In 2004, Fisher constructed a map on a 2-disc that admitted a hyperbolic set not contained in any locally maximal hyperbolic set. Furthermore, it was shown that this was an open property, and that it was embeddable into any smooth manifold of dimension greater than one. In the present work we show that analogous results hold for flows. Specifically, on any smooth manifold with dimension greater than or equal to three there exists an open set of flows such that each flow in the open set contains a hyperbolic set that is not contained in a locally maximal one.

In this thesis we consider a few results related to the Zermelo and Extended Zermelo Model as well as outline some partial results and open problems related thereto. First we will analyze a discrete dynamical system considering under what conditions the convergence of this dynamical system predicts the outcome of the Extended Zermelo Model. In the following chapter we will focus on the Zermelo Model by giving a method for simplifying the derivation of Zermelo ratings for tournaments in terms of specific types of strongly connected components. Following this, the idea of stability of a tournament will be discussed and an upper bound will be obtained on the stability of three-team tournaments. Finally, we will conclude with some partial results related to the topics presented in the previous chapters.

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We consider the problem of controlling a thermal convection flow by feedback. The system is governed by the Boussinesq approximation of the coupled set of Navier-Stokes and heat equations. The control is applied through Dirichlet boundary conditions. We concentrate on a two-dimensional mode and use a semidiscrete Galerkin scheme for numerical computations. We construct both a linear control and a non-linear quadratic control and apply them to the full non-linear model. First, we test these controllers on a one-mode approximation. The convergence of the numerical scheme is analyzed. We also consider LQR control for a two-dimensional heat equation. / Ph. D.

Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. In these formulations, motions take values in certain types of smooth manifolds which are coordinate-free and intrinsic. However, the dimension of the full configuration manifold can be large. The first study in this dissertation aims to build low-dimensional models models from motion capture data. This study also expands on the so-called learning problem from statistical learning theory over Euclidean spaces to estimating functions over manifolds. Experimental results are presented for estimating reptilian motion using motion capture data. The second study in this dissertation utilizes reproducing kernel Hilbert space (RKHS) formulations and Koopman theory, to achieve some of the advantages of learning theory for IID discrete systems to estimates generated over dynamical systems. Specifically, rates of convergence are determined for estimates generated via extended dynamic mode decomposition (EDMD) by relating them to estimates generated by distribution-free learning theory. Some analytical examples illustrate the qualitative behavior of the estimates. Additionally, a examination of the numerical stability of the estimates is also provided in this study. The approximation methods are then implemented to estimate forward kinematics using motion capture data of a human running along a treadmill. The final study of this dissertation contains an examination of the continuous time regression problem over subsets and manifolds. Rates of convergence are determined using a new notion of Persistency of Excitation over flows of manifolds. For practical considerations, two approximation methods of the exact solution to the continuous regression problem are introduced. Characteristics of these approximation methods are analyzed using numerical simulations. Implementations of the approximation schemes are also performed on experimentally collected motion capture data. / Doctor of Philosophy / Modeling the dynamics of complex multibody systems, such as those representing the motion of animals, can be accomplished through well-established geometric methods. However, many real-world systems, including those representing animal motion, are difficult to model from first principles. Machine learning, on the other hand, has proven to be extremely powerful in its ability to leverage "big data" to generate estimates from typically independent and identically distributed (IID) data. This dissertation expands on the so-called learning problem from statistical learning theory over Euclidean spaces to those over manifolds. This dissertation consists of three studies, the first of which aims to build low-dimensional models models from motion capture data. Using the distribution-free learning theory, estimates discussed in this dissertation minimize a proxy of the expected error, which cannot be calculated in closed form. This dissertation also includes a study into approximations of the so-called Koopman operator. This study determined that the rate of convergence of the estimate to the true operator depends on the reduced dimensionality of the embedded submanifold in the high-dimensional ambient input space. While most of the current work on machine learning focuses on cases where the samples used for learning or regression are generated from an IID, stochastic, discrete measurement process, this dissertation also contains a study of the regression problem in continuous time over subsets and manifolds. Additionally, two approximation methods of the exact solution to the continuous regression problem are introduced. Each of the aforementioned studies also includes several analytical results to illustrate the qualitative behavior of the approximations and, in each study, implementations of the estimation schemes are performed on experimentally collected motion capture data.

Much effort in recent times has been devoted to the study of low energy transport in multibody gravitational systems. Despite continuing advancements in computational abilities, such studies can often be demanding or time consuming in the three-body and four-body settings. In this work, the Hamiltonian describing the planar circular restricted three-body problem is rewritten for systems having small mass parameters, resulting in a 2D symplectic twist map describing the evolution of a particle's Keplerian motion following successive close approaches with the secondary. This map, like the true dynamics, admits resonances and other invariant structures in its phase space to be analyzed. Particularly, the map contains rotational invariant circles reminiscent of McGehee's invariant tori blocking transport in the true phase space, adding a new quantitative description to existing chaotic zone estimates about the secondary. Used in a patched three-body setting, the map also serves as a tool for investigating transfer trajectories connecting loose captures about one secondary to the other without any propulsion systems. Any identified initial conditions resulting in such a transfer could then serve as initial guesses to be iterated upon in the continuous system. In this work, the projection of the McGehee torus within the interior realm is identified and quantified, and a transfer from Earth to Venus is exemplified. / Master of Science / The transport of a particle between celestial bodies, such as planets and moons, is an important phenomenon in astrodynamics. There are multiple ways to mediate this objective; commonly, the motion can be influenced directly via propulsion systems or, more exotically, by utilizing the passive dynamics admitted by the system (such as gravitational assists). Gravitational assists are traditionally modelled using two-body dynamics. That is, a space- craft or particle performs a flyby within that body's sphere of influence where momentum is exchanged in the process. Doing so provides accurate and reliable results, but the design space effecting the desired outcome is limited when considering the space of all possibilities. Utilizing three-body dynamics, however, provides a significant improvement in the fidelity and variety of trajectories over the two-body approach, and thus a broader space through which to search. Through a series of approximations from the three-body problem, a discrete map describing the evolution of nearly Keplerian orbits through successive close encounters with the body is formed. These encounters occur outside of the body's sphere of influence and are thus uniquely formed from three-body dynamics. The map enables computation of a trajectory's fate (in terms of transit) over numerical integration and also provides a boundary for which transit is no longer possible. Both of these features are explored to develop an algorithm able to rapidly supply guesses of initial conditions for a transfer in higher fidelity models and further develop the existing literature on the chaotic zone surrounding the body.

