Submitted by GABRIEL DIAZ ITURRY null (gabriel.diaz.iturry@gmail.com) on 2017-05-04T14:59:10Z No. of bitstreams: 1 dissertacao_GDI.pdf: 7646479 bytes, checksum: 41982d1bb134afa5cd32f6db91fda5e9 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-05-05T14:41:22Z (GMT) No. of bitstreams: 1 diaziturry_g_me_rcla.pdf: 7646479 bytes, checksum: 41982d1bb134afa5cd32f6db91fda5e9 (MD5) / Made available in DSpace on 2017-05-05T14:41:22Z (GMT). No. of bitstreams: 1 diaziturry_g_me_rcla.pdf: 7646479 bytes, checksum: 41982d1bb134afa5cd32f6db91fda5e9 (MD5) Previous issue date: 2017-02-20 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Estudamos nessa dissertação algumas propriedades estatísticas no regime de equilibrio pós transitório para o modelo bouncer unidimensional considerando ambas versões completa e simplificada. O modelo consiste de uma partícula clássica movendo-se sob ação de uma força gravitacional constante e sofrendo colisões com uma plataforma móvel de massa muito maior que a massa da partícula. A versão completa leva em conta o movimento real da fronteira e o instante da colisão entre partícula e plataforma é obtido a partir da solução numérica de uma equação transcendental. Já o modelo simplificado, também conhecido como modelo de aproximação de fronteira fixa, assume que para o cálculo do instante da colisão a fronteira está parada, porém a partícula troca energia após a colisão ocorre como se a fonteira estivesse em movimento. Os comportamentos da velocidade média, velocidade quadrática média e desvio da velocidade quadrática média foram obtidos em função dos parâmetros de controle. Desenvolvemos um método semi-analítico permitindo-nos deduzir equações dos valores médios sem fazer simulações de larga escala. Em seguida, elaboramos uma simulação do tipo Monte-Carlo que nos permite obter os valores médios no estado estacionário sem resolver equações transcendentais, acelerando assim as simulações numéricas. O método de Monte-Carlo apresentado pode ser útil na investigação de sistemas mais complexos incluindo bilhares clássicos dependentes do tempo. / We studied some statistical properties in the stationary and post transitory state for the one-dimensional bouncer model considering wither complete and simplified versions. The model consists of a classical particle moving under the effect of a constant gravitational force and collides with a periodic moving platform whose mass is heavier as compared to the particle. The complete version takes into account the real motion of the moving wall. The instant of collision is obtained from the numerical solution of a transcendental equation. The simplified version, also called as a static wall approximation, takes into account to calculate the instant of the collisions as if the wall was fixed. However, the particle experiences an exchange of energy and momentum at the collision as if the wall were moving. The behavior for the average velocity, average squared velocity and deviation of the average squared velocity were obtained as a function of the control parameters. We developed a semi-analytic method allowing us to deduce equations for the average values without the need of doing large scale simulations. Using a Monte-Carlo-like simulation we obtained the average values for the stationary state without solving the transcendental equations. The Monte-Carlo method may have applications in the investigation of more complex systems including time dependent billiard systems.

