This dissertation presents geometric approaches of understanding chaotic transport in phase space that is fundamental across many disciplines in physical sciences and engineering. This approach is based on analyzing phase space transport using boundaries and regions inside these boundaries in presence of perturbation. We present a geometric view of defining such boundaries and study the transport that occurs by crossing such phase space structures. The structure in two dimensional non-autonomous system is the codimension 1 stable and unstable manifolds associated with the hyperbolic fixed points. The manifolds separate regions with varied dynamical fates and their time evolution encodes how the initial conditions in a given region of phase space get transported to other regions. In the context of four dimensional autonomous systems, the corresponding structure is the stable and unstable manifolds of unstable periodic orbits which reside in the bottlenecks of energy surface. The total energy and the cylindrical (or tube) manifolds form the necessary and sufficient condition for global transport between regions of phase space. Furthermore, we adopt the geometric view to define escaping zones for avoiding transition/escape from a potential well using partial control. In this approach, the objective is two fold: finding the minimum control that is required for avoiding escape and obtaining discrete representation called disturbance of continuous noise that is present in physical sciences and engineering. In the former scenario, along with avoiding escape, the control is constrained to be smaller than the disturbance so that it can not exactly cancel out the disturbances. / Ph. D.

This dissertation presents useful nonlinear models for fluid forcing on a moving body in two distinct contexts, and methods for analyzing the geometric structure within those and other mathematical models. This manuscript style dissertation presents three works within the theme of understanding fluid forcing and geometric structure. When a bluff body is free to move in the presence of an incoming bluff body wake, the average forcing on the body is dependent on its position relative to the upstream bluff body. This position-dependent forcing can be conceptualized as a stiffness, much like a spring. This work presents an updated model for the quasi-steady fluid forcing of a wake and extends the notion of wake stiffness to consider a nonlinear spring. These results are compared with kinematic experimental results to provide an example of the application of this framework. Fluid force models also play a role in understanding the behavior of passive aerodynamic gliders, such as gliding animals or plant material. The forces a glider experiences depend on the angle that its body makes with respect to its direction of motion. Modeling the glider as capable of pitch control, this work considers a glider with a fixed angle with respect to the ground. Within this model, all trajectories in velocity space collapse to a 1-dimensional invariant manifold known as the terminal velocity manifold. This work presents methods to identify the terminal velocity manifold, investigates its properties, and extends it to a 2-dimensional invariant manifold in a 3-dimensional space. Finally, in the search for manifolds such as the terminal velocity manifold, this dissertation introduces a new diagnostic for identifying the low dimensional geometric structure of models. The trajectory divergence rate uses instantaneous vector field information to identify regions of large normal stretching and strong normal convergence between nearby invariant manifolds. This work lays out the mathematical basis of the trajectory divergence rate and shows its application to approximate a variety of structures including slow manifolds and Lagrangian coherent structures. This dissertation applies nonlinear theoretical and numerical techniques to analyze models of fluid forcing and their geometric structure. The tools developed in this dissertation lay the groundwork for future research in the fields of flow-induced vibration, plant and animal biomechanics, and dynamical systems. / Ph. D. / When an object moves through a fluid such as air or water, the motion of the surrounding fluid generates forces on the moving object, affecting its motion. The moving object, in turn, affects the motion of the surrounding fluid. This interaction is complicated, nonlinear, and hard to even simulate numerically. This dissertation aims to analyze simplified models for these interactions in a way that gives a deeper understanding of the physics of the interaction between an object and a surrounding fluid. In order to understand these interactions, this dissertation looks at the geometric structure of the models. Very often, there are low-dimensional points, curves, or surfaces which have a very strong effect on the behavior of the system. The search for these geometric structures is another key theme of this dissertation. This dissertation presents three independent studies, with an introduction and conclusion to discuss the overall themes. The first work focuses on the forces acting on a cylinder in the wake of another cylinder. These forces are important to understand, because the vibrations that arise from wake forcing are important to consider when designing bridges, power cables, or pipes to carry oil from the ocean floor to offshore oil platforms. Previous studies have shown that the wake of a circular cylinder acts like a spring, pulling harder on the downstream cylinder the more it is moved from the center of the wake. In this work, I extend this idea of the wake as a spring to consider a nonlinear spring, which keeps the same idea, but provides a more accurate representation of the forces involved. The second work considers a simple model of gliding flight, relevant to understanding the behavior of gliding animals, falling leaves, or passive engineered gliders. Within this model, a key geometric feature exists on which the majority of the motion of the glider occurs, representing a 2-dimensional analogy to terminal velocity. In this work, I study the properties of this influential curve, show several ways to identify it, and extend the idea to a surface in a 3-dimensional model. The third study of this dissertation introduces a new mathematical quantity for studying models of systems, for fluid-body interaction problems, ocean flows, chemical reactions, or any other system that can be modeled as a vector field. This quantity, the trajectory divergence rate, provides an easily computed measurement of highly attracting or repelling regions of the states of a model, which can be used to identify influential geometric structures. This work introduces the quantity, discusses its properties, and shows its application to a variety of systems.

