In this dissertation, we demonstrate the importance of invariant objects in many areas of applied research. The areas of application we consider are chemistry, celestial mechanics and aerospace engineering, plasma physics, and coupled map lattices. In the context of chemical reactions, stable and unstable manifolds of fixed points separate regions of phase space that lead to a certain outcome of the reaction. We study how these regions change under the influence of exposing the molecules to a laser. In celestial mechanics and aerospace engineering, we compute periodic orbits and their stable and unstable manifolds for a object of negligible mass (e.g. a satellite or spacecraft) under the presence of Jupiter and two of its moons, Europa and Ganymede. The periodic orbits serve as convenient spot to place a satellite for observation purposes, and computing their stable and unstable manifolds have been used in constructing low-energy transfers between the two moons. In plasma physics, an important and practical problem is to study barriers for heat transport in magnetically confined plasma undergoing fusion. We compute barriers for which heat cannot pass through. However, such barriers break down and lead to robust partial barriers. In this latter case, heat can flow across the barrier, but at a very slow rate. Finally, infinite dimensional coupled map lattice systems are considered in a wide variety of areas, most notably in statistical mechanics, neuroscience, and in the discretization of PDEs. We assume that the interaction amont the lattice sites decays with the distance of the sites, and assume the existence of an invariant whiskered torus that is localized near a collection of lattice sites. We prove that the torus has invariant stable and unstable manifolds that are also localized near the torus. This is an important step in understanding the global dynamics of such systems and opens the door to new possible results, most notably studying the problem of energy transfer between the sites. / text

Dynamical Systems Theory (DST) (De Bot, Lowie, & Vespoor 2007; Larsen-Freeman 1997, 2007; Larsen-Freeman & Cameron 2008; Dörnyei 2009; and van Lier 2000) represents a scientific paradigm shift derived from the fields of physics, engineering and theoretical mathematics that attempts to solve real-world scenarios that do not respond to scientific reductionism, otherwise known as ‘analysis’. The purpose of this dissertation is to (re)frame foreign language learning/use as a dynamical process that that involves interplay among what Dörnyei (2009) terms the language, the agent and the environment. More specifically, this dissertation presents a quasi-experimental, psycholinguistic study that looks at the interface between language (in this case the talk that resulted from NS-NNS interactions) and agent (as defined by a set of personal traits, or Individual Differences [IDs], including motivation, attitudes, personality and aptitude) in order to answer the research question: Do IDs vary in conjunction with language learning/use, and if so, how? Eight tutored Spanish learners were followed over the course of 16 weeks during which time they participated in 8 chat sessions with a native Spanish-speaker. Their ID profiles were measured immediately before and after each session and sessions with significant pre- to post-session ID shifts were analyzed to determine to what extent such shifts correlated with certain types of talk and/or think-aloud sequences. Results indicated that all participants’ pre- and post-interactional ID profiles fluctuated measurably and significantly, even within the span of a single interaction. Moreover, those sessions with significantly positive ID shifts were qualitatively different in terms of language-related episodes (LREs), conversation management/pragmatic markers, and metacognition from those with significantly negative ID shifts. Other unexpected findings revealed, for example, that LREs (especially NS-initiated LREs) negatively impacted motivations and attitudes and, therefore, the language-learning process itself. Taken together, the results of this study indicate that the agent’s IDs and their (inter)language co-emerge; that is to say, they evolve simultaneously and in response to one another. Moreover, this study suggests that DST can indeed be quasi-experimentally applied to the study of SLA, thus necessitating further development in DST-oriented methodologies and research questions. / text

Efficient and robust numerical simulation of multiscale problems encountered in science and engineering is a formidable challenge. Full resolution of multiscale problems using direct numerical simulations requires enormous amounts of computational time and resources. This thesis develops seamless multiscale methods for ordinary and partial differential equations under the framework of the heterogeneous multiscale method (HMM). The first part of the thesis is devoted to the development of seamless multiscale integrators for ordinary differential equations. The first method, which we call backward-forward HMM (BFHMM), uses splitting and on-the-fly filtering techniques to capture slow variables of highly oscillatory problems without any a priori information. The second method, denoted by variable step size HMM (VSHMM), as the name implies, uses variable mesoscopic step sizes for the unperturbed equation, which gives computational efficiency and higher accuracy. VSHMM can be applied to dissipative problems as well as highly oscillatory problems, while BFHMM has some difficulties when applied to the dissipative case. The effect of variable time stepping is analyzed and the two methods are tested numerically. Multi-spatial problems and numerical methods are discussed in the second part. Seamless heterogeneous multiscale methods (SHMM) for partial differential equations, especially the parabolic case without scale separation are proposed. SHMM is developed first for the multiscale heat equation with a continuum of scales in the diffusion coefficient. This seamless method uses a hierarchy of local grids to capture effects from each scale and uses filtering in Fourier space to impose an artificial scale gap. SHMM is then applied to advection enhanced diffusion problems under incompressible turbulent velocity fields. / text

