Forest fires cause a significant amount of damage and destruction each year. Optimally dispatching resources reduces the amount of damage a forest fire can cause. Models predict the fire spread to provide the data required to optimally dispatch resources. However, the models are only as accurate as the data used to build them. Satellites are one valuable tool in the collection of data for the forest fire models. Satellites provide data on the types of vegetation, the wind speed and direction, the soil moisture content, etc. The current operating paradigm is to passively collect data when possible. However, images from directly overhead provide better resolution and are easier to process. Maneuvering a constellation of satellites to fly directly over the forest fire provides higher quality data than is achieved with the current operating paradigm. Before launch, the location of the forest fire is unknown. Therefore, it is impossible to optimize the initial orbits for the satellites. Instead, the expected cost of maneuvering to observe the forest fire determines the optimal initial orbits. A two-stage stochastic programming approach is well suited for this class of problem where initial decisions are made with an uncertain future and then subsequent decisions are made once a scenario is realized. A repeat ground track orbit provides a non-maneuvering, natural solution providing a daily flyover of the forest fire. However, additional maneuvers provide a second daily flyover of the forest fire. The additional maneuvering comes at a significant cost in terms of additional fuel, but provides more data collection opportunities. After data are collected, ground stations receive the data for processing. Optimally selecting the ground station locations reduce the number of built ground stations and reduces the data fusion issues. However, the location of the forest fire alters the optimal ground station sites. A two-stage stochastic programming approach optimizes the selection of ground stations to maximize the expected amount of data downloaded from a satellite. The approaches of selecting initial orbits and ground station locations including uncertainty will provide a robust system to reduce the amount of damage caused by forest fires.

Thesis (PhD (Electrical and Electronic Engineering))--University of Stellenbosch, 1997. / This dissertation considers the feasibility of using atmospheric drag to control constellations of micro-satellites in low Earth orbits. The constellation control requirements include an acquisition phase and a maintenance phase. Optimal strategies are designed to control the relative positions of the satellites during these two phases. It is shown that the feasibility and success of the strategies depend on many factors, including the satellite properties and orbital configuration. A nominal test constellation is presented and used as a generic example for the application of the control strategies. The dissertation also focuses on the accurate modelling and simulation of a typical low Earth orbit satellite, moving under the influence of a variety of significant orbit perturbation forces. The simulations form an integral part of the study and are used to verify the application of all the proposed control strategies.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

[pt] O compartilhamento de frequências entre sistemas de comunicações por satélite e sistemas terrestres tem sido objeto de estudos desde o aparecimento dos primeiros sistemas comerciais de comunicações por satélite. Um caso particular que tem despertado o interesse dos operadores de sistemas terrestres diz respeito à interferência produzida pelos enlaces descendentes de sistemas de satélite que utilizam orbitas elípticas altamente inclinadas (Highly Elliptical Orbit - HEO) nos receptores do Serviço Fixo Terrestre (Fixed Service - FS). Os trabalhos encontrados na literatura, envolvendo este tipo de interferência, apresentam análises que consideram a hipótese simplificadora de que todos os receptores vítima do Serviço Fixo Terrestre têm suas antenas receptoras com elevação zero graus. No presente trabalho é feita uma análise probabilística da interferência agregada produzida por satélites de múltiplos sistemas HEO sobre receptores do Serviço Fixo Terrestre na faixa de 18 GHz, na qual o ângulo de elevação da antena receptora dos sistemas FS considerados é modelado por uma variável aleatória com função densidade de probabilidade conhecida. A modelagem matemática desenvolvida é aplicada a dois cenários envolvendo múltiplos sistemas HEO interferentes. Mais especificamente, um dos cenários considera três sistemas HEO com satélites operando apenas no hemisfério norte e o outro, três sistemas HEO com satélites operando tanto no hemisfério norte quanto no sul. Os resultados obtidos são comparados àqueles que utilizam a hipótese de que as antenas receptoras de todos os sistemas FS considerados têm angulo de elevação zero. / [en] Frequency sharing between satellite systems and terrestrial fixed service (FS) systems has been object of studies since the onset of commercial satellite communication systems. A particular case of interest refers to the interference produced by the downlink of highly elliptical orbit satellite systems (HEO) into fixed service receivers. In the literature, studies involving this type of interference have presented analyzes that consider the simplifying assumption that all victim fixed service receivers have their receiving antennas with zero degree elevation angle. In this work a probabilistic analysis is used to evaluate the aggregate interference produced by the satellites of multiple highly elliptical orbit satellite systems into fixed service receivers operating in the 18 GHz frequency band. The FS receiving antenna elevation angle is modeled as a random variable with known probability density function. The proposed mathematical model is applied to two scenarios involving multiple interfering HEO systems. More specifically, the first scenario considers three interfering HEO systems having satellites that operate only in the northern hemisphere. In the second scenario, three HEO systems with satellites that operate in both the northern and southern hemisphere are considered. The obtained results are compared to those resulting from analyses that use the hypothesis that the receiving antennas of all the FS victim systems have a zero degree elevation angle.