Η παρούσα διατριβή έχει ως θέμα της το κβαντικό χάος σε μπιλλιάρδα. Ειδικότερα, εισάγεται ένας μετασχηματισμός (Μετασχηματισμός Εφελκυσμού), ο οποίος προβάλλει το σύνορο ενός μπιλλιάρδου πάνω στον μοναδιαίο κύκλο. Αυτό εισάγει μια μη-Ευκλείδια μετρική στο επίπεδο και έναν διαφορικό τελεστή, ο οποίος περιέχει όλη την πληροφορία σχετικά με το σχήμα του συνόρου και τις ιδιότητες, ως προς την ολοκληρωσιμότητα ή μη, του μπιλλιάρδου. Κλασικά οι ευθείες γραμμές της ελεύθερης κίνησης αντιστοιχούν σε γεωδαισιακές, και κβαντομηχανικά το ενεργειακό φάσμα είναι αυτό του τελεστή Laplace-Beltrami με Dirichlet συνοριακές συνθήκες στον μοναδιαίο κύκλο. Οι γεωδαισιακές εξισώσεις είναι μη-γραμμικές, ομως στο διάστημα μεταξύ δύο διαδοχικών σκεδάσεων υπάρχουν δύο ολοκληρώματα κίνησης, αυτό της κινητικής ενέργειας και αυτό της στροφορμής, οπότε είναι δυνατή η λύση τους. Οι λύσεις αυτές μπορούν να χρησιμοποιηθούν στο κλασικό πρόβλημα σκέδασης. Κβαντικά παίρνουμε το φάσμα των μπιλλιάρδων: Έλλειψη, στάδιο, Robnik και τετράγωνο, για διάφορες τιμές μιας παραμέτρου διαταραχής. Το φάσμα υπολογίζεται διαταρακτικά για μικρές τιμές της παραμέτρου διαταραχής και με διαγωνοποίηση για πιο μεγάλες τιμές της. Η μέθοδος αυτή μπορεί να εφαρμοστεί σε οποιοδήποτε σχήμα συνόρου μπιλλιάρδου, αρκεί ο μετασχηματισμός να είναι αντιστρέψιμος, και μπορεί να χρησιμοποιηθεί σαν ένας γρήγορος τρόπος προσδιορισμού του φάσματος καθώς και σαν ένα θεωρητικό εργαλείο για την ανάλυση θεμελιακών ιδιοτήτων της ολοκληρωσιμότητας, του χάους και της ενδιάμεσης αυτών περιοχής, μέσω του τελεστή Laplace-Beltrami. Σαν ένδειξη των δυνατοτήτων της μεθόδου παραθέτουμε ένα γραφικό τεστ, όπου για πολύ μικρές αποκλίσεις από τον μοναδιαίο κύκλο ένα ολοκληρώσιμο και δύο εν-δυνάμει χαοτικά μπιλλιάρδα διακρίνονται καθαρά μεταξύ τους από τις κατανομές των διαφορών της διόρθωσης πρώτης τάξης στην ενέργεια. Το τεστ αυτό εμφανίζεται για πρώτη φορά στη βιβλιογραφία και έρχεται να συμπληρώσει την γνωστή κατανομή αποστάσεων εγγυτάτων γειτόνων, η οποία για τόσο μικρές αποκλίσεις από το κυκλικό μπιλλιάρδο δεν καταφέρνει να διαχωρίσει τα ολοκληρώσιμα από τα μη-ολοκληρώσιμα σχήματα. Τέλος εισάγεται η έννοια του ανοικτού μπιλλιάρδου, στο οποίο θεωρείται ότι το σύνορο βρίσκεται στο άπειρο. Τα ανοικτά μπιλλιάρδα αν και είναι ολοκληρώσιμα, περιέχουν εντούτοις την πληροφορία για την ολοκληρωσιμότητα ή μη των αντιστοίχων κλειστών σχημάτων. Για την εξαγωγή της τελευταίας πληροφορίας χρησιμοποιούνται διάφοροι μέθοδοι όπως συναρτήσεις αυτο- και ετερο- συσχέτισης. / For a billiard of a general shape a transformation is introduced (Stretching Transformation) which projects the boundary on the unit circle. This introduces a non-Euclidean metric on the plane, which contains all relevant information of the shape of the boundary. Classically the straight lines of the free motion correspond to geodesics and quantum mechanically the energy spectrum is that of Laplace-Beltrami operator with Dirichlet boundary conditions on the unit circle. The geodesic equations are highly non-linear. Nevertheless for the interval between two consecutive scatterings we have two integrals of motion, the kinetic energy and the angular momentum. This fact helps to solve explicitly the geodesic equations. These solutions can be used to derive interesting properties for the classical scattering. Quantum mechanically the spectrum of the above billiards is obtained for certain parameter values both perturbatively for small values of the parameter and also using a diagonalisation procedure. This method is applicable to any particular form of a billiard for which the transformation is invertible and can be used on one hand as a quick method of approximate spectral determination and as a theoretical tool to analyze specific properties of integrability and chaos through the associated connection form and the Laplace-Beltrami operator. As aν indication of the potentiality of this method we present a graphical test where for very small deviations from the circular billiard an integrable and two non-integrable billiards can be distinguished by the distribution of the differences of the first order corrections while this distinction is not evident by the usual test for the nearest neighbor level spacing. Furthermore the open billiard concept is being introduced. An open billiard is one whose boundary is assumed to be at infinity, thus being classified as an integrable billiard, which contains nevertheless the information about potential non-integrability within. Various methods for the extraction of this hidden information are being investigated.

No description available.

国立情報学研究所で電子化したコンテンツを使用している。

Thesis (Ph. D.)--University of Alabama at Birmingham, 2007. / Additional advisors: Nikolai Chernov, S. S. Ravindran, Alan Sprague, Min Sun. Description based on contents viewed Feb. 6, 2008; title from title screen. Includes bibliographical references.

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This Bachelor's thesis deals with fractals and orbits on Sierpinski carpets. We present the fundamental theory regarding fractals and some illustrative examples together with fractal billiards. In the latter part of the thesis we use elementary methods to present an original proof concerning the closure of some billiard orbits on Sierpinski carpets. A survey of the article Periodic Billiard orbits of self-similar Sierpinski Carpets, see [8], has been done, in which we make a discussion about one open question regarding reflections on the carpet. Furthermore, we state and prove some propositions related to this open question.

