This research project explores and experimentally tests a radically new mathematical framework for understanding and predicting complicated behaviors in numerous fundamental and practical problems in science, engineering, and medicine (e.g., weather forecasting, characterization of cardiac arrhythmias, etc.). Complex behaviors in many such problems are often governed by patterns that appear fleetingly but repeatedly. The research develops general, powerful techniques for identifying and quantifying key patterns, including the temporal sequences in which the patterns may appear; knowledge of the patterns and sequences can then be harnessed to construct "road maps" for predicting future behaviors. This study will focus on demonstrating "proof of principle" by constructing road maps of complex behavior observed in turbulent fluid flow in laboratory experiments. If successful, the results of this study will lead directly to the development of faster and more accurate ways to make predictions of complicated behavior in large real world problems. For example, the ability to identify and quantify important patterns and sequences in atmospheric turbulence should enable weather forecasts that are better and more rapid than those currently possible today. All software and useful solution data produced by the research activities will be made publicly available. The research program tightly integrates with teaching and learning at the undergraduate and graduate levels and includes activities to increase participation of underrepresented groups.
The primary goal of this research program is to develop a novel geometrical/topological approach to modeling and prediction of turbulent flows and to validate it experimentally. Investigation will focus on a weakly turbulent flow in a shallow electrolyte fluid layer. A combination of existing numerical methods and new methods developed as a part of this program will be used to compute a large set of unstable states (known as exact coherent states in fluid dynamics) and the network of connections between these states, given by the numerically exact solutions of the mathematical model of the flow. Temporal averages will be compared with state averages in experiment and simulations to verify the statistical predictions of periodic orbit theory. A low-dimensional predictive model for the dynamics based on the topology of the network of connections will similarly be validated against experiment and simulations. Understanding, prediction, and control of spatiotemporally chaotic dynamics, in general, and of turbulent fluid flows, in particular, is largely an open problem of both practical and fundamental significance. The geometrical/topological framework that will be developed and tested under this project will provide a novel reduced-order, predictive, dynamical description of turbulent fluid flows. This framework will also provide a connection between the dynamical description and the conventional statistical description of fluid turbulence. In addition, this framework should serve as a foundation for a radically new way to control complex dynamical regimes in a wide range of applications.