The majority of cognitive scientists today view the mind as a computer, instantiating some function mapping the inputs it gets from the environment to the gross behaviors of the organism. As a result, the emphasis in most ongoing research programmes is on finding that function, or some part of that function. Moreover, the types of functions considered are limited somewhat by the preconception that the mind must be instantiating a function that can be expressed as a computer program. I argue that research done in the last two decades suggests that we should approach cognition with as much consideration to the environment as to the inner workings of the mind. Our cognition is often shaped by the constraints the environment places on us, not just by the "inputs" we receive from it. I argue also that there is a new approach to cognitive science, viewing the mind not as a computer but as a dynamical system, which captures the shift in perspective while eliminating the requirement that cognitive functions be expressable as computer programs. Unfortunately, some advocates of this dynamical perspective have argued that we should replace all of traditional psychology and neuroscience with their new approach. In response to these advocates, I argue that we cannot develop an adequate dynamical picture of the mind without engaging in precisely those sorts of research and hypothesizing that traditional neuroscience and psychology engage in. In short, I argue that we require certain types of explanations in order to get our dynamical (or computational) theories off the ground, and we cannot get those from other dynamical (or computational) theories. / Master of Arts

Lake Erie, one of the Great Lakes in North America, has witnessed recurrent summertime low oxygen dead zones for decades. This is a yearly phenomenon that causes microbial production of the greenhouse gas nitrous oxide from denitrification. Complete denitrification is a microbial process of reduction of nitrate to nitrogen gas via nitrite, nitric oxide, and greenhouse gas nitrous oxide. After scanning the microbial community in Lake Erie, Pseudomonas aeruginosa is decided to be examined, not because it is abundant in Lake Erie, but because it can perform denitrification under anaerobic conditions. This study focuses on a mathematical model of the metabolic network in Pseudomonas aeruginosa under denitrification and testable hypotheses generation using polynomial dynamical systems and stochastic discrete dynamical systems. Analysis of the long-term behavior of the system changing the concentration level of oxygen, nitrate, and phosphate suggests that phosphate highly affects the denitrification performance of the network. / Master of Science

Implementing high-fidelity quantum control and suppressing the unwanted environmental noise has been one of the essential challenges in developing quantum information technologies. In the past, driving pulse sequences based on Dirac delta functions or square wave functions, such as Hahn spin echo or CPMG, have been developed to dynamically correcting the noise effects. However, implementing these ideal pulses with high fidelity is a challenging task in real experiments. In this thesis, we provide a new and simple method to explore the entire solution space of driving pulse shapes that suppress environmental noise in the evolution of the system. In this method, any single-qubit phase gate that is first-order robust against quasi-static transversal noise corresponds to a closed curve on a two-dimensional plane, and more general first-order robust single-qubit gates correspond to closed three-dimensional space curves. Second-order robust gates correspond to closed curves having the property that their projection onto any two-dimensional planes shall enclose a zero net area. The driving pulse shapes that implement the gates can be determined by the curvature, torsion, and the length of the curve. By utilizing the framework it is possible to obtain globally optimal solutions in pulse shaping in respect of experimental constraints by mapping them into geometrical optimization problems. One such problem we solved is to prove that the fastest possible single-qubit phase gates that are second-order noise-resistant shall be implemented using sign-flipping square functions. Since square waves are not experimentally feasible, we provide a method to smooth these pulses with minimal loss in gate speed while maintaining the robustness, based on the geometrical framework. This framework can also be useful in diagnosing the noise-cancellation properties of pulse shapes generated from numerical methods such as GRAPE. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation. / Doctor of Philosophy / Controlling a quantum system with high-fidelity is one of the main challenges in developing quantum information technologies, and it is an essential task to reduce the error caused by unwanted environmental noise. In this thesis, we developed a new geometrical formalism that enables us to explore all possible driving fields and provides a simple recipe to generate an infinite number of experimentally feasible driving pulse shapes for implementing quantum gates. We show that single-qubit operations that could suppress quasi-static noise to first-order correspond to closed three-dimensional space curves, and single-qubit gates that are second-order robust correspond to closed curves with zero enclosed net area. This simple geometrical framework can be utilized to obtain optimal solutions in quantum control problems, and can also be used as a method to diagnose driving pulse shapes generated from other means. We show that this method for pulse shaping can significantly improve the fidelity of single-qubit gates through numerical simulation.