Submitted by Luciana Ferreira (lucgeral@gmail.com) on 2014-08-01T15:13:17Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Disseertacao Luciano Paulo de Araujo Maia.pdf: 1329124 bytes, checksum: b9e2abb00156f6dd57d05cd14dbacbb1 (MD5) / Made available in DSpace on 2014-08-01T15:13:17Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Disseertacao Luciano Paulo de Araujo Maia.pdf: 1329124 bytes, checksum: b9e2abb00156f6dd57d05cd14dbacbb1 (MD5) Previous issue date: 2002 / In this paper we present a new quantum state of light, obtained from an asymmetric superposition of coherent states in circular stationary mode. The generation of this state was proposed and the experimental apparatus required is presented. General expressions describing various properties (statistical distribution, variances, atomic inversion, etc..) Were obtained for arbitrary generations. We observed how they behave their properties by varying the phase between the components of superpo- sition states. We show that the statistical properties do not fully characterize the resulting state, showing different states with the same statistics. Alternative ways to distinguish these states were considered. Based on quantum interference in phase space, we show how we can generate an approximation of the number state N2 thru a process called "quantum sculpture,"and check the influence of the stage in this process. Finally, we measured the classicalidade not analyze this state and its representation in phase space. / Neste trabalho, apresentamos um novo estado quântico da luz, obtido de uma su- perposição assimétrica de estados coerentes circulares no modo estacionário. A ger- ação deste estado foi proposta e o aparato experimental necessário é apresentado. Expressões gerais descrevendo diversas propriedades (distribuição estatística, vari- anças, inversão atômica, etc.) foram obtidas para arbitrárias gerações. Verificamos como se comportam suas propriedades, variando a fase entre os estados componentes da superposição. Mostramos que as propriedades estatísticas não caracterizam com- pletamente o estado resultante, exibindo estados distintos com mesma estatística. Formas alternativas para distinguir estes estados foram consideradas. Baseado na interferência quântica no espaço de fase, mostramos como podemos gerar uma aprox- imação ao estado de número N2 , atravéz de um processo denominado “escultura quântica” e verificamos a influência da fase neste processo. Finalmente, medimos a não classicalidade deste estado e analisamos a sua representação no espaço de fase.

No description available.

Berry curvature appears in the semi-classical theory of Bloch electrons already to first order in electromagnetic fields, resulting in profound modification of the carrier velocity and phase space density of states. Here we derive the equations of motion for the physical position and crystal momentum to second order in the fields. The dynamics still has a Hamiltonian structure, albeit with noncanonical Poisson brackets between the physical variables. We are able to expand both the carrier energy and the Poisson brackets to second order in the fields with terms of clear physical meaning. To demonstrate the utility of our theory, we obtain with much ease the electromagnetic response and orbital magnetic susceptibility. / text

This research uses a Bayesian approach to study the biases that may occur when kinematic data is used to estimate the mass of a galaxy. Data is simulated from the Hernquist (1990) distribution functions (DFs) for velocity dispersions of the isotropic, constant anisotropic, and anisotropic Osipkov (1979) and Merritt (1985) type, and then analysed using the isotropic Hernquist model. Biases are explored when i) the model and data come from the same DF, ii) the model and data come from the same DF but tangential velocities are unknown, iii) the model and data come from different DFs, and iv) the model and data come from different DFs and the tangential velocities are unknown. Mock observations are also created from the Gauthier (2006) simulations and analysed with the isotropic Hernquist model. No bias was found in situation (i), a slight positive bias was found in (ii), a negative bias was found in (iii), and a large positive bias was found in (iv). The mass estimate of the Gauthier system when tangential velocities were unknown was nearly correct, but the mass profile was not described well by the isotropic Hernquist model. When the Gauthier data was analysed with the tangential velocities, the mass of the system was overestimated. The code created for the research runs three parallel Markov Chains for each data set, uses the Gelman-Rubin statistic to assess convergence, and combines the converged chains into a single sample of the posterior distribution for each data set. The code also includes two ways to deal with nuisance parameters. One is to marginalize over the nuisance parameter at every step in the chain, and the other is to sample the nuisance parameters using a hybrid-Gibbs sampler. When tangential velocities, v(t), are unobserved in the analyses above, they are sampled as nuisance parameters in the Markov Chain. The v(t) estimates from the Markov chains did a poor job of estimating the true tangential velocities. However, the posterior samples of v(t) proved to be useful, as the estimates of the tangential velocities helped explain the biases discovered in situations (i)-(iv) above. / Thesis (Master, Physics, Engineering Physics and Astronomy) -- Queen's University, 2013-03-26 17:23:14.643

