Characterization of Spatiotemporal Chaos

9722814亨利格林赛德 “时空混沌的表征” 要求继续支持跨学科研究 （非平衡物理，非线性动力学和计算） 数学）关于时空的表征 混沌在大型非平衡系统，如那些发现在 生物、化学、流体和激光系统。研究 我将使用数值方法，并行计算机，并仔细 选择数学模型探讨两个问题：（1）如何 时空混沌是否与数学结构有关 相空间中的吸引子（2）无限 不稳定周期轨道族 奇异吸引子被用来表征和控制 高维混沌拟议的研究将：（a）产生 新的方法来分析积累的时空数据， 实验学家和计算科学家;（B）提供 关于不稳定周期轨道（UPO）的新信息 并提出了新的战略， （c）提供有关平行的新见解 求非稳定周期轨道的数值算法 大空间域中的偏微分方程;以及 (d)为杜克的学生和博士后提供有价值的培训， 数值分析、模拟和复杂理论 非线性动力系统 基础科学和应用科学中的许多问题需要更深入的研究。 理解持续的非平衡系统， 复杂的随时间变化的行为是由一个稳定的流量 能量和物质的混合体。实例可见于 几乎所有的科学领域，但可能包括流体 动荡，心脏颤动，大脑癫痫， 高功率激光器的设计，以及电 电网当前高度关注的一个基本问题 对于许多科学领域来说， 一种复杂非平衡介质结构及其时间特性 状态，特别是当系统非周期性变化时 （混乱地）及时。拟议中的研究提出了两种方法 解决这个尚未完全解决的 通过以前的研究探索。本研究将应用 数学和计算机模拟研究简化 数学模型中的时空结构 可以详细检查，然后将各种见解 用实验数据进行检验。最终，这项研究 应提高工程控制和复杂的设计 系统，并为开发 持续非平衡系统的基本理论。

9722814 Henry Greenside "CHARACTERIZATION OF SPATIOTEMPORAL CHAOS" Renewal of support is requested for interdisciplinary research (nonequilibrium physics, nonlinear dynamics, and computational mathematics) concerning the characterization of spatiotemporal chaos in large nonequilibrium systems such as those found in biological, chemical, fluid, and laser systems. The research will use numerical methods, parallel computers, and carefully chosen mathematical models to explore two questions: (1) how is spatiotemporal chaos related to the mathematical structure of attractors in phase space? and, (2), can the infinite hierarchy of unstable periodic orbits associated with a strange attractor be used to characterize and control high-dimensional chaos? The proposed research will: (a) yield new methods for analyzing spatiotemporal data accumulated by experimentalists and by computational scientists; (b) provide new information about unstable periodic orbits (UPOs) associated with extensive chaos and suggest new strategies for their stabilization; (c) provide new insights about parallel numerical algorithms for finding unstable periodic orbits of partial differential equations in large spatial domains ; and (d) give valuable training to students and postdocs at Duke in the numerical analysis, simulation, and theory of complex nonlinear dynamical systems. Many problems in basic and applied science require a deeper understanding of sustained nonequilibrium systems in which complicated time-dependent behavior is caused by a steady flux of energy and matter into a system. Examples can be found in almost all areas of science but might include fluid turbulence, fibrillation in a heart, epilepsy in a brain, the design of high-power lasers, and instabilities of electrical power grids. A fundamental question of high current interest for many areas of science is how to re late the spatial structure of a complex nonequilibrium medium to its temporal state, especially when the system varies nonperiodically (chaotically) in time. The proposed research suggests two ways of addressing this question that have yet been incompletely explored by previous research. This research will use applied mathematics and computer simulation to study simplified mathematical models in which spatial and temporal structure can be examined in detail, and then various insights will be tested with experimental data. Eventually, this research should improve engineering control and design of complex systems as well as provide a foundation for developing a fundamental theory of sustained nonequilibrium systems.