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

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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.