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21	[en] QUASIPERIODICITY AND THE POSITIVITY OF LYAPUNOV EXPONENTS / [pt] QUASE PERIODICIDADE E A POSITIVIDADE DOS EXPOENTES DE LYAPUNOV LUCAS BARBOSA GAMA 11 January 2019 (has links) [pt] O teorema de Benedicks e Carleson afirma que para a família quadrática existe um conjunto de parâmetros, com medida positiva, para os quais o expoente de Lyapunov é positivo no ponto crítico. Nesta dissertação apresentamos uma demonstração rigorosa e detalhada desse célebre resultado. Uma parte importante da demonstração é o estudo do comportamento quase periódico de um conjunto de órbitas. Além disso, um argumento de grandes desvios é utilizado para mostrar que os parâmetros que não satisfazem a propriedade desejada formam um conjunto pequeno. Tais técnicas apresentam um interesse intrínseco, já que têm se mostrado muito proveitosas para o estudo de outros problemas em sistemas dinâmicos. Combinando o teorema de Benedicks e Carleson ao teorema de Singer, conclui-se que para um conjunto de parâmetros com medida positiva, a função quadrática correspondente não admite atratores periódicos, indicando um comportamento caótico. Neste trabalho, também são estudados critérios para a positividade do expoente de Lyapunov de cociclos quase periódicos de Schrodinger, como o teorema de Herman. O estudo de cociclos de Schrodinger representa um importante tópico na área de física matemática. Mais ainda, algumas das generalizações de tais critérios utilizam as técnicas de Benedicks-Carleson. / [en] The Benedicks and Carleson theorem states that for the quadratic family there exists a set of parameters, with positive measure, for which the Lyapunov exponent is positive at the critical point. In this dissertation we present a rigorous and detailed proof of this famous result. An important part of the proof is the study of the quasi periodic behavior of a set of orbits. In addition, a large deviation argument is used to show that parameters which do not satisfy the desired property form a small set. Such techniques have an intrinsic interest, as they have proven fruitful in the study of other problems in dynamical systems. Combining Benedicks-Carlesons theorem with Singers theorem, we conclude that for a set of parameters with positive measure, the corresponding quadratic function does not admit periodic attractors, indicating its chaotic behavior. In this work we also study criteria for the positivity of the Lyapunov exponent of quasi-periodic Schrodinger cocycles, such as Hermans theorem. The study of the Schrodinger cocycles represents an important topic in mathematical physics. Moreover, some of the generalizations of such criteria use the techniques of Benedicks-Carleson. [pt] A FAMILIA QUADRATICA [en] THE QUADRATIC FAMILY [pt] O TEOREMA DE SINGER [en] SINGER S THEOREM [pt] O TEOREMA DE BENEDICKS-CARLESON [en] BENEDICKS-CARLESON S THEOREM [pt] EXPOENTES DE LYAPUNOV [en] LYAPUNOV EXPONENTS [pt] GRANDES DESVIOS [en] LARGE DEVIATIONS [en] QUASI-PERIODIC SCHRUDINGER COCYCLES
22	A influ?ncia do esqueleto a??car-fostato no transporte da mol?cula de DNA Sarmento, Ricardo Gondim 30 September 2008 (has links) Made available in DSpace on 2014-12-17T15:14:49Z (GMT). No. of bitstreams: 1 RicardoGS.pdf: 538526 bytes, checksum: 84faa5411061c273b56347caa03d1fac (MD5) Previous issue date: 2008-09-30 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / This dissertation analyses the influence of sugar-phosphate structure in the electronic transport in the double stretch DNA molecule, with the sequence of the base pairs modeled by two types of quasi-periodic sequences: Rudin-Shapiro and Fibonacci. For the sequences, the density of state was calculated and it was compared with the density of state of a piece of human DNA Ch22. After, the electronic transmittance was investigated. In both situations, the Hamiltonians are different. On the analysis of density of state, it was employed the Dyson equation. On the transmittance, the time independent Schr?dinger equation was used. In both cases, the tight-binding model was applied. The density of states obtained through Rudin-Shapiro sequence reveal to be similar to the density of state for the Ch22. And for transmittance only until the fifth generation of the Fibonacci sequence was acquired. We have considered long range correlations in both transport mechanism / Esta disserta??o analisa a influ?ncia do esqueleto a??car-fosfato no transporte eletr?nico na mol?cula de DNA de fita dupla, com o sequenciamento dos pares de base modelado por dois tipos de seq??ncias quasi-peri?dicas: Rudin-Shapiro e Fibonacci. Para ambas as seq??ncias, foram calculadas as densidades de estado e comparadas com a densidade de estado de um trecho do DNA humano Ch 22. Em seguida, foi investigada a transmit?ncia eletr?nica. Nos dois casos, as Hamiltonianas s?o distintas. Na an?lise da densidade de estado foi empregada a equa??o de Dyson. Na transmit?ncia foi feito uso da equa??o de Schr?dinger independente do tempo. Em ambos os casos, foi utilizado o modelo tight-binding. Os resultados para a densidade de estado foram mais satisfat?rios para a seq??ncia de Rudin-Shapiro, que forneceu um perfil muito pr?ximo do perfil da densidade de estado para o Ch22. A transmit?ncia foi calculada somente para a seq??ncia de Fibonacci at? a quinta gera??o. Nestes dois mecanismos de transporte, as correla??es s?o de longo alcance DNA Esqueleto a??car-fosfato Transporte eletr?nico Fitas duplas Seq??ncia quasi-peri?dica Cromossomo 22 Genoma humano DNA Sugar-phosphate structure Electronic transport Double stretch Quasi-periodic sequences Ch22 Human genome CNPQ::CIENCIAS EXATAS E DA TERRA::FISICA
