Invariant Manifolds and Complex Behavior in Nonlinear Physical Systems

在物理的非线性演化问题中，人们通常关注影响大类初始状态的可观测动力学行为的分析。动力系统理论为解决这类问题提供了一个强大的几何工具--不变流形。不变流形能够作为系统相空间中的组织结构，并且能够为相当大的初始条件集合创造复杂或混沌的行为。这些现象的例子包括有限维系统中的混沌混合、快速能量转移和扩散，以及演化方程中的跳跃行为和脉冲产生。这一建议提出了不变流形理论在一系列领域的新扩展和新应用，从应用分析的困难开放问题到激光光学、海洋学和大气科学的具体问题。自然界中的大多数现象都是动态的，即人们观察它们随着时间的变化。这些现象包括洋流、大气事件和行星运动。描述这些时变现象的数学学科称为动力系统理论。这一理论主要关注的是用数学公式来理解所研究现象的演变，并在很长一段时间内预测它的未来。这一理论已经在几个应用中显示出它的力量，但还不够成熟，不足以处理大规模的物理过程。这一建议建议开发几个新的工具，以便使动力系统理论适用于激光光学、海洋学和大气科学等非常重要的问题。例如，建议的研究方向之一是分析和预测海洋中的大涡运动。拟议工作的另一个方面将为南极极地涡旋附近的臭氧消耗速度提供科学可靠的估计。

In nonlinear evolutionary problems of physics, one is typically concerned with the analysis of observable dynamical behavior, which impacts large classes of initial states. Dynamical systems theory offers a powerful geometric tool, invariant manifolds, to tackle such problems. Invariant manifolds are capable of acting as organizing structures in the phase space of a system, as well as creating complex or chaotic behavior for sizable sets of initial conditions. Examples of these phenomena include chaotic mixing, fast energy transfer, and diffusion in finite dimensional systems, and jumping behavior and pulse generation in evolution equations. This proposal suggests novel extensions and new applications of invariant manifold theory in an array of areas, ranging from difficult open questions of applied analysis to concrete problems in laser optics, oceanography, and atmospheric science.Most phenomena in nature are dynamic in nature, i.e., one observes them change in time. Such phenomena include ocean currents, atmospheric events, and the motion of planets. The mathematical discipline that deals with the description of these time-varying phenomena is called dynamical systems theory. The main concern of this theory is to understand the evolution of the studied phenomenon in terms of mathematical formulas and predict its future for long times. This theory has already showed its power in several applications, but has not been mature enough to deal with large-scale physical processes. This proposal suggests the development of several new tools in order to make dynamical systems theory applicable to highly important problems of laser optics, oceanography and atmospheric science. For example, one of the suggested directions of research is aimed at the analysis and prediction of the motion of large eddies in the ocean. Another aspect of the proposed work will provide scientifically solid estimates for the rate of ozone depletion near the Antarctic polar vortex.