Earthquakes, floods, rainfall represent a class of nonlinear systems termed chaotic, in which the relationships between variables in a system are dynamic and disproportionate, however completely deterministic. Classical linear time series models have proved inadequate in analysis and prediction of complex geophysical phenomena. Nonlinear approaches such as Artificial Neural Networks, Hidden Markov Models and Nonlinear Prediction are useful in forecasting of daily discharge values in a river. The focus of these methods is on forecasting magnitudes of future discharge values and not the prediction of floods. Chaos theory provides a structured explanation for irregular behavior and anomalies in systems that are not inherently stochastic. Time Series Data Mining methodology combines chaos theory and data mining to characterize and predict complex, nonperiodic and chaotic time series. Time Series Data Mining focuses on the prediction of events.

This dissertation details the development and experimental implementation of single-photon atomic cooling. In this scheme atoms are transferred from a large-volume magnetic trap into a small-volume optical trap via a single spontaneous Raman transition that is driven near each atom's classical turning point. This arrangement removes nearly all of an atomic ensemble's kinetic energy in one dimension. This method does not rely on a transfer of momentum from photon to atom to cool. Rather, single-photon atomic cooling achieves a reduction in temperature and an increase in the phase-space density of an atomic ensemble by the direct reduction of the system's entropy. Presented here is the application of this technique to a sample of magnetically trapped ⁸⁷Rb. Transfer efficiencies between traps of up to 2.2% are demonstrated. It is shown that transfer efficiency can be traded for increased phase-space compression. By doing so, the phase-space density of a magnetically trapped ensemble is increased by a factor of 350 by the single-photon atomic cooling process. / text

Many seemingly simple systems can display extraordinarily complex dynamics which has been studied and uncovered through nonlinear dynamical theory. The leitmotif of this thesis is changing phase-space structures and their (linear or nonlinear) stabilities by adding control functions (which act on the system as external perturbations) to the relevant Hamiltonians. These phase-space structures may be periodic orbits, invariant tori or their stable and unstable manifolds. One-electron systems and diatomic molecules are fundamental and important staging ground for new discoveries in nonlinear dynamics. In past years, increasing emphasis and effort has been put on the control or manipulation of these systems. Recent developments of nonlinear dynamical tools can provide efficient ways of doing so. In the first subtopic of the thesis, we are adding a control function to restore tori at prescribed locations in phase space. In the remainder of the thesis, a control function with parameters is used to change the linear stability of the periodic orbits which govern the processes in question. In this thesis, we report our theoretical analyses on multiphoton ionization of Rydberg atoms exposed to strong microwave fields and the dissociation of diatomic molecules exposed to bichromatic lasers using nonlinear dynamical tools. This thesis is composed of three subtopics. In the first subtopic, we employ local control theory to reduce the stochastic ionization of hydrogen atom in a strong microwave field by adding a relatively small control term to the original Hamiltonian. In the second subtopic, we perform periodic orbit analysis to investigate multiphoton ionization driven by a bichromatic microwave field. Our results show quantitative and qualitative agreement with previous studies, and hence identify the mechanism through which short periodic orbits organize the dynamics in multiphoton ionization. In addition, we achieve substantial time savings with this approach. In the third subtopic we extend our periodic orbit analysis to the dissociation of diatomic molecules driven by a bichromatic laser. In this problem, our results based on periodic orbit analysis again show good agreement with previous work, and hence promise more potential applications of this approach in molecular physics.