23	Cislunar Trajectory Design Methodologies Incorporating Quasi-Periodic Structures With Applications Brian P. McCarthy (5930747) 29 April 2022 (has links) <p> </p> <p>In the coming decades, numerous missions plan to exploit multi-body orbits for operations. Given the complex nature of multi-body systems, trajectory designers must possess effective tools that leverage aspects of the dynamical environment to streamline the design process and enable these missions. In this investigation, a particular class of dynamical structures, quasi-periodic orbits, are examined. This work summarizes a computational framework to construct quasi-periodic orbits and a design framework to leverage quasi-periodic motion within the path planning process. First, quasi-periodic orbit computation in the Circular Restricted Three-Body Problem (CR3BP) and the Bicircular Restricted Four-Body Problem (BCR4BP) is summarized. The CR3BP and BCR4BP serve as preliminary models to capture fundamental motion that is leveraged for end-to-end designs. Additionally, the relationship between the Earth-Moon CR3BP and the BCR4BP is explored to provide insight into the effect of solar acceleration on multi-body structures in the lunar vicinity. Characterization of families of quasi-periodic orbits in the CR3BP and BCR4BP is also summarized. Families of quasi-periodic orbits prove to be particularly insightful in the BCR4BP, where periodic orbits only exist as isolated solutions. Computation of three-dimensional quasi-periodic tori is also summarized to demonstrate the extensibility of the computational framework to higher-dimensional quasi-periodic orbits. Lastly, a design framework to incorporate quasi-periodic orbits into the trajectory design process is demonstrated through a series of applications. First, several applications were examined for transfer design in the vicinity of the Moon. The first application leverages a single quasi-periodic trajectory arc as an initial guess to transfer between two periodic orbits. Next, several quasi-periodic arcs are leveraged to construct transfer between a planar periodic orbit and a spatial periodic orbit. Lastly, transfers between two quasi-periodic orbits are demonstrated by leveraging heteroclinic connections between orbits at the same energy. These transfer applications are all constructed in the CR3BP and validated in a higher-fidelity ephemeris model to ensure the geometry persists. Applications to ballistic lunar transfers are also constructed by leveraging quasi-periodic motion in the BCR4BP. Stable manifold trajectories of four-body quasi-periodic orbits supply an initial guess to generate families of ballistic lunar transfers to a single quasi-periodic orbit. Poincare mapping techniques are used to isolate transfer solutions that possess a low time of flight or an outbound lunar flyby. Additionally, impulsive maneuvers are introduced to expand the solution space. This strategy is extended to additional orbits in a single family to demonstrate "corridors" of transfers exist to reach a type of destination motion. To ensure these transfers exist in a higher fidelity model, several solutions are transitioned to a Sun-Earth-Moon ephemeris model using a differential corrections process to show that the geometries persist.</p> Aerospace Engineering trajectory design quasi-periodic orbits cislunar Multi-Body Dynamics astrodynamics Three-body problem four-body problem ballistic lunar transfers Dynamical Systems Theory Poincare Map orbit mechanics spacecraft path planning invariant curves
