Orbit of a spacecraft in three dimensional Inertial Reference Frame is in general represented by a standard set of six parameters like Keplerian Orbital Elements namely semimajor axis, eccentricity, inclination, argument of perigee, right ascension of ascending node, and true anomaly. An orbit can also be represented by an equivalent set of six parameters namely the position and velocity vectors, hereafter referred as orbit-vectors. The process of determining the six orbital parameters from redundant set of observations (more than the required minimum observations) is known as Orbit Determination (OD) process. This is, in general, solved using Least Squares principle. Availability of accurate, almost continuous, space borne observations provide tremendous scope for simplifications and new directions in Autonomous OD (AOD). The objective of this thesis is to develop a suitable scheme for onboard autonomy in OD, specifically for low-earth-orbit-missions that are in high demand in the immediate future. The focus is on adopting a simple orbit model by a thorough study and analysis by considering the individual contributions from the different force models or component accelerations acting on the spacecraft. Second step in this work is to address the application of an onboard estimation scheme like Kalman Filter for onboard processing. The impact of the approximation made in the orbit model for filter implementation manifests as propagation error or estimation residuals in the estimation. The normal procedure of tuning the filter is by getting an appropriate state and measurement noise covariance matrices by some means, sometimes through trial and error basis. Since this tuning is laborious and the performance may vary with different contexts, it is attempted to propose a scheme on a more general footing, with dynamically compensating for the model simplification. There are three parts of this problem namely (i) Analysis of different Orbit Dynamics Models and selection of a simplified Onboard Model (ii) Design of an Estimator Filter based on Kalman Filter approach for Onboard Applications and (iii) Development of a suitable Filter Compensation procedure to ensure best estimates of orbit vectors even with the simplified orbit model. Development of a Numerical Integration scheme (and a software tool) and extensive simulation exercises to justify the conclusion on the simple model to be used in the estimation procedure forms the first part of the thesis. Tables quantify the effect of individual accelerations and demonstrate the effects of various model components on orbit propagation. In general, it is well known that the atmospheric drag is a non-conservative force and reduces energy; it is also known that the effect of first zonal harmonic term is predominant than any other gravity parameters; such anticipated trends in the accuracies are obtained. This particular exercise is carried out for orbits of different altitudes and different inclinations. The analysis facilitates conclusions on a limited model orbit dynamics suitable for onboard OD. Procedures and results of this model selection analysis is published in Journal of Spacecraft Technology, Vol. 16, No.1,pp 8-30, Jan 2006, titled “Orbit Model Studies for Onboard Orbit Estimation” [69]. Design of Estimator based on Kalman Filter There are two steps involved in dealing with the next part of the defined work: • Design and implementation of Extended Kalman Filter Estimation (EKF) scheme • Steps to compensate for approximation made in the reduced orbit dynamics The GPS receivers on board some of the IRS satellites (for example, the Resource-Sat-1), output the GPS Navigation Solutions (GPSNS) namely the position and velocity vectors of the IRS satellite along with the Pseudo-range measurements. These are recorded onboard for about two orbits duration, and are down loaded. An Extended Kalman Filter Algorithm for the estimation of the orbit vectors using these GPSNS observations is developed. Estimation is carried out assuming a Gaussian white noise models for the state and observation noises. The results show a strong dependence on the initial covariance of the noise involved; reconstruction of the observations results only if the assumption of realistic noise characteristics (which are unknown) is strictly adhered. Hence this simple non-adaptive EKF is found inadequate for onboard OD scheme. Development of the Dynamics Filter Compensation (DFC) Scheme In next part of the thesis, the problem of dealing with the un-modeled accelerations has been addressed. A suitable model-compensation scheme that was first developed by D.S Ingram el at [60] and successfully applied to Lunar missions, has been modified suitably to treat the problem posed by the reduced orbit dynamics. Here, the un-modeled accelerations are approximated by the OU stochastic process described as the solution of the Langavin stochastic differential equation. A filter scheme is designed where the coefficients of the un- modeled acceleration components are also estimated along with the system state yielding a better solution. Further augmentation to the filter include a standard Adaptive Measurement Noise covariance update; results are substantiated with actual data of IRS-P6 (Resource–Sat 1, see chapter 4). Classified as the Structured Adaptive Filtering Scheme, this results in a Dynamic Filter Compensation(DFC) Scheme which provides distinctly improved results in the position of the state. First, the estimation is carried out using actual GPS Navigation Solutions as observations. What is to be estimated itself is observed; the State-Observation relation is simple. The results are seen to improve the orbit position five times; bringing down the position error from 40 meters to about 8 meters. However, this scheme superimposes an extra factor of noise in the velocity vector of the GPSNS solutions. It is noted that this scheme deals only with the process noise covariance. To tackle the noise introduced in the velocity components, modifications of the original scheme by introducing an adaptive measurement noise covariance update is done. This improves the position estimate further by about 2 meters and also removes the noise introduced in the velocity components and reconstructs the orbit velocity vector output of the GPSNS. The results are confirmed using one more set of actual data corresponding to a different date. This scheme is shown to be useful for obtaining continuous output –without data gaps- of the GPSNS output. Next, the estimation is carried out taking the actual GPS observations which are the Pseudo Range, Range rate measurements from the visible GPS satellites (visible to the GPS receiver onboard ). Switching over to the required formulation for this situation in the state-measurement relation profile, estimation is carried out. The results are confirmed in this case also. Clear graphs of comparisons with definitive orbital states (considered as actual) versus estimated states show that the model reduction attempted at the first part has been successfully tackled in this method. In this era of space-borne GPS observations, where frequent sampling of the orbiting body is suggestive of reduced orbit models, an attempt for replacement of the conventional treatment of expensive and elaborate OD procedure is proved feasible in this thesis work.