Desde que surgiu, o computador vem sendo utilizado na modelagem de fenômenos em todas as áreas do conhecimento. Em neurociências, a modelagem computacional é utilizada para descrever, reproduzir e fazer previsões sobre o comportamento dos diferentes componentes do sistema nervoso. Assim como em outras áreas das ciências, este procedimento tem-se mostrado eficiente no estudo e aprimoramento das teorias a respeito da função cerebral. Com o crescente aumento do poder computacional, maiores e mais detalhados modelos podem ser construídos com um grau de realismo biológico cada vez maior. Neste trabalho, apresentamos modelos computacionais biologicamente plausíveis de neurônios corticais do sistema somestésico primário. Os modelos foram construídos com base no formalismo de Hodgkin-Huxley para a implementação de canais iônicos e na técnica de compartimentalização de Rall para modelar sua extensão espacial. Os parâmetros foram ajustados a partir de resultados experimentais 'in vivo' e 'in vitro' com neurônios, retirados da literatura. Os resultados das simulações mostraram que os modelos são biologicamente aceitáveis e de qualidade superior a de outros modelos construídos anteriormente, possibilitando a construção de modelos de redes neuronais em larga escala mais precisos. / Since its appearance, the computer has been used to model phenomena in all areas of knowledge. In neuroscience, computer modeling is used to describe, reproduce and predict behaviors of different components of the nervous system. As well as in other areas of sciences, this procedure has been shown to be efficient in the study and improvement of theories on brain function. With the increasing power of computers, larger and more detailed models can be constructed with an increasing degree of biological realism. In this work, we present biologically plausible computer models of cortical neurons from the primary somatosensory system. The models have been implemented based on the Hodgkin-Huxley formalism for ionic channels and the Rall´s compartmental technique for spatial extent. The parametrs have been adjusted based on in vivo and in vitro experimental results taken from the literature. Simulation results have shown that the models are biologically acceptable and of superior quality in comparison with previous models, allowing the construction of more precise large-scale neuronal network models.

A time-reversible Maxwell's demon is demonstrated which creates a density difference between two chambers initialized to have equal density. The density difference is estimated theoretically and confirmed by computer simulations. It is found that the reversible Maxwell's demon compresses phase space volume even though its dynamics are time reversible. The significance of phase space volume compression in operating a microscopic heat engine is also discussed.

