CAREER: Integrating Geometric, Probabilistic, and Topological Methods for Phase Space Transport in Dynamical Systems

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The research objective of this Faculty Early Career Development (CAREER) project is to develop a firm unified foundation for the theoretical and computational analysis of the phase space transport problem. Many physical systems can be modeled as having an underlying dynamical skeleton which organizes how all the possible behaviors are related. For systems known analytically with simple time dependence, this skeleton consists of well-known mathematical structures such as invariant sets and sets which asymptotically approach or depart from such sets. But for systems of increasing interest which are non-periodic, based on data, and observed over only a short time, there are no such structures in general. The proposed research will find appropriate analogs to these structures to expand the applicability of dynamical systems results to real world data. Working in the context of fluid flows, we will develop tools using geometric, probabilistic, and topological approaches, including coherent sets, almost-invariant sets, and symbolic dynamics.Successful completion of this project will provide a new and fruitful approach for conceptualization, visualization, and extraction of information regarding the possible behaviors of a system, with applicability to fluid mechanics and beyond, such as boundaries and transitions between qualitatively different kinds of behavior in data sets defined by meteorological, financial, psychological, or population observations. Education, outreach and dissemination are integral to the project activities, aimed at four audiences: (1) undergraduates will participate in research and develop learning materials for (2) high school students, who will learn about the role of chaotic transport and its relevance in the environment and biology; (3) graduate students and faculty will explore applications of tools from dynamical systems to new problem domains through a project-based course; and the (4) global community of researchers will benefit from numerical tools disseminated online which may reveal previously hidden structure in dynamic data sets.