Orientador: Lee Luan Ling / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-10T06:59:04Z (GMT). No. of bitstreams: 1 Perlingeiro_FirmianoRamos_D.pdf: 3410071 bytes, checksum: c0484605d794231363118e1130a5e764 (MD5) Previous issue date: 2006 / Resumo: O provisionamento de QoS garantida é de extrema importância no desenvolvimento das futuras redes. Os recentes avanços tecnológicos em comutação e em transmissão permitem a implementação de redes com velocidades extremamente altas que podem transportar grandes quantidades de tráfego geradas por aplicações mais sensíveis aos requisitos de qualidade de serviço. A próxima geração de redes deverá suportar novas aplicações multimídia em um ambiente global e disponibilizar novos serviços em plataformas flexíveis sem necessidade de alteração da infra-estrutura. Isto requer uma nova arquitetura de redes capaz de oferecer serviços de transporte e de processamento para aplicações de comunicação com fortes requisitos de QoS. No amplo escopo da engenharia de tráfego de redes e do provimento de serviços com qualidade assegurada, esta tese se dedica a propor algumas soluções para os problemas de alocação de recursos de rede, em especial soluções para a estimação da banda efetiva. Para tanto, se utiliza de forma intensiva a caracterização de tráfego, métodos analíticos, heurísticos e de simulação. Os métodos propostos de alocação de banda neste estudo estão fundamentados na Teoria dos Grandes Desvios, aproximação Gaussiana e de caracterização de tráfego. Em termos de caracterização de tráfego, além de vários parâmetros já adotados na literatura é abordada a teoria fractal, incluindo mono e multifractais em seus diferentes aspectos, e ainda, é introduzido um novo parâmetro de tráfego que inclui as características mono e multifractal. Adicionalmente são consideradas as restrições de atraso e jitter, através de adoção de critérios para validação da estimação da banda efetiva, para tráfego em tempo real. A validação da metodologia proposta neste trabalho foi efetivada através de exaustivos testes de simulação com arquivos de tráfego real / Abstract: The assured QoS provisioning has great importance in the development of future networks. Recently, the technological advances in transmission and switching has allowed the implementation of very high speed networks which can transport a huge amount of traffic generated by QoS sensitive applications. The next generation networks must support new multimedia applications in a global environment and deliver new services over flexible platforms without the need of change in the infrastructure. That means that the new network architecture has to be able to transport and process information with strong QoS requirements. Under the wide scope of teletraffic engineering and assured quality of service provisioning, this thesis proposes solutions for some open problems of network resource allocation, especially bandwidth allocation. In order to get reliable solutions, we use intensive traffic characterization, analytical and heuristical methods and simulations. The proposed bandwidth allocation methods in this study are based on the Large Deviation Theory, Gaussian Approximation and traffic characterization. In terms of traffic characterization, in addition to the well known traffic parameters, the fractal theory, including mono and multifractals, are considered. Besides, we introduce a new traffic parameter that takes the mono and multifractal characteristics into account. The proposed bandwidth estimation approaches were tested with real real time traffic under both delay and jitter criteria. All proposed methodologies in this work have been validated by exhaustive simulation tests with real traffic traces / Doutorado / Telecomunicações e Telemática / Doutor em Engenharia Elétrica

Orientador: Lee Luan Ling / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de Computação / Made available in DSpace on 2018-08-06T02:50:26Z (GMT). No. of bitstreams: 1 Jorge_Christian_M.pdf: 1384136 bytes, checksum: 21169a9d54dc981f5cfe38cdbaf733e5 (MD5) Previous issue date: 2006 / Resumo: Nas redes de comunicações, a atual integração de vários tipos de serviços, cada qual com características estatísticas e requisitos de qualidade de serviço distintos, traz consigo a necessidade de esquemas eficientes de gerenciamento e controle de congestionamento do tráfego presente. Em pequenas escalas de tempo, os esquemas atuais podem ter sua eficiência reduzida devido à alta irregularidade do tráfego. Desta forma, neste presente trabalho, tendo como base à disciplina de escalonamento Generalized Processor Sharing (GPS), propõe-se um esquema de escalonamento de fluxos de dados que utiliza o expoente de Hölder pontual para caracterização local de cada fluxo. Para isso, propõe-se conjuntamente um estimador dinâmico destes expoentes e um preditor. Os expoentes de Hölder pontuais são estimados dinamicamente por meio do decaimento dos coeficientes wavelets em janelas de tempo. O preditor proposto possui características adaptativas e baseia-se no filtro de Kalman e no filtro de Mínimos Médios Quadrados Normalizado (Normalized Least-Mean-Square - NLMS). As avaliações realizadas mostram que este esquema de escalonamento contribui para o controle dinâmico preventivo no sentido de se obter uma menor perda de dados e um melhor uso da taxa de transmissão do enlace, em comparação com o GPS convencional / Abstract: Today network traffic is composed of many services with different statistical characteristics and quality of service requirements. This integration needs efficient traffic congestion control and management schemes. Dynamic and preventive schemes usually anticipate traffic conditions by means of a prediction process. Nevertheless, at fine-grained time scales, traffic exhibits strong irregularities and more complex scaling law that make this prediction process a non-trivial task. In this work we model network traffic flows as multifractal processes and introduce the pointwise Hölder exponent as an indicator of the local regularity degree. Also we propose a new traffic flow scheduling scheme based on the Generalized Processor Sharing (GPS) discipline that incorporate the pointwise Hölder exponent to locally characterize each data flow. For this end we explicitly present both dynamic pointwise Hölder exponent estimation and prediction mechanisms. The pointwise Hölder estimation is carried out dynamically based on the decay of the wavelet coefficients in the selected time windows. The proposed predictor is adaptive and implemented with both Kalman and Normalized Least Mean Squares (NLMS) filters. Experimental evaluations have validated the proposed scheduling scheme, resulting in low data loss rate and a better sharing of the network resources in comparison with the usual GPS scheme / Mestrado / Telecomunicações e Telemática / Mestre em Engenharia Elétrica