Die Arbeit befasst sich mit zweierleiZum einen wird der STM-Abbildungsprozess simuliert, indem Probe und Spitze durch zweidimensionale Sommerfeld-Metalle frei waehlbarer Geometrie beschrieben werden und der Tunnelstrom im Transfer-Hamiltonian-Formalismus bestimmt wird. Die Berechnung der Eigenzustaende der Elektroden erfolgt numerisch durch Diskretisierung der Schroedingergleichung im Differenzenverfahren. Ueber die geometrische Entfaltung der erhaltenen Konstantstromprofile mit der Spitzengeometrie werden der Vergleich zum geometrischen (mechanischen) Abtasten gezogen und Moeglichkeiten einer Vermessung von Spitze und Probe diskutiert. Zum anderen wird durch Berechnung von Eigenzustaenden in grossen zweidimensionalen Potentialkaesten (Quantenbillards) endlicher Wandhoehe der Frage nachgegangen, welchen Einfluss klassisch verbotene Gebiete (Aussenraum, Tunnelbarriere) auf Eigenfunktionen in semiklassisch grossen Systemen haben. Betrachtet wird insbesondere ein Gesamtsystem bestehend aus zwei Potentialkaesten, die ueber eine Tunnelbarriere koppeln (¨Quantenbillards endlicher Wandhoehe im Tunnelkontakt¨). Bei einer Reihe von Zustaenden zeigen sich Scars, die aus der Barriere austreten und in diese zuruecklaufen. Das Gesamtsystem ist in hohem Masse nichtintegrabel, ¨sichtbar¨ wird dieses aber nur fuer Bahnen entweder des Kontinuums oder fuer komplexe Orbits. Eine semiklassische Beschreibung dieses Phaenomens mit der gegenwaertigen, auf klassischen Orbits fussenden Theorie periodischer Bahnen ist nicht mehr moeglich. Die Einbeziehung komplexer Orbits oder Bahnen des Kontinuums (¨ungebundener Orbits¨) wird durch diese Ergebnisse angemahnt.