Phase Space Reconstruction is a method that allows to reconstruct the phase space of a system using only an one dimensional time series as input. It can be used for calculating Lyapunov-exponents and detecting chaos. It helps to understand complex dynamics and their behavior. And it can reproduce datasets which were not measured. There are many different methods which produce correct reconstructions such as time-delay, Hilbert-transformation, derivation and integration. The most used one is time-delay but all methods have special properties which are useful in different situations. Hence, every reconstruction method has some situations where it is the best choice. Looking at all these different methods the questions are: Why can all these different looking methods be used for the same purpose? Is there any connection between all these functions? The answer is found in the frequency domain : Performing a Fourier transformation all these methods getting a similar shape: Every presented reconstruction method can be described as a multiplication in the frequency domain with a frequency-depending reconstruction function. This structure is also known as a filter. From this point of view every reconstructed dimension can be seen as a filtered version of the measured time series. It contains the original data but applies just a new focus: Some parts are amplified and other parts are reduced. Furthermore I show, that not every function can be used for reconstruction. In the thesis three characteristics are identified, which are mandatory for the reconstruction function. Under consideration of these restrictions one gets a whole bunch of new reconstruction functions. So it is possible to reduce noise within the reconstruction process itself or to use some advantages of already known reconstructions methods while suppressing unwanted characteristics of it. / Attraktorrekonstruktion („Phase Space Reconstruction“) ist eine Technik, die es ermöglicht, aus einer einzelnen Zeitreihe den vollständigen Phasenraum eines Systems zu rekonstruieren und somit Rückschlüsse auf topologische Eigenschaften dieses dynamischen Systems zu ziehen. Sie findet Verwendung in der Bestimmung von Lyapunov-Exponenten und zur Reproduktion von unbeobachteten Systemgrößen. Es gibt viele verschiedene Methoden zur Attraktorrekonstruktion wie z.B. die Time-Delay-Methode or Rekonstruktion durch Ableitung, Integration oder mithilfe einer Hilbert-Transformation. Zumeist wird der Time-Delay-Ansatz verwendet, es gibt jedoch auch diverse Problemstellungen, in welchen die alternativen Methoden bessere Ergebnisse liefern. Die Kernfragen, die beim Vergleich dieser Methoden entsteht, sind: Wie kommt es, dass alle Ansätze, trotz ihrer teilweise sehr unterschiedlichen Struktur, denselben Zweck erfüllen? Gibt es Übereinstimmungen zwischen all diesen Methoden? Die Antwort lässt sich im Frequenzraum finden: Nach einer Fourier-Transformation besitzen alle genannten Methoden plötzlich eine sehr ähnliche Struktur. Jede Methode transformiert sich im Frequenzraum zu einer einfachen Multiplikation des Eingangssignals mit einer frequenzabhängigen Rekonstruktionsfunktion. Diese Struktur ist in der Datenanalyse auch bekannt als Filter. Aus dieser Perspektive lässt sich jede Rekonstruktionsdimension als gefilterte Zeitreihe der ursprünglichen Zeitreihe interpretieren: Sie enthält den Originaldatensatz, allerdings mit einem verschobenen Fokus: Einige Eigenschaften der Originalzeitreihe werden unterdrückt, während andere Teile verstärkt wiedergegeben werden. Des weiteren zeige ich in der Diplomarbeit, dass nicht jede beliebige Funktion im Frequenzraum zur Rekonstruktion verwendet werden kann. Ich stelle drei Eigenschaften vor, welche jede Rekonstruktionsfunktion erfüllen muss. Unter Beachtung dieser Bedingungen ergeben sich nun diverse Möglichkeiten für neue Rekonstruktionsfunktionen. So ist es z.B. möglich gleichzeitig mit der Rekonstruktion das Ursprungssignal auch zu filtern, oder man kann bereits bestehende Rekonstruktionsfunktionen so abwandeln, dass unerwünschte Nebeneffekte der Rekonstruktion abgemildert oder gar ganz unterdrückt werden.

Josephson junction qubits are promising candidates for a scalable quantum processor. Such qubits are commonly manipulated by means of sequences of rf-pulses and different methods are used to determine their quantum state. The readout should be able to distinguish the two qubit states with high accuracy and be faster than the relaxation time of the qubit. We discuss and experiment with a readout method based on the switching of a Josephson junction from the zero voltage state to a finite voltage state. The Josephson junction circuit has a non-linear dynamics and when it is brought to a bifurcation point, it can be made arbitrarily sensitive to small perturbations. This extreme sensitivity at a bifurcation point can be used to distinguish the two quantum states if the topology of the phase space of the circuit leads to a quick separation into the final states where re-crossings of the bifurcation point are negligible. We optimize a switching current detector by analyzing the phase space of a Josephson junction circuit with frequency dependent damping. A pulse and hold technique is used where an initial current pulse brings the junction close to its bifurcation point and the subsequent hold level is used to give the circuit enough time to evolve until the two states can be distinguished by the measuring instrument. We generate the pulse and hold waveform by a new technique where a voltage step with following linear voltage rise is applied to a bias capacitor. The frequency dependent damping is realized by an on-chip RC-environment fabricated with optical lithography. Josephson junction circuits are added on by means of e-beam lithography. Measurements show that switching currents can be detected with pulses as short as 5 ns and a resolution of 2.5% for a sample directly connected to the measurement leads of the cryostat. Detailed analysis of the switching currents in the RC-environment show that pulses with a duration of 20 us can be explained by a generalization of Kramers' escape theory, whereas switching the same sample with 25 ns pulses occurs out of thermal equilibrium, with sensitivity and speed adequate for qubit readout. / QC 20100924