Orientador: Nelson Luis Sadanha da Fonseca / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-06T04:44:40Z (GMT). No. of bitstreams: 1 Ostrowsky_LarissaOliveira_M.pdf: 1550430 bytes, checksum: 823c0f45a69f06a55a90d4abbffa7e24 (MD5) Previous issue date: 2005 / Resumo: Uma característica importante do tráfego gerado pelo protocolo Internet Protocol (IP) é a existência de padrões scaling, que impactam significantemente o desempenho dos mecanismos de controle de tráfego e que, por isto vem sendo foco de atenção de diversas pesquisas. A natureza scaling do tráfego IP tem sido alvo de bastante polêmica. Em pequenas escalas de tempo o tráfego IP é altamente variável e a variabilidade difere da natureza fractal encontrada em grandes escalas de tempo, não existindo ainda um consenso em relação a natureza do tráfego IP nestas pequenas escalas de tempo. No presente estudo, foram revisadas as evidências da multifractalidade nas pequenas escalas de tempo, através da análise experimental de diversos traços de tráfego real. Constatou-se que não se pode generalizar a natureza dos fluxos do protocolo IP e do protocolo Transmission Control Protocol (TCP) como monofractal ou como multifractal, enquanto que a natureza do fluxo do protocolo User Datagram Protocol (UDP) é sempre multifractal. O crescente uso do protocolo UDP pelas emergentes aplicações que necessitam requisitos de tempo real altera consideravelmente a natureza scaling do tráfego IP, dado que este tipo de tráfego n¿ao reage a situações de congestionamento. Apesar de existirem diversos modelos para tráfego TCP, pouca atenção tem sido dada a modelagem de tráfego UDP. Outra contribuição desta dissertação é a proposta de um modelo de tráfego para fluxos UDP, desenvolvido a partir da caracterização das distribuições estatísticas de traços reais. Este modelo consegue reproduzir com precisão várias características do tráfego UDP real, inclusive a natureza scaling. O modelo foi validado via simulação e a precisão dos resultados do modelo para avaliar um sistema de filas foi comparada às precisões do Modelo Wavelet Multifractal (MWM) e do Modelo MAP Multifractal. Resultados indicam que o modelo proposto reproduz melhor a ocupação em uma fila para diferentes capacidades de armazenamento e taxas de serviço do que os outros modelos avaliados. A geração de traços sintéticos a partir do modelo adotado pode ser realizada em pequenos intervalos de tempo. Assim, o modelo proposto é adequado para o dimensionamento de recursos da rede e provisionamento de Qualidade de serviço / Abstract: An important characteristic of the traffic generated by IP protocol is the existence of scaling, since it has great impact on the performance of traffic control mechanism. Therefore, it has been the focus of attention in many researches. The scaling nature of IP traffic has generated lots of controversy. At small time scales the traffic is very irregular and the variability is different from that found in fractal nature at larges scales. There is no general agreement in relation to the nature of the traffic at these small time scales. In the present study, the evidences of multifractality at small time scales were revised via experimental analyses of several real traffic traces. It was concluded that is not possible to generalize that the nature of IP and TCP flows is either monofractal or multifractal, while the nature of the UDP flows is always multifractal. The increasing use of the UDP protocol by real time applications is changing substantially the scaling nature of IP traffic, since this type of traffic does not react to congestion situations. Although there are models for TCP traffic, not much attention has been given to the modeling of UDP flows. Another contribution of this dissertation is a proposition of UDP traffic model, developed from the characterization of the statistics distributions of real traces. This model can reproduce with precision several characteristics of UDP real traffic, including the scaling nature. The model was validated via simulation and the precision of results of the model was compared with the precisions of the MWM and MAP models. Results indicate that the proposed model better reproduces the queue occupation for different storage capacities and service rates then the others evaluated models. The generation of synthetic traces using the adopted model can be realized with low execution times. Thus, the proposed model is acceptable to the dimension of network resources measurement and to provide Quality of Service / Mestrado / Redes e Multimidia / Mestre em Ciência da Computação

