Mathematical Sciences: Cellular Modeling and Dynamical- Systems Analysis of Spatial Patterns in Sedimentary Systems

9316078多个不同领域的Paola研究人员已经证明，细胞计算(“格子气体”)模型在研究抵抗传统分析形式的复杂的、空间扩展的系统时可以发挥巨大的作用。在这样的模型中，晶格中的细胞根据基于支配系统的物理抽象的规则相互作用。这种方法检查系统各部分之间的相互作用--导致系统作为一个整体的行为的相互作用--并允许评估潜在的详细物理的哪些方面对整体动力学最重要。这项拟议研究的一个主要目标是开发一种编织流的细胞计算模型，该模型结合了可移动微珠上自由表面流动的基本物理。到目前为止，由于缺乏模型与真实系统数据之间的定量比较方法，此类模型通常只被定性地评估。该项目的第二个主要目标是在动力系统数学的最新进展的基础上开发新的数据分析技术，以促进细胞计算方法的定量测试。我们的方法是处理空间扩展系统中的下游测量序列，就像处理动力系统的时间序列一样。描述各种几何变量下游发展的空间“吸引子”图，以及显示这些变量的值如何影响下游值的离散迭代图，都量化了空间扩展系统各部分之间的相互作用，并表征了整个系统的行为。像这样的分析，应用于自然系统、实验室实验和计算模型的数据，将提供比现有技术更有效的模型定量测试。在拟议的项目中，这种测试将应用于上述辫状流模型以及现有的风成和水下河床模型，使用从所有三个系统的实验室和现场实例收集的数据。***

9316078 Paola Researchers in various fields have shown that cellular computational ("lattice gas") models can be of great utility in the study of complex, spatially extended systems that resist conventional forms of analysis. In such models the cells of a lattice interact according to rules based on abstractions of the physics governing a system. This approach examines the interactions between the parts of a system--interactions that lead to the behavior of the system as a whole--and allows evaluation of what aspects of the underlying detailed physics are most import to the overall dynamics. One main goal of the proposed research is the development of a cellular-computational model of braided streams that incorporates the basic physics of free-surface flow over a movable bead. To date such models have usually been evaluated only qualitatively due to a lack of a quantitative method of comparison between the models and data from real systems. The second main goal of this project is to develop new data-analysis techniques base=d on recent advances in dynamical-systems mathematics that will facilitate quantitative testing of cellular-computational methods. Our approach is to treat downstream series of measurements in spatially extended systems like the time series of dynamical systems. Graphs of spatial "attractors" that characterize the downstream development of various geometrical variables, and discrete iterative plots that show how the value of these variables influence the values further downstream, both quantify the interactions between the parts of a spatially extended system and characterize the behavior of a system as a whole. Analyses such as these, applied to data from natural systems, laboratory experiments, and computational models, will provide a more effective quantitative test of the models that existing techniques. In the proposed projects such tests will be applied to the braided-stream model mentioned above and to existing models of aeolian and subaqueous bedforms, using data collected form laboratory and field examples of all three systems. ***