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.

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

To the pair (E,c), where E is a countable Borel equivalence relation on a standard Borel space (X,A) and c a normalized Borel T-valued 2-cocycle on E, we associate a sequentially weakly closed Borel *-algebra Br*(E,c), contained in the bounded linear operators on L^2(E). Associated to Br*(E,c) is a natural (Borel) Cartan subalgebra (Definition 6.4.10) L(Bo(X)) isomorphic to the bounded Borel functions on X. Then L(Bo(X)) and its normalizer (the set of the unitaries u in Br*(E,c) such that u*fu in L(Bo(X)), f in L(Bo(X))) countably generates the Borel *-algebra Br*(E,c). In this thesis, we study Br*(E,c) and in particular prove that: i) If E is smooth, then Br*(E,c) is a type I Borel *-algebra (Definition 6.3.10). ii) If E is a hyperfinite, then Br*(E,c) is a Borel AF-algebra (Definition 7.5.1). iii) Generalizing Kumjian's definition, we define a Borel twist G over E and its associated sequentially closed Borel *-algebra Br*(G). iv) Let a Borel Cartan pair (B, Bo) denote a sequentially closed Borel *-algebra B with a Borel Cartan subalgebra Bo, where B is countably Bo-generated. Generalizing Feldman-Moore's result, we prove that any pair (B, Bo) can be realized uniquely as a pair (Br*(E,c), L(Bo(X))). Moreover, we show that the pair (Br*(E,c), L(Bo(X))) is a complete invariant of the countable Borel equivalence relation E. v) We prove a Krieger type theorem, by showing that two aperiodic hyperfinite countable equivalence relations are isomorphic if and only if their associated Borel *-algebras Br*(E1) and Br*(E2) are isomorphic.

This work considers the effect of symmetries on analysing bifurcations in dynamical systems. We consider an example of a laser with strong optical feedback which is modelled using coupled non-linear differential equations. A stationary point can be found in space, which can then be continued in parameter space using software such as AUTO. This software will then detect and continue bifurcations which indicate change in dynamics as parameters are varied. Due to symmetries in the equations, using AUTO may require the system of equations to be reduced in order to study periodic orbits of the original system as (relative) equilibria of the reduced system. Reasons for this are explored as well as considering how the equations can be changed or reduced to remove the symmetry. Invariant and Equivariant theory provide the tools for reducing the system of equations to the orbit space, allowing further analysis of the lasers dynamics.

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations

Newton-Picard methods are iterative methods that work well for computing roots of nonlinear equations within a continuation framework. This project presents one of these methods and includes the results of a computation involving the Brusselator problem performed by an implementation of the method. This work was done in collaboration with Andrew Salinger at Sandia National Laboratories.

Este trabalho consiste do estudo das propriedades de transporte de sistemas dinâmicos caóticos por meio do uso de técnicas de operadores de projeção. Tais sistemas podem exibir difusão determinística e relaxação para o equilíbrio. Mostramos que esse comportamento difusivo pode ser visto como uma propriedade espectral do operador de Perron-Frobenius associado. Em particular, a ressonância dominante de Policott-Ruelle é calculada analiticamente para uma classe geral de mapas que preservam área. Sua dependência do número de onda determina os coeficientes de transporte normais. Calculamos uma fórmula geral exata para o coeficiente de difusão, obtida sem qualquer aproximação de alta estocasticidade, e um novo efeito emergiu: a evolução angular pode induzir modos rápidos ou lentos de difusão mesmo no regime de alta estocasticidade. Os aspectos não-Gaussianos do transporte caótico são também investigados para esses sistemas. O estudo é realizado por meio de uma relação entre a curtose, o coeficiente de difusão e o coeficiente de Burnett de quarta ordem, os quais são calculados analiticamente. Uma escala de tempo característica que delimita os regimes Gaussiano e Markoviano para a função densidade foi estabelecida. À parte os modos acelerados, cujas propriedades cinéticas são anômalas, todo os resultados estão em excelente acordo com as simulações numéricas / This work consists in the study of the transport properties of chaotic Hamiltonian systems by using projection operator techniques. Such systems can exhibit deterministic diffusion and display an approach to equilibrium. We show that this diffusive behavior can be viewd as a spectral property of the associated Perron-Frobenius operator. In particular, the leading Pollicott-Ruelle resonance is calculated analytically for a general class of two-dimensional area-preserving maps. Its wavenumber dependence determines the normal transport coefficients. We calculate a general exact formula for the diffusion coefficient, derived without any high stochasticity approximation and a new effect emerges: the angular evolution can induce fast or slow modes of diffusion even in the high stochasticity regime. The non-Gaussian aspects of the chaotic transport are also investigated for this systems. This study is done by means of a relationship between kurtosis and diffusion coefficient and fourth order Burnett coefficient, which are calculated analytically. A characteristic time scale which delimits the Markovian and Gaussian regimes for the density function was established. Despite the accelerator modes, whose kinetics properties are anomalous, all theoretical results are in excellent agreement with the numerical simulations