24	Neural basis and behavioral effects of dynamic resting state functional magnetic resonance imaging as defined by sliding window correlation and quasi-periodic patterns Thompson, Garth John 20 September 2013 (has links) While task-based functional magnetic resonance imaging (fMRI) has helped us understand the functional role of many regions in the human brain, many diseases and complex behaviors defy explanation. Alternatively, if no task is performed, the fMRI signal between distant, anatomically connected, brain regions is similar over time. These correlations in “resting state” fMRI have been strongly linked to behavior and disease. Previous work primarily calculated correlation in entire fMRI runs of six minutes or more, making understanding the neural underpinnings of these fluctuations difficult. Recently, coordinated dynamic activity on shorter time scales has been observed in resting state fMRI: correlation calculated in comparatively short sliding windows and quasi-periodic (periodic but not constantly active) spatiotemporal patterns. However, little relevance to behavior or underlying neural activity has been demonstrated. This dissertation addresses this problem, first by using 12.3 second windows to demonstrate a behavior-fMRI relationship previously only observed in entire fMRI runs. Second, simultaneous recording of fMRI and electrical signals from the brains of anesthetized rats is used to demonstrate that both types of dynamic activity have strong correlates in electrophysiology. Very slow neural signals correspond to the quasi-periodic patterns, supporting the idea that low-frequency activity organizes large scale information transfer in the brain. This work both validates the use of dynamic analysis of resting state fMRI, and provides a starting point for the investigation of the systemic basis of many neuropsychiatric diseases. fMRI Dynamic analysis Sliding window Sliding windows Resting state Default mode Functional connectivity Functional network Correlation Coherence Functional magnetic resonance imaging Local field potentials LFP Band-limited power BLP Inter-individual Intra-individual Spontaneous activity Brain Primary somatosensory cortex Rodent Rat Human Simultaneous experiments Neural basis Glial basis Neurons Astrocytes Vasomotion Hemodynamic response Dynamic Spatiotemporal dynamics Quasi-periodic patterns Brain networks Resting state networks Anesthesia Dexmedetomidine Isoflurane Awake Global signal Magnetic resonance imaging Brain Pathophysiology Brain mapping
25	Characterization of Quasi-Periodic Orbits for Applications in the Sun-Earth and Earth-Moon Systems Brian P. McCarthy (5930747) 17 January 2019 (has links) <div>As destinations of missions in both human and robotic spaceﬂight become more exotic, a foundational understanding the dynamical structures in the gravitational environments enable more informed mission trajectory designs. One particular type of structure, quasi-periodic orbits, are examined in this investigation. Speciﬁcally, eﬃcient computation of quasi-periodic orbits and leveraging quasi-periodic orbits as trajectory design alternatives in the Earth-Moon and Sun-Earth systems. First, periodic orbits and their associated center manifold are discussed to provide the background for the existence of quasi-periodic motion on n-dimensional invariant tori, where n corresponds to the number of fundamental frequencies that deﬁne the motion. Single and multiple shooting diﬀerential corrections strategies are summarized to compute families 2-dimensional tori in the Circular Restricted Three-Body Problem (CR3BP) using a stroboscopic mapping technique, originally developed by Howell and Olikara. Three types of quasi-periodic orbit families are presented: constant energy, constant frequency ratio, and constant mapping time families. Stability of quasi-periodic orbits is summarized and characterized with a single stability index quantity. For unstable quasi-periodic orbits, hyperbolic manifolds are computed from the diﬀerential of a discretized invariant curve. The use of quasi-periodic orbits is also demonstrated for destination orbits and transfer trajectories. Quasi-DROs are examined in the CR3BP and the Sun-Earth-Moon ephemeris model to achieve constant line of sight with Earth and avoid lunar eclipsing by exploiting orbital resonance. Arcs from quasi-periodic orbits are leveraged to provide an initial guess for transfer trajectory design between a planar Lyapunov orbit and an unstable halo orbit in the Earth-Moon system. Additionally, quasi-periodic trajectory arcs are exploited for transfer trajectory initial guesses between nearly stable periodic orbits in the Earth-Moon system. Lastly, stable hyperbolic manifolds from a Sun-Earth L<sub>1</sub> quasi-vertical orbit are employed to design maneuver-free transfer from the LEO vicinity to a quasi-vertical orbit.</div> Aerospace Engineering aerospace engineering trajectory trajectory design astrodynamics three body three body mechanics circular restricted three body problem CR3BP quasi-periodic orbits periodic orbits orbit mechanics celestial mechanics dynamical systems dynamical systems theory dynamics applications orbits astronautics differential corrections invariance invariant circle invariant torus torus invariant tori stroboscopic mapping manifolds Sun-Earth system Earth-Moon system stability multiple shooting stroboscopic map eclipse avoidance vertical orbit halo orbit distant retrograde orbit quasi-halo quasi-vertical lissajous quasi-DRO DRO NRHO quasi-NRHO synodic resonance trajectory arcs mission design astromechanics spaceflight mechanics dynamical structures

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