii)某些类型晶体的导电性；以及(Iii)某些约瑟夫森结阵列的电行为，这些阵列被称为超导量子干涉器件(所谓的鱿鱼)。

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).