On étudie les résonances semi-excitées pour un Opérateur h-Pseudo-différentiel (h-PDO)H(x, hDx) sur L2(M) induites par une orbite périodique de type hyperbolique à l’énergie E = 0. Par exemple M = Rn et H(x, hDx; h) est l’opérateur de Schrödinger avec effet Stark, ouH(x, hDx; h) est le flot géodesique sur une variété axi-symétrique M, généralisant l’exemplede Poincaré de systèmes Lagrangiens à 2 degrés de liberté. On étend le formalisme de Gérard and Sjöstrand, au sens où on autorise des valeurs propres hyperboliques et elliptiques del’application de Poincaré, et où l’on considère des résonances dont la partie imaginaire est del’ordre de hs, pour 0 < s < 1.On établit une règle de quantification de type Bohr-Sommerfeld au premier ordre en fonction des nombres quantiques longitudinaux (réels) et transverses (complexes), incluantl’intégrale d’action le long de l’orbite, la 1-forme sous-principale, et l’indice de Conley-Zehnder. / In this Thesis we consider semi-excited resonances for a h-Pseudo-Differential Operator (h-PDO for short) H(x, hDx; h) on L2(M) induced by a periodic orbit of hyperbolic type at energy E = 0, as arises when M = Rn and H(x, hDx; h) is Schrödinger operator withAC Stark effect, or H(x, hDx; h) is the geodesic flow on an axially symmetric manifold M,extending Poincaré example of Lagrangian systems with 2 degree of freedom. We generalizethe framework of Gérard and Sjöstrand, in the sense that we allow for hyperbolic and ellipticeigenvalues of Poincaré map, and look for (excited) resonances with imaginary part of magnitude hs, with 0 < s < 1,It is known that these resonances are given by the zeroes of a determinant associatedwith Poincaré map. We make here this result more precise, in providing a first order asymptoticsof Bohr-Sommerfeld quantization rule in terms of the (real) longitudinal and (complex)transverse quantum numbers, including the action integral, the sub-principal 1-form and Gelfand-Lidskii index.

[pt] Seja (Es)t o espaço de germes na origem de funções suaves entre os espaços euclidianos de dimensões e t. Nesta dissertação, apresentamos a parte da Teoria de Mather que descreve hipóteses suficientes para k-determinação em (Es)t sob duas ações diferentes, induzindo as chamadas R- e K-equivalências. Um germe é k-determinado se é equivalente a qualquer perturbação que deixa invariante seu k-jato, os termos de ordem até k de sua expansão de Taylor na origem. A R-equivalência consiste em compor germes com germes de difeomorfismos µa direita. A K- equivalência é mais difícil de descrever. / [en] Let (Es)t be the space of smooth map-germs at the origin between Euclidian spaces of dimensions s and t. In this dissertation, we present a section of Mather theory describing su±cient conditions for k- determinacy of this map-germs under two different actions, inducing the so called R- e K- equivalences. A map-germ is k-determined if it is equivalent to any perturbation that leaves invariant its k-jet, i.e., the terms up to order k of its Taylor expansion at the origin. The R-equivalence consists of compositions with germs of diffeomorphisms to the right. The K- equivalence is harder to describe.

