Characterization of Spatiotemporal Chaos

9722814 Henry Greenside“时空混沌的特征”要求为跨学科研究(非平衡物理、非线性动力学和计算数学)更新支持，涉及大型非平衡系统(如生物、化学、流体和激光系统)中的时空混沌的特征。这项研究将使用数值方法、并行计算机和精心选择的数学模型来探索两个问题：(1)时空混沌与相空间中吸引子的数学结构是如何关联的？以及，(2)与奇怪吸引子相关的不稳定周期轨道的无穷级数是否可以用来刻画和控制高维混沌？拟议的研究将：(A)提供新的方法来分析由实验者和计算科学家积累的时空数据；(B)提供与广泛的混沌相关的不稳定周期轨道(UPO)的新信息，并为其稳定化提出新的策略；(C)为寻找大空间区域中偏微分方程的不稳定周期轨道的并行数值算法提供新的见解；(D)在复杂的非线性动力系统的数值分析、模拟和理论方面为杜克大学的学生和博士后提供有价值的培训。基础科学和应用科学中的许多问题需要对持续的非平衡系统有更深入的了解，在持续的非平衡系统中，复杂的依赖时间的行为是由稳定的能量和物质流入系统引起的。在几乎所有的科学领域都可以找到这样的例子，但可能包括流体湍流、心脏纤颤、大脑癫痫、高功率激光的设计，以及电网的不稳定性。当前许多科学领域感兴趣的一个基本问题是如何将复杂的非平衡介质的空间结构恢复到其时间状态，特别是当系统在时间上非周期(混沌)变化的时候。这项拟议的研究提出了两种解决这个问题的方法，但之前的研究尚未完全探索这些方法。这项研究将利用应用数学和计算机模拟来研究简化的数学模型，在这些模型中可以详细地考察时空结构，然后用实验数据来检验各种见解。最终，这项研究将改进复杂系统的工程控制和设计，并为发展持续非平衡系统的基本理论提供基础。

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.