Made available in DSpace on 2016-12-12T20:15:51Z (GMT). No. of bitstreams: 1 Ana Carolina.pdf: 30877153 bytes, checksum: df1e32c0dad640cd008d62b73fa92072 (MD5) Previous issue date: 2014-07-30 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work/study, have showed the results of the investigation about dynamics standard map in the forms; conservative, dissipative and dissipative relativistic, through a numerical and analytical approach for the last case. For the three abovementioned cases, have been build and inspected the respective phase spaces, parameter spaces and spaces of periods, by which has been found the coexistence of chaotic and periodic domains. It has still been characterized and discussed the influence of the parameters responsibles for the dissipation and damping or by a relativistic component in the dynamics of a particle subjected to periodic pulses, as well as the influence of the parameter responsible for noise in the system. The results showed the existence of periodic structures that are organized from the smallest to highest period immersed in chaotic regions. Finally, it has showed that the self-similar structures were gradually disturbed by the noise, as well as the less stable regions, where there are bifurcations, even were hit and porteriormente are totally destroyed and replaced by a chaotic structure. / Neste trabalho apresentaram-se os resultados obtidos da investigação da dinâmica do mapa padrão nas formas; conservativa, dissipativa e dissipativa relativística, através de uma abordagem numérica e analítica para o último caso. Para os três casos supracitados, construíram-se e inspecionaram-se os respectivos espaços de fases, espaços de parâmetros e espaços de períodos, pelos quais constatou-se a coexistência de domínios caóticos e periódicos. Caracterizou-se e discutiu-se ainda a influência dos parâmetros responsáveis pelo amortecimento ou dissipação e por uma componente relativística na dinâmica de uma partícula submetida a pulsos periódicos, assim como a influência do parâmetro responsável pelo ruído no sistema. Os resultados evidenciaram a existência de estruturas periódicas que se organizam do menor para o maior período imersas em regiões caóticas. Finalmente , mostrou-se que as estruturas auto similares foram gradativamente perturbadas pelo ruído, assim como as regiões menos estáveis onde há bifurcações, também foram atingidas e porteriormente são totalmente destruídas e substituídas por uma estrutura caótica.

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Tunneling in 1D describes the effect that quantum particles can penetrate a classically insurmountable potential energy barrier. The extension to classically forbidden transitions in phase space generalizes the tunneling concept. A typical 1D Hamiltonian system has a mixed phase space. It contains regions of regular and chaotic dynamics, the so-called regular islands and the chaotic sea. These different phase space components are classically separated by dynamically generated barriers. Quantum mechanically they are, however, connected by dynamical tunneling. We perform a semiclassical quantization of almost resonance-free regular islands and transporting island chains of quantum maps. This yields so-called quasimodes, which are used for the investigation of direct dynamical tunneling from an almost resonance-free regular island to the chaotic sea. We derive a formula which allows for the determination of dynamical tunneling rates. Good agreement between this analytical prediction and numerical results is found over several orders of magnitude for two example systems. / Der 1D Tunneleffekt bezeichnet das Durchdringen einer klassisch nicht überwindbaren potentiellen Energiebarriere durch Quantenteilchen. Eine Verallgemeinerung des Tunnelbegriffs ist die Erweiterung auf jegliche Art von klassisch verbotenen Übergangsprozessen im Phasenraum. Der Phasenraum eines typischen 1D Hamiltonschen Systems ist gemischt. Er besteht aus Bereichen regulärer und chaotischer Dynamik, den sogenannten regulären Inseln und der chaotischen See. Während diese verschiedenen Phasenraumbereiche klassisch durch dynamisch generierte Barrieren voneinander getrennt sind, existiert quantenmechanisch jedoch eine Verknüpfung durch den dynamischen Tunnelprozess. In dieser Arbeit wird eine semiklassische Quantisierung von praktisch resonanz-freien regulären Inseln und transportierenden Inselketten von Quantenabbildungen durchgeführt. Die daraus folgenden sogenannten Quasimoden werden für die Untersuchung des direkten dynamischen Tunnelns aus einer praktisch resonanz-freien regulären Insel in die chaotische See verwendet, was auf eine Tunnelraten vorhersagende Formel führt. Ihre anschlie?ßende Anwendung auf zwei Modellsysteme zeigt eine gute Übereinstimmung zwischen Numerik und analytischer Vorhersage über viele Größenordnungen.