We describe a procedure for classifying 4D N=2 superconformal theories of the type introduced by Davide Gaiotto. Any punctured curve, C, on which the 6D (2,0) SCFT is compactified, may be decomposed into 3-punctured spheres, connected by cylinders. The 4D theories, which arise, can be characterized by listing the ``matter" theories corresponding to 3-punctured spheres, the simple gauge group factors, corresponding to cylinders, and the rules for connecting these ingredients together. Different pants decompositions of C correspond to different S-duality frames for the same underlying family of 4D \mathcal{N}=2 SCFTs. We developed such a classification for the A_{N-1} and the D_N series of 6D (2,0) theories. We outline the procedure for general A_{N-1} and D_N, and construct, in detail, the classification through A_4 and D_4, respectively. / text

Στην συγκεκριμένη διπλωματική εργασία ασχολήθηκα με το φωτοβαρυτικό περιορισμένο πρόβλημα των τριών σωμάτων.Εστίασα στις τρισδιάστατες περιοδικές κινήσεις και περιέγραψα αναλυτικά οικογένειες τρισδιάστατων περιοδικών τροχιών. / At this master I work on the photogravitational restricted 3-orbit problem. Espesially I focus on the 3-dimensional orbits solutions and I describe families of 3-dimensional orbits.

Paper discusses the expected uncertainty of orbital parameters of binary stars as measured by the space-based gravitational wave observatory LISA (Laser Interferometer Space Antenna) and how the inclusion of spin in the model of the binary stars affects the uncertainty. The uncertainties are found by calculating the received gravitational wave from a binary pair and then performing a linear least-squares parameter estimation. The case of a 1500 solar mass black hole that is 20 years from coalescing with a 1000 solar mass black hole--both of which are 50 x 10^6 light years away--is analyzed, and the results show that the inclusion of spin has a negligible effect upon the angular resolution of LISA but can increase the accuracy in mass and distance measurements by factors of 15 and 65, respectively.

Mon travail de thèse a porté sur la compréhension «systèmes dynamiques de la dynamique à grande échelle dans l’écoulement pleinement développé de cisaillement turbulent. Dans le plan écoulement de Couette, simulation des grandes échelles (LES) est utilisée pour modéliser petits mouvements d’échelle et de ne résoudre mouvements à grande échelle afin de calculer non linéaire ondes progressives (SNT) et orbites périodiques relatives (RPO). Artificiel sur-amortissement a été utilisé pour étancher une gamme croissante de petite échelle motions et prouvent que les motions grande échelle sont auto-entretenue. Les solutions d’onde inférieure branche itinérantes qui se trouvent sur le bassin laminaire turbulent limite sont obtenues pour ces simulation sur-amortie et continue encore dans l’espace de paramètre à des solutions de branche supérieure. Cette approche ne aurait pas été possible si, comme supposé dans certains enquêtes précédentes, les mouvements à grande échelle dans le mur bornées flux de cisaillement sont forcée par un mécanisme fondé sur l’existence de structures actives à plus petite échelle. En flux Poseuille, orbites périodiques relatives à décalage réflexion symétrie sur la limite du bassin laminaire turbulent sont calculés en utilisant DNS. Nous montrons que le RPO trouvé sont connectés à la paire de voyager vague (TW) solution via bifurcation mondiale (noeud-col-période infinie bifurcation). La branche inférieure de cette solution TW évoluer dans un état de l’envergure localisée lorsque le domaine de l’envergure est augmentée. La solution de branche supérieure développe plusieurs stries avec un espacement de l’envergure compatible avec des mouvements à grande échelle en régime turbulent. / My thesis work focused on ‘dynamical systems’ understanding of the large-scale dynamics in fully developed turbulent shear flow. In plane Couette flow, large-eddy simulation (L.E.S) is used to model small scale motions and to only resolve large-scale motions in order to compute nonlinear traveling waves (NTW) and relative periodic orbits (RPO). Artificial over-damping has been used to quench an increasing range of small-scale motions and prove that the motions in large-scale are self-sustained. The lower-branch traveling wave solutions that lie on laminar-turbulent basin boundary are obtained for these over-damped simulation and further continued in parameter space to upper branch solutions. This approach would not have been possible if, as conjectured in some previous investigations, large-scale motions in wall bounded shear flows are forced by mechanism based on the existence of active structures at smaller scales. In Poseuille flow, relative periodic orbits with shift-reflection symmetry on the laminar-turbulent basin boundary are computed using DNS. We show that the found RPO are connected to the pair of traveling wave (TW) solution via global bifurcation (saddle-node-infinite period bifurcation). The lower branch of this TW solution evolve into a spanwise localized state when the spanwise domain is increased. The upper branch solution develops multiple streaks with spanwise spacing consistent with large-scale motions in turbulent regime.

