Collaborative Research: Revealing the Geometry of Spatio-temporal Chaos with Computational Topology: Theory, Numerics and Experiment

我们经历的天气是由对流驱动的，阳光使地球变暖，而地球又加热了大气，而大气则因外层空间的寒冷温度而冷却。大多数人对微观行为不感兴趣，例如空气中单个分子的行为，也不对宏观行为感兴趣，例如全球平均温度。令人感兴趣的是介观模式，例如导致局部温度变化的天气锋面。相对于大规模系统的微观或宏观特征，人们对介观的这种兴趣出现在各种复杂的大规模物理现象中，例如发动机的燃烧、海洋中的生物质动力学、人体心脏的心室颤动等。这些介观模式具有许多不同的形状和大小，并随时间变化，有时缓慢，有时迅速。这些模式的形式以及它们如何在时间上演变通常非常依赖于参数。新技术正在极大地提高我们测量和模拟这些物理现象的能力，产生了巨大的数据集，但我们提取和量化这些信息的能力并没有跟上步伐，这些信息能够导致对这些系统的理解、预测和控制。我们将探索使用新的数学工具来解决这个问题。Rayleigh-Bénard对流的时空复杂性产生了高维时间序列数据。一个相对较新的称为持久同调的代数拓扑工具将被用来为非线性降维提供新的工具。为了确保这些方法的适用性和保留动力学的物理上重要的介观特征，它们将与精心控制的高精度对流实验和最先进的、大规模、高分辨率的Boussinesq方程数值模拟的进一步发展结合在一起开发。这包括协变Lyapunov指数的几何分析。在这项工作中开发的新的计算工具应该在涉及自然界(海洋和大气流动、气候和天气预报)和技术(非线性光学系统、燃烧和化学反应)中复杂的非平衡系统的各种问题中得到广泛的应用，其中需要了解和预测复杂的行为。

The weather we experience is driven by convection, sunlight warms the earth which heats the atmosphere which is cooled by the cold temperatures of outer space. Most people are not interested in microscopic behavior, for example the behavior of the individual molecules in the air, nor macroscopic behavior, such as worldwide average temperature. What is of interest are mesoscopic patterns, for example weather fronts which result in local changes in temperature. This interest in mesoscopic, as opposed to micro- or macroscopic features, of large scale systems occurs in a wide variety of complex large scale physical phenomena such as combustion in engines, dynamics of biomass in the oceans, ventricle fibrillation in a human heart, etc. These mesoscopic patterns take on many different shapes and sizes and change with time, sometimes slowly and sometimes rapidly. The form of these patterns and how they evolve in time is often very dependent on parameters. New technologies are greatly increasing our abilities to measure and simulate these physical phenomena, resulting in enormous data sets, but our ability to extract and quantify this information in a way that leads to understanding, predictability, and control of these systems is not keeping pace. We will explore the use of new mathematical tools to address this problem.The spatial and temporal complexity of Rayleigh-Bénard convection produces high dimensional time series data. A relatively new algebraic topological tool called Persistent Homology will be used to provide new tools for nonlinear dimension reduction. To ensure the applicability of these methods and that physically important mesoscopic features of the dynamics are preserved they will be developed in conjunction with the further development of carefully controlled high precision convection experiments and state-of-the-art, large scale, high-resolution numerical simulations of the Boussinesq equations. This includes the analysis of the geometry of covariant Lyapunov exponents. The new computational tools developed in this work should find broad application in a wide variety of problems involving complex nonequilibrium systems in nature (oceanic and atmospheric flows, climate and weather forecasting) and in technology (nonlinear optical systems, combustion and chemical reactions) where understanding and prediction of complex behavior is desired.