UC Santa Barbara

During the 1930s, researchers realized that an abstract dynamical system induces a group of linear operators acting on the space of square-integrable functions. For measure-preserving systems, the induced operator is unitary and self-adjoint. As such, its spectrum is restricted to the unit circle and has been shown to encode many important statistical and geometric properties of the dynamical system. Since then, the induced Koopman group of operators' spectral properties have drawn an immense amount of research interest over the last decade. Due to the rise of computing capabilities and data availability, there has been an explosive amount of research into developing data-driven algorithms that can compute the spectrum numerically from data.

In the first part of this dissertation, we demonstrate how Koopman operator methods can offer a model-free, data-driven approach to analyze and forecast highway traffic dynamics. By obtaining a decomposition of data sets collected by the Federal Highway Administration and the California Department of Transportation, we can reconstruct observed data, distinguish any growing or decaying patterns, and obtain a hierarchy of previously identified and never before identified spatiotemporal patterns. Furthermore, it is demonstrated how this methodology can be utilized to forecast highway network conditions. The developed forecasting scheme readily generalizes to the much-needed scenario of multi-lane highway networks without any loss to its performance or efficiency. Also, we do not rely on large historical training data nor parameter tuning or selection. Thereby providing a completely efficient and accurate method of analyzing and forecasting traffic patterns at the levels required by modern intelligent transportation systems.

In the second part of this dissertation, we consider the equivalent induced linear operators acting on the space of sections of the tangent, cotangent, and tensor bundles of the state space. We begin by first demonstrating how these operators are indeed natural generalizations of Koopman operators acting on functions. The fundamental insight lies in understanding the connection between the differential geometric concept of pulling back objects (functions, vector fields, covector fields, tensor fields) under a diffeomorphism and how their pull-back relates to the Lie derivative of that object. We then draw connections between the various operators' spectrum and characterize the algebraic and differential topological properties of their spectrum. We describe these operators' discrete spectrum for linear dynamical systems and derive spectral type expansions for linear vector fields. The expansions derived resemble the familiar spectral expansion of functions under the Koopman operator. We define the notion of an "eigendistribution" and provide conditions for when an eigendistribution is integrable. We then demonstrate how to recover the foliations arising from their integral manifolds via the level sets of Koopman eigenfunctions. Many of the results presented in the second part of this dissertation stem from well-known differential geometric concepts. Prior work on such generalized operators on vector fields exists but has remained mostly unnoticed by the growing Koopman operator community.

We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized. Specifically, we show that the flows of two vector fields commute, subject to an appropriate rescaling of the flow time, if and only if one vector field is an eigensection of the other vector field. The eigenvalue prescribes the required time scaling, and we recover the original statement that the flows of commuting vector fields commute as a particular case of our result. We also apply our results to the study of a hyperbolic toral automorphism known as Arnold's Cat map. We demonstrate that the Lyapunov exponents are contained within the spectrum of the induced operator on vector fields, and we recover the stable and unstable foliations via the level sets of the joint eigenfunctions of the stable and unstable eigendistributions.