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-10T12:10:27Z (GMT). No. of bitstreams: 1 Roberto_LuciAnyFrancisco_D.pdf: 627926 bytes, checksum: 0c8cb4e26df805282fa716847859d82f (MD5) Previous issue date: 2008 / Resumo: Neste trabalho estudamos sistemas dinâmicos possuindo estruturas Hamiltonianas e reversíveis( / Abstract: In this work we study dynamical systems possessing Hamiltonian and time-reversible structures. The reversibility concept is de¯ned in terms of an involution. Initially we discuss the dynamics of Hamiltonian vector ¯elds with 2 and 3 degrees of freedom around an elliptic equilibrium in the presence of an involution which preserves the symplectic structure. The main results discuss the existence of one-parameter families of reversible periodic solutions terminating at the equilibrium. The main techniques that are used in the proofs are Belitskii and Birkho® normal forms and the Liapunov-Schmidt Reduction. Next we consider a case of the 3-body restricted problem in rotating coordinates. In this case the two primaries are oving in an elliptic collision orbit. By the continuation method of Poincare we characterize that the periodic circular orbits and the symmetric periodic elliptic orbits from the Kepler problem which can be prolonged to pseudo periodic orbits of the planar restricted 3{body problem in rotating coordinates with the two primaries moving in an elliptic collision orbit / Doutorado / Topologia e Geometria / Doutor em Matemática

Orientador: Marco Antonio Teixeira / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T21:55:10Z (GMT). No. of bitstreams: 1 Lima_MauricioFirminoSilva_D.pdf: 1170094 bytes, checksum: 090e81187787a7a96591621e58ae7742 (MD5) Previous issue date: 2006 / Resumo: Neste trabalho tratamos de famílias a um-parâmetro de campos vetoriais R-reversíveis definidos em uma vizinhança de um ponto de equilíbrio ressonante em R2n. Focalizamos a atenção às 0:p:q-ressonâncias. Inicialmente estudamos a existência/bifurcação de órbitas periódicas simétricas para tais sistemas. A existência e rigidez de famílias de órbitas homoclínicas também são discutidas. Além disso, também analisamos, para n = 3, a rigidez de famílias de Cantor¿ de dois-toros invariantes por meio da Teoria KAM / Abstract: In this work we deal with one parameter families of R-reversible vector fields defined around a resonant equilibrium point in R2n. We focus our attention to 0:p:q resonances. First of all we study the existence/bifurcation of symmetric periodic orbits for such systems. The existence and rigidity of families of homoclinic orbits are also discussed. We also analyze for n = 3 the rigidity of ¿Cantor families¿ of invariant two-torus by means of KAM Theory / Doutorado / Sistemas Dinamicos / Doutor em Matemática