Quantum systems with finite Hilbert space where position x and momentum p take values in Z(d) (integers modulo d) are considered. Symplectic tranformations S(2¿,Z(p)) in ¿-partite finite quantum systems are studied and constructed explicitly. Examples of applying such simple method is given for the case of bi-partite and tri-partite systems. The quantum correlations between the sub-systems after applying these transformations are discussed and quantified using various methods. An extended phase-space x¿p¿X¿P where X, P ¿ Z(d) are position increment and momentum increment, is introduced. In this phase space the extended Wigner and Weyl functions are defined and their marginal properties are studied. The fourth order interference in the extended phase space is studied and verified using the extended Wigner function. It is seen that for both pure and mixed states the fourth order interference can be obtained. / Ministry of Higher Education, Sultanate of Oman

In many problems in dynamical systems one is interested in the identification of sets which have qualitatively different fates. The finite-time Lyapunov exponent (FTLE) method is a general and equation-free method that identifies codimension-one sets which have a locally high rate of stretching around which maximal exponential expansion of line elements occurs. These codimension-one sets thus act as transport barriers. This geometric framework of transport barriers is used to study various problems in phase space transport, specifically problems of separation in flows that can vary in scale from the micro to the geophysical. The first problem which we study is of the nontrivial motion of inertial particles in a two-dimensional fluid flow. We use the method of FTLE to identify transport barriers that produce segregation of inertial particles by size. The second problem we study is the long range advective transport of plant pathogen spores in the atmosphere. We compute the FTLE field for isobaric atmospheric flow and identify atmospheric transport barriers (ATBs). We find that rapid temporal changes in the spore concentrations at a sampling point occur due to the passage of these ATBs across the sampling point. We also investigate the theory behind the computation of the FTLE and devise a new method to compute the FTLE which does not rely on the tangent linearization. We do this using the 925 matrix of a probability density function. This method of computing the geometric quantities of stretching and FTLE also heuristically bridge the gap between the geometric and probabilistic methods of studying phase space transport. We show this with two examples. / Ph. D.

Seit Beginn der Pandemie Anfang des Jahres 2020 werden statistische Daten erhoben mit dem Ziel, die Ausbreitung von COVID-19 zu charakterisieren. Grundlage der statistischen Analysen bilden die Zeitreihen der täglich erfassten Anzahl von Neuinfektionen. Die Dynamik der Pandemie lässt sich als Trajektorie in einem Phasenraum visualisieren. Dieser Zugang und ein Vergleich mit dem mathematischen Modell des logistischen Wachstums ermöglicht eine Analyse der Wirksamkeit von Maßnahmen und liefert Hinweise für eine Optimierung von Strategien zur Eindämmung der Virusausbreitung.:1. Einleitung 2. Zeitliche Entwicklung der Infektionszahlen und logistisches Wachstum 3. Pandemie im Phasenraum und log-log-Darstellung 3.1 Dynamik der Pandemie bis 02.03.2020 bis 27.06.2021 3.2 Dynamik der Pandemie von 28.06.2021 bis 02.06.2023 4. Diskussion 4.1 Datenerhebung und statistische Auswertung 4.2 Zeitliche, räumliche und sachliche Analyse der Daten 4.3 Schlussfolgerungen und Ausblick 5. Anhang 6. Literatur / Since the start of the pandemic at the beginning of 2020, statistical data have been collected with the aim of characterizing the spread of COVID-19. The basis of the statistical analyzes is the time series of the number of new infections recorded daily. The dynamics of the pandemic can be visualized as a trajectory in a phase space. This approach and a comparison with the mathematical model of logistic growth enables us an analysis of the effectiveness of measures and provides evidence for optimizing strategies for containment of the virus.:1. Einleitung 2. Zeitliche Entwicklung der Infektionszahlen und logistisches Wachstum 3. Pandemie im Phasenraum und log-log-Darstellung 3.1 Dynamik der Pandemie bis 02.03.2020 bis 27.06.2021 3.2 Dynamik der Pandemie von 28.06.2021 bis 02.06.2023 4. Diskussion 4.1 Datenerhebung und statistische Auswertung 4.2 Zeitliche, räumliche und sachliche Analyse der Daten 4.3 Schlussfolgerungen und Ausblick 5. Anhang 6. Literatur