Neste trabalho, apresentamos um método computacional rigoroso para a demonstração de existência de órbitas periódicas de alguns sistemas de equações diferenciais ordinárias com campo autônomo do tipo polinomial. Mostraremos que o problema de encontrar órbitas periódicas para esses sistemas de equações é equivalente a buscar por raízes de certas funções definidas no espaço de Banach das sequências com decaimento algébrico. O método pode ser dividido em duas etapas. Na primeira, buscamos numericamente por soluções periódicas aproximadas. Na segunda, mostraremos a existência de uma órbita periódica numa vizinhança da curva encontrada numericamente. O rigor das verificações computacionais é garantido pelo uso de aritimética intervalar. / In this work, we present a rigorous computational method for proving the existence of periodic orbits of some systems of ordinary differential equations with autonomous vector field of polynomial type. We show that the problem of finding periodic orbits for these systems is equivalent to check for roots of certain functions defined in the Banach space of sequences with algebraic decay. The method can be divided into two steps. First, we seek, numerically, to approximated periodic solutions. Then, we show the existence of a periodic orbit in a neighborhood of the curve numerically found in the previous stage. The accuracy of the computational verifications is guaranteed by the use of interval arithmetic.

Neste trabalho, estudamos a dinâmica de Reeb associada a uma forma de contato $\\lambda$ definida numa 3-variedade compacta e conexa M. Assumimos que $\\lambda$ é tight e a primeira classe de Chern da estrutura de contato $\\xi=\\ker\\lambda$ se anula sobre $\\pi_2(M)$. No nosso primeiro resultado, supomos que M é fechada e existe uma órbita fechada L do fluxo de Reeb que é um p-nó trivial com número de auto-enlaçamento $-1/p$. Supomos, além disso, que o número de rotação transversal da p-ésima iterada de L é estritamente menor do que 1. Nestas condições, provamos que existe uma órbita fechada (de Reeb) contrátil geometricamente distinta de L e não-enlaçada em L cujo número de rotação transversal é 1. Apresentamos também uma versão deste resultado para o caso em que M é uma 3-variedade cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb e não existem órbitas fechadas contidas no bordo. Nosso segundo resultado é uma caracterização dinâmica do toro sólido. Seja $\\lambda$ uma forma de contato não-degenerada definida em uma 3-variedade M cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb. Se o fluxo de Reeb satisfaz certas hipóteses de torção sobre o bordo, então ou existe uma órbita fechada contrátil com índice de Conley-Zehnder 2 ou M é folheada por discos transversais ao campo de Reeb. Neste último caso, M é difeomorfa a um toro sólido e existe uma órbita fechada não-contrátil em M que é ponto fixo da aplicação de retorno induzida pela folheação. / In this work, we study the Reeb dynamics associated to a tight contact form $\\lambda$ defined on a compact, connected 3-manifold M. Suppose that the first Chern class of $\\xi=\\ker\\lambda$ vanish on $\\pi_2(M)$. In our first result, we assume that M is closed and there exists a closed Reeb orbit L which is a p-unknotted, has self-linking number $-1/p$ and the transverse rotation number of the p-th iterate of L is less than 1. Under these conditions, we verify that there exists a contractible closed Reeb orbit which is geometrically distinct from L and not linked to L with transverse rotation number 1. We also prove a version of this result when M is a compact 3-manifold M whose boundary is diffeomorphic to a torus and invariant by the flow and, moreover, there does not exist closed Reeb orbits on the boundary. Our second result is a dynamical characterization of the solid torus. We assume that $\\lambda$ is a contact form on a compact 3-manifold M whose boundary is diffeomorphic to a torus. Under the hypothesis of $\\lambda$ being non-degenerate, if the flow is tangent to $\\partial M$ and satisfies some twist conditions on the boundary, then either there exists a contractible closed Reeb orbit which has Conley-Zehnder index 2 or M is foliated by disks transverse to the Reeb flow. In this last case, we see that M is diffeomorphic to a solid torus and there exists a non-contractible closed Reeb orbit M which is a fixed point of the return map induced by the foliation.

