Committee:
Dr. Roman Grigoriev, Department of Physics, Georgia Institute of Technology (advisor)
Dr. Michael Schatz, Department of Physics, Georgia Institute of Technology
Dr. Kurt Wiesenfeld, Department of Physics, Georgia Institute of Technology
Dr. Predrag Cvitanović, Department of Physics, Georgia Institute of Technology
Dr. Luca Dieci, Department of Mathematics, Georgia Institute of Technology

Abstract
Chaos is an intrinsic property of many real world systems, impacting a number of today's open research questions. While many chaotic systems have known governing equations and are deterministically ``solved," we still lack a comprehensive framework for predicting, controlling, and simply making sense of such systems. And while recent advances in technology allow us to explore these systems through direct numerical simulation better than ever before, the need for an insightful theoretical framework is still very much alive. 

Such a framework exists in a subset of chaotic systems, known as Axiom A chaotic systems and, as a result, Axiom A systems are understood quite well. However, the requirements for a system to be Axiom A are quite strict, and the overlap between systems that are Axiom A and those that are physically significant is quite small.

A very important concept in Axiom A systems is the notion of shadowing, which allows the chaotic dynamics to be decomposed piecewise-in-time in terms of much easier to analyze solutions known as periodic orbits. Periodic orbits are solutions to the governing equations that, unlike chaos, repeat in time. Their compactness make periodic orbits very simple objects to manipulate, both numerically and theoretically. A decomposition in terms of periodic orbits results in a predictive theory of Axiom A systems, both deterministically and statistically.

In this talk, I will discuss how to generalize aspects of this decomposition to a broader class of (non Axiom A) chaotic systems, specifically, fluid turbulence. Although recent studies suggest that Exact Coherent Structures (ECS)---e.g., repeating solutions to the Navier-Stokes equation---are descriptive of turbulence, it is an open question whether they are to turbulence what periodic orbits are to Axiom A chaos. I will present evidence of ECS being shadowed by turbulence in various fluid geometries and, additionally, present preliminary work suggesting a statistical picture of turbulence in terms of ECS may also be feasible. "