Invariant Manifolds and Complex Behavior in Nonlinear Physical Systems

在物理学的非线性演化问题中，人们通常关注可观察的动力学行为的分析，这影响了大量的初始状态。动力系统理论提供了一个强大的几何工具，不变流形，来解决这些问题。不变流形能够在系统的相空间中充当组织结构，以及在相当大的初始条件下创建复杂或混沌行为。这些现象的例子包括有限维系统中的混沌混合、快速能量传递和扩散，以及演化方程中的跳跃行为和脉冲产生。这一建议提出了新的扩展和新的应用不变流形理论在一系列领域，从困难的开放问题的应用分析，以具体的问题，在激光光学，海洋学和大气科学。大多数现象在性质上是动态的，即，人们观察到它们随时间而变化。这些现象包括洋流、大气事件和行星的运动。描述这些时变现象的数学学科称为动力系统理论。该理论的主要关注点是用数学公式来理解所研究现象的演变，并预测其未来很长一段时间。这个理论已经在几个应用中显示了它的力量，但还不够成熟，无法处理大规模的物理过程。该建议建议开发几种新的工具，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大涡旋的运动。拟议工作的另一个方面将提供南极极涡附近臭氧消耗速率的科学可靠估计。

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.