Neste trabalho, estudamos a dinâmica de Reeb associada a uma forma de contato $\\lambda$ definida numa 3-variedade compacta e conexa M. Assumimos que $\\lambda$ é tight e a primeira classe de Chern da estrutura de contato $\\xi=\\ker\\lambda$ se anula sobre $\\pi_2(M)$. No nosso primeiro resultado, supomos que M é fechada e existe uma órbita fechada L do fluxo de Reeb que é um p-nó trivial com número de auto-enlaçamento $-1/p$. Supomos, além disso, que o número de rotação transversal da p-ésima iterada de L é estritamente menor do que 1. Nestas condições, provamos que existe uma órbita fechada (de Reeb) contrátil geometricamente distinta de L e não-enlaçada em L cujo número de rotação transversal é 1. Apresentamos também uma versão deste resultado para o caso em que M é uma 3-variedade cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb e não existem órbitas fechadas contidas no bordo. Nosso segundo resultado é uma caracterização dinâmica do toro sólido. Seja $\\lambda$ uma forma de contato não-degenerada definida em uma 3-variedade M cujo bordo é difeomorfo a um toro e invariante pelo fluxo de Reeb. Se o fluxo de Reeb satisfaz certas hipóteses de torção sobre o bordo, então ou existe uma órbita fechada contrátil com índice de Conley-Zehnder 2 ou M é folheada por discos transversais ao campo de Reeb. Neste último caso, M é difeomorfa a um toro sólido e existe uma órbita fechada não-contrátil em M que é ponto fixo da aplicação de retorno induzida pela folheação. / In this work, we study the Reeb dynamics associated to a tight contact form $\\lambda$ defined on a compact, connected 3-manifold M. Suppose that the first Chern class of $\\xi=\\ker\\lambda$ vanish on $\\pi_2(M)$. In our first result, we assume that M is closed and there exists a closed Reeb orbit L which is a p-unknotted, has self-linking number $-1/p$ and the transverse rotation number of the p-th iterate of L is less than 1. Under these conditions, we verify that there exists a contractible closed Reeb orbit which is geometrically distinct from L and not linked to L with transverse rotation number 1. We also prove a version of this result when M is a compact 3-manifold M whose boundary is diffeomorphic to a torus and invariant by the flow and, moreover, there does not exist closed Reeb orbits on the boundary. Our second result is a dynamical characterization of the solid torus. We assume that $\\lambda$ is a contact form on a compact 3-manifold M whose boundary is diffeomorphic to a torus. Under the hypothesis of $\\lambda$ being non-degenerate, if the flow is tangent to $\\partial M$ and satisfies some twist conditions on the boundary, then either there exists a contractible closed Reeb orbit which has Conley-Zehnder index 2 or M is foliated by disks transverse to the Reeb flow. In this last case, we see that M is diffeomorphic to a solid torus and there exists a non-contractible closed Reeb orbit M which is a fixed point of the return map induced by the foliation.

