Nonlinear Dynamics with Applications to Physical Systems

Proposal #0205128

PI: Mark Levi

Institution: Penn State University

Title: Nonlinear Dynamics with Applications to Physical Systems

ABSTRACT

This 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).