Estudamos o comportamento dinâmico de séries temporais experimentais obtidas de um circuito de Chua quando dois parâmetros de controle, $Delta R_1$ e $Delta R_2$, são variados.Investigamos os comportamentos caótico e periódico, analisando as séries temporais ao redor e no interior de duas janelas periódicas presentes no espaço de parâmetros $(Delta R_1,Delta R_2)$ do circuito. Na vizinhança da janela de período três, analisamos como a dinâmica simbólica se altera quando construída em diferentes seções de Poincaré de um mesmo atrator, e investigamos a dimensão dos mapas de retorno, uni ou bidimensional, para diferentes atratores caóticos presentes nessa região do espaço de parâmetros. Ainda nessa vizinhança, empregamos técnicas de caracterização topológica para confirmar a existência de fibras caóticas, que são curvas de codimensão um no espaço de parâmetros onde as propriedades caóticas dos atratores são preservadas.Ao redor da janela de período quatro, investigamos a transição entre os três comportamentos caóticos para os quais construímos os respectivos moldes topológicos. Propusemos também um molde topológico para o regime caótico após a crise por fusão ocorrer no circuito. Finalizando, investigamos as bifurcações e a estrutura topológica das órbitas periódicas que formam as janelas de período três e de período quatro, construindo um espaço de parâmetros topológico, baseado em um mapa bi-modal, para descrever as duas janela periódicas. / We have studied the dynamical behavior of experimental time series obtained from a Chua's circuit by variation of two parameter control, $Delta R_1$ and $Delta R_2$. We investigated the chaotic and periodic behaviors of the circuit, analyzing temporal series around and inside of two periodic windows in the two-parameter space $(Delta R_1,Delta R_2)$. In the period-three window neighborhood, we analyzed how the symbolic dynamics changes when it is built by different Poincaré sections of an attractor, and we studied the dimension of return map, one- or two-dimensional, for many chaotic attractors in this region of the parameter space. In this neighborhood, we also applied topological techniques to confirm the existence of chaotic fibers: codimension one curves where the chaotic properties of the attractors remain unchanged in the two-parameter space.Around the period-four window, we investigated, by template analysis, the transition between three chaotic attractors found in the Chua's circuit. We proposed a template for chaotic regime of the circuit after merge-crisis. Finally, we investigated the bifurcations and topological structure of periodic orbits in period-three and period-four windows and also proposed a topological parameter space, based in a bimodal map model, that describe these two periodic windows.

<div>Over the past twenty years, ridesharing opportunities for smallsats, i.e., secondary payloads, has increased with the introduction of Evolved Expendable Launch Vehicle (EELV) Secondary Payload Adapter (ESPA) rings. However, the orbits available for these secondary payloads is limited to Low Earth Orbits (LEO) or Geostationary Orbits (GEO). By incorporating a propulsion system, propulsive ESPA rings offer the capability to transport a secondary payload, or a collection of payloads, to regions beyond GEO. In this investigation, the ridesharing scenario includes a secondary payload in a dropped-off Geosynchronous Transfer Orbit (GTO) and the region of interest is the vicinity near the Sun-Earth Lagrange points. However, mission design for secondary payloads faces certain challenges. A significant mission constraint for a secondary payload is the drop-off orbit orientation, as it is dependent on the primary mission. To address this mission constraint, strategies leveraging dynamical structures within the Circular Restricted Three-Body Problem (CRTBP) are implemented to construct efficient and flexible transfers from GTO to orbits near Sun-Earth Lagrange points. First, single-maneuver ballistic transfers are constructed from a range of GTO departure orientations. The ballistic transfer utilize trajectories within the stable manifold structure associated with periodic and quasi-periodic orbits near the Sun-Earth L1 and L2 points. Numerical differential corrections and continuation methods are leveraged to create families of ballistic transfers. A collection of direct ballistic transfers are generated that correspond to a region of GTO departure locations. Additional communications constraints, based on the Solar Exclusion Zone and the Earth’s penumbra shadow region, are included in the catalog of ballistic transfers. An integral-type path condition is derived and included throughout the differential corrections process to maintain transfers outside the required communications restrictions. The ballistic transfers computed in the CRTBP are easily transitioned to the higher-fidelity ephemeris model and validated, i.e., their geometries persist in the ephemeris model. To construct transfers to specific orbits near Sun-Earth L1 or L2, families of two-maneuver transfers are generated over a range of GTO departure locations. The two-maneuver transfers consist of a maneuver at the GTO departure location and a Deep Space Maneuver (DSM) along the trajectory. Families of two-maneuver transfers are created via a multiple- shooting differential corrections method and a continuation process. The generated families of transfers aid in the rapid generation of initial guesses for optimized transfer solutions over a range of GTO departure locations. Optimized multiple-maneuver transfers into halo and Lissajous orbits near Sun-Earth L1 and L2 are included in this analysis in both the CRTBP model and the higher-fidelity ephemeris model. Furthermore, the two-maneuver transfer strategy employed in this analysis are easily extended to other Three-Body systems. </div>