Complex Singularities in Numerical Analysis and Nonlinear Dynamics

本计画将探讨Lorenz系统之复杂奇异性之精确解析形式、三体问题之8字形解，以及由Navier-Stokes边界层所获得之讯号。分析和数值技术都将被使用。研究Lorenz系统的目的是理解Lorenz解在整个复平面上的解析延拓。关于8字形的解决方案，该项目将表明，这个系统的复杂奇点有一个吸引人的简单结构。Navier-Stokes边界层主要是用谱分析技术研究的。我们将在复平面上定位和解释信号解析延拓的奇异点，并获得测量以及理解外部流所施加的时间尺度。首先，我们的目标是获得新的见解纳维尔-斯托克斯边界层，这是非常重要的工程和气象。作为其重要性的一个例子，我们提到汽车的能量摄入的一个主要部分是在边界层耗散。其次，我们将写一本新书，把计算机体系结构放在科学计算的核心。这本书将介绍一种科学计算的风格，它深深地被计算机体系结构的最新进展所告知。

This project will investigate the precise analytic form of the complex singularities ofthe Lorenz system, the figure-eight solution of the three-body problem, and signals obtainedfrom the Navier-Stokes boundary layer. Both analytic and numerical techniqueswill be used. Research on the Lorenz system has the aim of understanding the analyticcontinuation of the Lorenz solutions to the entire complex plane. With regard to thefigure-eight solution, the project will show that the complex singularities of this systemhave an appealingly simple structure. The Navier-Stokes boundary layer has beenstudied primarily using techniques from spectral analysis. We will locate and elucidatesingularities of the analytic continuation of the signal in the complex plane, and andobtain a measurement as well as an understanding of the time scales imposed by theouter flow.This project endeavors to benefit the public in two respects. Firstly, we aim to obtainnew insights into the Navier-Stokes boundary layer which is of immense importance inengineering and meteorology. As an example of its importance, we mention that amajor part of the energy intake of automobiles is dissipated in the boundary layer.Secondly, we will write a new book that puts computer architecture at the heart ofscientific computing. This book will introduce a style of scientific computing that isdeeply informed by recent progress in computer architecture to a wider audience.