Mathematical Sciences: New Resonance Phenomena and Adiabatic Chaos

几何奇异摄动理论的最新进展为分析具有两个时间或长度尺度的常微分方程组的共振现象和同宿轨建立了一套强有力的工具。在这些新的工具中，指数小交换引理(称为EXSEL)是作者与C.Jones和N.Kopell共同发展的，它已经成功地用于证明多自由度扰动哈密顿系统的多个快脉冲同宿轨道的存在性，以及在耦合反应扩散方程的行波问题中的应用。本文从四个方面分析了奇摄动系统中的共振现象：(1)共振亚谐轨道和超谐轨道的存在；(2)阻碍绝热混沌混合的稳定岛的几何形状；(3)强迫混沌中同宿结构的共振响应；(4)共振通过问题的周期轨道的存在性。多个时间尺度的物理问题出现在许多科学和技术分支中，包括流体力学、加速器动力学、等离子体物理、神经生理学和生物学。这些问题在数学上被建模为奇异摄动或绝热系统。尽管在这一领域取得了很长的进步历史，并继续引起人们的极大兴趣，但仍然存在大量悬而未决的问题。这一建议涉及四个基本的数学问题，涉及在应用中具有直接意义的共振现象和同宿行为。最近出现了几种新的数学技术，其中一些是作者开发的，它们为解决这些问题提供了实质性的希望。本文提出的工作的主要目标是使用和扩展这些新工具的范围，同时也力求从技术应用的角度回答上述问题。为此，提出者指出，他成功地开发了增强的流体混合技术，这是他早期关于奇异扰动系统中同宿纠缠几何的工作的应用。

Recent advances in geometric singular perturbation theory have established a powerful set of tools for analyzing resonance phenomena and homoclinic orbits in systems of ordinary differential equations with two time or length scales. Among these new tools is the Exponentially Small Exchange Lemma (called EXSEL), which was developed by the author with C. Jones and N. Kopell, and which has been used successfully to prove the existence of multiple fast-pulse homoclinic orbits in perturbed multi-degree-of-freedom Hamiltonian systems, as well as in traveling wave problems for coupled reaction diffusion equations. It is proposed here to analyze four fundamental aspects of resonance phenomena in singularly-perturbed systems: (1) the existence of resonant sub- and super-harmonic orbits, (2) the geometry of islands of stability, which are obstructions to mixing in adiabatic chaos, (3) the resonant response of homoclinic structures in the forced, damped sine-Gordon equation which are conjectured to be sources of chaos, (4) the existence of periodic orbits in problems of passage throughresonance. Physical problems with multiple time scales arise in many branches of science and technology, including fluid mechanics, accelerator dynamics, plasma physics, neurophysiology, and biology. These problems are modeled mathematically as singularly-perturbed or adiabatic systems. Despite the long history of progress and the considerable continuing interest in this area, a wealth of open problems exists. This proposal concerns a group of four fundamental mathematical questions concerning resonance phenomena and homoclinic behavior that are of direct significance in the applications. Several new mathematical techniques, some developed by the author, have recently become available which offer substantial promise of solving these problems. While the main goal of the work proposed here is to use and extend the scope of these new tools, it is also endeavored to answer the above questions with an eye toward the technological applications. Toward this end, the proposer points to his successful development of enhanced fluid-mixing technology as an application of his earlier work on the geometry of homoclinic tangles in singularly-perturbed systems.