Neste trabalho, mostramos que os bilhares hiperbólicos construídos originalmente por Bussolari- Lenci têm a propriedade de Bernoulli. Tais bilhares não satisfazem as técnicas standard de Wojtkowski-Markarian-Donnay-Bunimovich para bilhares focalizadores hiperbólicos, a qual requer que o diâmetro da mesa do bilhar seja de mesma ordem que o maior raio de curvatura ao longo da componente focalizadora. Nossa prova, utiliza um teorema ergódico local que nos diz que sob certas condições, existe um conjunto de medida total do espaço de fase do bilhar tal que cada ponto desse conjunto possui uma vizinhança contida (mod 0) em uma componente Bernoulli da aplicação do bilhar. / In this work, we show that hyperbolic billiards constructed originally by Bussolari-Lenci has the Bernoulli property. These billiards do not satisfy the standard Wojtkowski-Markarian-Donnay- Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane, which requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. Our proof employs a locally ergodic theorem which says that under a few conditions, there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained, up to a zero measure set, in one Bernoulli component of the billiard map.

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Um bilhar consiste basicamente de uma partÃcula confinada em uma regiÃo do espaÃo. Trataremos apenas de bilhares em duas dimensÃes na ausÃncia de campos externos e desprezaremos qualquer tipo de forÃas dissipativas, de modo que as colisÃes da partÃcula com as fronteiras do bilhar sÃo elÃsticas. AlÃm disso, as fronteiras sÃo fixas, ou seja, respeitam uma equaÃÃo do tipo $R = R(r, heta)$, onde r e $ heta$ sÃo as coordenadas polares planas. O bilhar à um modelo interessante por vÃrios motivos. Primeiro, à um sistema muito simples (tem poucos graus de liberdade) e de fÃcil visualizaÃÃo. No entanto, possui uma dinÃmica nÃo-trivial com grande riqueza de comportamentos (podendo apresentar comportamento regular, caÃtico ou atà mesmo misto, caso em que coexistem no espaÃo de fase de um Ãnico bilhar regiÃes caÃticas e regulares). Segundo, o tratamento numÃrico desses sistemas nÃo requer integraÃÃo numÃrica de equaÃÃes diferenciais e, portanto, nÃo consume muito tempo de execuÃÃo. AlÃm disso, os bilhares permitem que realizemos investigaÃÃes de carÃter fundamental, por exemplo, podemos estudar como sistemas regulares reagem ao serem levemente perturbados. Especificamente, iremos aplicar uma rugosidade na fronteira do bilhar circular e elÃptico e observar como o espaÃo de fase irà mudar ao sofrer tal perturbaÃÃo. / In this work we are going to study a physical system known as billiard. A billiard is defined to be basically a confined particle in a closed region of the space. We are going to deal with only two-dimensionals billiards in the absence of extern fields and to neglect any kind of dissipative forces, in a way that the colisions of the particle with the boundary are elastics. Beyond that, the boundary are fixed, it means they respect an equation of kind $R(r, heta)$, where $r$ and $ heta$ are the polar coordinates on a plan. A billiard is a very interesting model by several reasons. First, it is a simple system (it has a few degree of freedom) and it is of easy visualization. However, it has a non-trivial dynamics with a big richness of behaviors (from a billiard it could appear regular behavior, chaotic behavior, or even a mixed behavior, where coexist in the phase space of one billiard chaotics and regular regions). Second, the numerical approach of these systems does not require numerical integration of diferential equations and, therefore, does not take too much time of execution. Furthermore, the billiards allow us to perform investigations of fundamental nature, for example, we can study how regular systems react by being slightly disturbed. Especificaly, we perform a rugosity perturbation on the billiard surface and observe how the phase space is going to change.

Neste trabalho, mostramos que os bilhares hiperbólicos construídos originalmente por Bussolari- Lenci têm a propriedade de Bernoulli. Tais bilhares não satisfazem as técnicas standard de Wojtkowski-Markarian-Donnay-Bunimovich para bilhares focalizadores hiperbólicos, a qual requer que o diâmetro da mesa do bilhar seja de mesma ordem que o maior raio de curvatura ao longo da componente focalizadora. Nossa prova, utiliza um teorema ergódico local que nos diz que sob certas condições, existe um conjunto de medida total do espaço de fase do bilhar tal que cada ponto desse conjunto possui uma vizinhança contida (mod 0) em uma componente Bernoulli da aplicação do bilhar. / In this work, we show that hyperbolic billiards constructed originally by Bussolari-Lenci has the Bernoulli property. These billiards do not satisfy the standard Wojtkowski-Markarian-Donnay- Bunimovich technique for the hyperbolicity of focusing or mixed billiards in the plane, which requires the diameter of a billiard table to be of the same order as the largest ray of curvature along the focusing boundary. Our proof employs a locally ergodic theorem which says that under a few conditions, there exists a full measure set of the billiard phase space such that each of its points has a neighborhood contained, up to a zero measure set, in one Bernoulli component of the billiard map.