This thesis presents results based on data collected with the 930 MHz EISCAT UHF radar system and three SKiYMet specular meteor radars. It describes in detail a method for meteoroid orbit calculation. The EISCAT UHF system comprises three identical 32 m parabolic antennae: one high-power transmitter/receiver and two remote receivers. Precise meteoroid deceleration and radar cross section are determined from 410 meteor head echoes simultaneously observed with all three receivers between 2002 and 2005, during four 24h runs at the summer/winter solstice and the vernal/autumnal equinox. The observations are used to calculate meteoroid orbits and estimate meteor visual magnitudes. None of the observed meteors appear to be of extrasolar or asteroidal origin; comets, particularly short period (&lt;200 years) ones, may be the dominant source for the particles observed. About 40% of the radiants are associated with the north apex sporadic meteor source and 58% of the orbits are retrograde. The geocentric velocity distribution is bimodal with a prograde population centred around 38 km/s and a retrograde population peaking at 59 km/s. The absolute visual magnitudes of meteors are estimated to be in the range of +9 to +5 using a single-object numerical ablation model. They are thus observable using intensified CCD cameras with telephoto lenses. The thesis also investigates diurnal meteor rate differences and sporadic meteor radiant distributions at different latitudes using specular meteor trail radar measurements from 68°N, from 55°N and from 8°S. The largest difference in amplitude of the diurnal flux variation is at equatorial latitudes, the lowest variation is found at high latitudes. The largest seasonal variation of the diurnal flux is observed with the high-latitude meteor radar. The investigations show a variation in the sources with both latitude and time of day. The EISCAT UHF system and the high-latitude meteor radar are located close to the Arctic Circle. Such a geographical position means that zenith points towards the North Ecliptic Pole (NEP) once every day all year round. This particular geometry allows the meteoroid influx from the north ecliptic hemisphere to be compared throughout the year as the ecliptic plane coincides with the local horizon. Considering only the hour when NEP is closest to zenith, the EISCAT UHF head echo rate is about a factor of three higher at summer solstice than during the other seasons, a finding which is consistent with the high-latitude meteor radar measurements. / Avhandlingens resultat är baserade på mätningar med den trestatiska EISCAT UHF-radarn och tre SKiYMet meteorradarsystem. En metod för meteoroidbanberäkning presenteras i detalj. EISCAT UHF-systemet består av tre identiska, 32 m stora parabolantenner: en högeffektssändare/mottagare och två fjärrstyrda mottagare. Under fyra 24-timmarsmätningar vid vår-/höstdagjämning och sommar-/vintersolstånd mellan 2002 och 2005 detekterades 410 meteoriska huvudekon simultant med alla tre mottagare. Dessa trestatiska meteorers atmosfärsinbromsning och radartvärsnitt har fastställts mycket noggrant och använts till att beräkna meteoroidernas banor samt uppskatta meteorernas luminositeter. Ingen av de observerade meteoroiderna verkar vara av interstellärt eller asteroidursprung. Deras troligaste ursprung är kometer, framför allt kortperiodskometer (&lt;200 år). Ungefär 40% av meteorradianterna kan associeras till norra apex, ett källområde för sporadiska meteorer, och totalt är 58% av partiklarnas banor retrograda. Meteoroidernas geocentriska hastighetsfördelning har två lokala maxima: ett för den prograda populationen vid 38 km/s och ett för den retrograda vid 59 km/s. Genom att anpassa datat till en numerisk ablationsmodell som simulerar meteoroidernas färd genom atmosfären har de detekterade meteorernas absoluta visuella magnituder uppskattats till mellan +9 och +5. Detta innebär att de är observerbara med bildförstärkta, teleskopiska CCD-kameror. Avhandlingen diskuterar även hur sporadiska meteorers dygns- och säsongsinflöde beror på geografisk latitud och meteorradianternas distribution på himmelssfären. Detta utreds med hjälp av spårekon detekterade under perioden 1999-2004 med tre meteorradarsystem på latituderna 68°N, 55°N och 8°S. Dygnsinflödet varierar mest på låga latituder och minst på höga. Ju högre latitud, desto mer förändras däremot dygnsinflödet över året. Avhandlingen visar att de dominerande källområdena varierar med säsong, över dygnet och med latitud. Både EISCAT UHF-systemet och meteorradarn på 68°N är belägna nära polcirkeln. Detta innebär att norra ekliptiska polen (NEP) är i zenit en gång per dygn, året om. Vid just denna tidpunkt sammanfaller ekliptikan med den lokala horisonten, vilket möjliggör att det observerade meteorinflödet från norra ekliptiska hemisfären kan jämföras över året. Under timmen då NEP är närmast zenit har EISCAT UHF uppmätt ett ungefär tre gånger högre meteorinflöde vid sommarsolståndet än under de andra säsongerna, vilket överensstämmer med resultaten från meteorradarn på 68°N.