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This is a thesis on general relativity. It analyzes dynamical properties of Einstein's field equations in cosmology and in the vicinity of spacetime singularities in a number of different situations. Different techniques are used depending on the particular problem under study; dynamical systems methods are applied to cosmological models with spatial homogeneity; Hamiltonian methods are used in connection with dynamical systems to find global monotone quantities determining the asymptotic states; Fuchsian methods are used to quantify the structure of singularities in spacetimes without symmetries. All these separate methods of analysis provide insights about different facets of the structure of the equations, while at the same time they show the relationships between those facets when the different methods are used to analyze overlapping areas. The thesis consists of two parts. Part I reviews the areas of mathematics and cosmology necessary to understand the material in part II, which consists of five papers. The first two of those papers uses dynamical systems methods to analyze the simplest possible homogeneous model with two tilted perfect fluids with a linear equation of state. The third paper investigates the past asymptotic dynamics of barotropic multi-fluid models that approach a `silent and local' space-like singularity to the past. The fourth paper uses Hamiltonian methods to derive new monotone functions for the tilted Bianchi type II model that can be used to completely characterize the future asymptotic states globally. The last paper proves that there exists a full set of solutions to Einstein's field equations coupled to an ultra-stiff perfect fluid that has an initial singularity that is very much like the singularity in Friedman models in a precisely defined way. / <p>Status of the paper "Perfect Fluids and Generic Spacelike Singularities" has changed from manuscript to published since the thesis defense.</p>

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy

We provide a unifying projection-based framework for structure-preserving interpolatory model reduction of parameterized linear dynamical systems, i.e., systems having a structured dependence on parameters that we wish to retain in the reduced-order model. The parameter dependence may be linear or nonlinear and is retained in the reduced-order model. Moreover, we are able to give conditions under which the gradient and Hessian of the system response with respect to the system parameters is matched in the reduced-order model. We provide a systematic approach built on established interpolatory $\mathcal{H}_2$ optimal model reduction methods that will produce parameterized reduced-order models having high fidelity throughout a parameter range of interest. For single input/single output systems with parameters in the input/output maps, we provide reduced-order models that are \emph{optimal} with respect to an $\mathcal{H}_2\otimes\mathcal{L}_2$ joint error measure. The capabilities of these approaches are illustrated by several numerical examples from technical applications.

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Computational efficiency of solving the dynamics of highly oscillatory systems is an important issue due to the requirement of small step size of explicit numerical integration algorithms. A system is considered to be highly oscillatory if it contains a fast solution that varies regularly about a slow solution. As for multibody systems, stiff force elements and contacts between bodies can make a system highly oscillatory. Standard explicit numerical integration methods should take a very small step size to satisfy the absolute stability condition for all eigenvalues of the system and the computational cost is dictated by the fast solution. In this research, a new hybrid integration scheme is proposed, in which the local linearization method is combined with a conventional integration method such as the fourth-order Runge-Kutta. In this approach, the system is partitioned into fast and slow subsystems. Then, the two subsystems are transformed into a reduced and a boundary-layer system using the singular perturbation theory. The reduced system is solved by the fourth-order Runge-Kutta method while the boundary-layer system is solved by the local linearization method. This new hybrid scheme can handle the coupling between the fast and the slow subsystems efficiently. Unlike other multi-rate or multi-method schemes, extrapolation or interpolation process is not required to deal with the coupling between subsystems. Most of the coupling effect can be accounted for by the reduced (or quasi-steady-state) system while the minor transient effect is taken into consideration by averaging. In this research, the absolute stability region for this hybrid scheme is derived and it is shown that the absolute stability region is almost independent of the fast variables. Thus, the selection of the step size is not dictated by the fast solution when a highly oscillatory system is solved, in turn, the computational efficiency can be improved. The advantage of the proposed hybrid scheme is validated through several dynamic simulations of a vehicle system including a flexible tire model. The results reveal that the hybrid scheme can reduce the computation time of the vehicle dynamic simulation significantly while attaining comparable accuracy.

The main goal of this dissertation was to study the bifurcation structure underlying families of low dimensional dynamical systems that model cellular excitability. One of the main contributions of this work is a mathematical characterization of profiles of electrophysiological activity in excitable cells of the same identified type, and across cell types, as a function of the relative levels of expression of ion channels coded by specific genes. In doing so, a generic formulation for transmembrane transport was derived from first principles in two different ways, expanding previous work by other researchers. The relationship between the expression of specific membrane proteins mediating transmembrane transport and the electrophysiological profile of excitable cells is well reproduced by electrodiffusion models of membrane potential involving as few as 2 state variables and as little as 2 transmembrane currents. Different forms of the generic electrodiffusion model presented here can be used to study the geometry underlying different forms of excitability in cardiocytes, neurons, and other excitable cells, and to simulate different patterns of response to constant, time-dependent, and (stochastic) time- and voltage-dependent stimuli. In all cases, an initial analysis performed on a deterministic, autonoumous version of the system of interest is presented to develop basic intuition that can be used to guide analyses of non-autonomous or stochastic versions of the model. Modifications of the biophysical models presented here can be used to study complex physiological systems involving single cells with specific membrane proteins, possibly linking different levels of biological organization and spatio-temporal scales.