Collaborative Research: Topics in Infinite-Dimensional and Stochastic Dynamical Systems

合作研究：无限维和随机动力系统主题

• 批准号: 1413603
• 负责人: Kening Lu
• 金额: $ 16万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-15 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1413603&HistoricalAwards=false
• 关键词: Collaborative; Research; Topics; Infinite; Dimensional

While many physical, biological, climatological, and financial processes appear to be subject to random or stochastic forces, there are also coherent structures underlying these processes that give some measure of predictability. This project is laying the groundwork both for the determination of these hidden structures and for analyzing specific situations arising in several applications. Among these is the embryonic development of the wing of a fruit fly and its newly discovered relation to current through potassium channels in the cell membrane. Part of this project is to develop and use sophisticated mathematical techniques to understand that ionic current. The fruit fly model has implications for mammalian development and may lead to an understanding of the cause of some serious birth defects. Applications of the abstract mathematical investigations also include understanding of other dynamical systems subject to random perturbations, including the density distribution in highly excited plasmas or the fine structure of an alloy and how defects are distributed and evolve in time. This project builds upon the past work of the principal investigators and others to establish the existence of coherent structures embedded in the phase space of complex dynamical systems, both finite- and infinite-dimensional and both deterministic or subject to random forcing. The fundamental and abstract theory to be developed during the course of the project lies behind concrete and observed phenomena in the physical and biological sciences, particularly at the molecular, microscopic, or nano-scale. Infinite-dimensional dynamical systems are required to represent the temporal and spatial fluctuations of quantities subject to physical laws or biochemical processes, such as the distribution of bone morphogenic protein in a developing embryonic fly wing, the current through an ion channel in a cell membrane, the density of a relativistic plasma, or the motion of microscopic defects in an alloy, to name just a few of the systems considered in this project. Furthermore, as complex as these may be, stochastic perturbations must be considered due to thermal or other fluctuations in the environment and imprecise measurement of quantities at small scales. While one cannot hope to give exact representations of all states subject to complex spatial and temporal interaction, one can sometimes glean information due to the presence of robust, but possibly hidden, structures such as invariant manifolds and their invariant foliations whose existence is implied by the laws governing the processes under investigation. The goals of this project are to discover the conditions under which such structures exist, even when stochastically forced, and to examine the implications in the particular physical and biological systems underlying the equations.

虽然许多物理、生物、气候和金融过程似乎受到随机或随机力量的影响，但这些过程背后也存在连贯的结构，可以提供一定程度的可预测性。该项目为确定这些隐藏结构和分析多个应用中出现的具体情况奠定了基础。其中包括果蝇翅膀的胚胎发育及其新发现的与细胞膜钾通道电流的关系。 该项目的一部分是开发和使用复杂的数学技术来理解离子电流。 果蝇模型对哺乳动物的发育具有重要意义，并可能有助于了解一些严重出生缺陷的原因。抽象数学研究的应用还包括理解受随机扰动影响的其他动力系统，包括高度激发等离子体中的密度分布或合金的精细结构以及缺陷如何随时间分布和演变。 该项目建立在主要研究人员和其他人过去的工作基础上，以确定嵌入复杂动力系统相空间中的相干结构的存在，包括有限维和无限维，以及确定性或随机强迫。 该项目过程中要发展的基本和抽象理论隐藏在物理和生物科学中具体和观察到的现象背后，特别是在分子、微观或纳米尺度上。 需要无限维动力系统来表示受物理定律或生化过程影响的量的时间和空间波动，例如发育中的胚胎蝇翼中骨形态发生蛋白的分布、通过细胞膜中离子通道的电流、相对论等离子体的密度，或合金中微观缺陷的运动，仅举几个本项目中考虑的系统。 此外，尽管这些可能很复杂，但由于环境中的热波动或其他波动以及小尺度数量的不精确测量，必须考虑随机扰动。 虽然人们不能希望给出受复杂空间和时间相互作用影响的所有状态的精确表示，但有时可以由于存在鲁棒但可能隐藏的结构（例如不变流形及其不变叶状结构）而收集信息，这些结构的存在隐含在管辖调查过程的法律。 该项目的目标是发现此类结构存在的条件（即使是随机强制的），并检查方程背后的特定物理和生物系统的含义。