Mis en évidence dans les années 80 dans les domaines de la turbulence et des attracteurs étranges, les multifractals ont rapidement gagné en popularité. On les trouve aujourd'hui en finance, en géophysique, dans l'étude du trafic internet et dans bien d'autres domaines des sciences appliquées. Cet essor s'est accompagné de la nécessité de construire des modèles théoriques adaptés. La Mesure Aléatoire Multifractale de Bacry et Muzy est l'un de ces modèles. Du fait de son caractère très général, de sa grande souplesse et de sa relative simplicité, elle est devenue un outil central du domaine des multifractals depuis dix ans. Après un chapitre introductif, on propose dans cette thèse la construction de deux familles de processus multifractals. Ces constructions reposent sur les travaux de Schmitt et de ses co-auteurs et sur ceux de Bacry et Muzy. Dans le chapitre 2, on construit des processus multifractals à partir de moyennes mobiles alpha-stables, tandis que le chapitre 3 est consacré à la construction des Marches Aléatoires Fractionnaires Multifractales d'indice de Hurst 0<H<1/2. Ces travaux sont complétés par l'étude de versions affines par morceaux et par des simulations numériques. De nombreux problèmes connexes sont également étudiés. / Since their emergence in the 80's in the areas of turbulence and of strange attractors, multifractals have gained popularity. They appear now in finance, geophysics, study of network traffic and in many other areas of applied sciences. This development required adapted theoretical models. Bacry and Muzy's Multifractal Random Measure is one of these models. Thanks to its generality, its flexibility and to its relative simplicity, it became central in the domain of multifractals over the past ten years.In this PhD thesis, two families of multifractal processes are proposed. Their construction is based on the works of Schmitt and co-authors and of those of Bacry and Muzy. After the introduction (chapter 1), we use in chapter 2 alpha-stable moving averages to build multifractal processes; whereas chapter 3 is devoted to the construction of Multifractal Fractional Random Walks with Hurst index 0<H<1/2. This work is complemented by the study of linear versions and by numerical simulations. We study also numerous related problems.

No description available.

The problem of establishing the correct approach to complexity is a very hot and crucial issue to which this dissertation gives some contributions. This dissertation considers two main possibilities, one, advocated by Tsallis and co-workers, setting the foundation of complexity on a generalized, non-extensive , form of thermodynamics, and another, proposed by the UNT Center for Nonlinear Science, on complexity as a new condition that, for physical systems, would be equivalent to a state of matter intermediate between dynamics and thermodynamics. In the first part of this dissertation, the concept of Kolmogorov-Sinai entropy is introduced. The Pesin theorem is generalized in the formalism of Tsallis non-extensive thermodynamics. This generalized form of Pesin theorem is used in the study of two major classes of problems, whose prototypes are given by the Manneville and the logistic map respectively. The results of these studies convince us that the approach to complexity must be made along lines different from those of the non-extensive thermodynamics. We have been convinced that the Lévy walk can be used as a prototype model of complexity, as a condition of balance between order and randomness that yields new phenomena such as aging, and multifractality. We reach the conclusions that these properties must be studied within a dynamic rather than thermodynamic perspective. The second part focuses on the study of the heart beating problem using a dynamic model, the so-called memory beyond memory, based on the Lévy walker model. It is proved that the memory beyond memory effect is more obvious in the healthy heart beating sequence. The concepts of fractal, multifractal, wavelet transformation and wavelet transform maximum modulus (WTMM) method are introduced. Artificial time sequences are generated by the memory beyond memory model to mimic the heart beating sequence. Using WTMM method, the multifratal singular spectrums of the sequences are calculated. It is clear that the sequence with strong memory beyond memory effect has broader singular spectrum.2003-08

Mis en évidence dans les années 80 dans les domaines de la turbulence et des attracteurs étranges, les multifractals ont rapidement gagné en popularité. On les trouve aujourd'hui en finance, en géophysique, dans l'étude du trafic internet et dans bien d'autres domaines des sciences appliquées. Cet essor s'est accompagné de la nécessité de construire des modèles théoriques adaptés. La Mesure Aléatoire Multifractale de Bacry et Muzy est l'un de ces modèles. Du fait de son caractère très général, de sa grande souplesse et de sa relative simplicité, elle est devenue un outil central du domaine des multifractals depuis dix ans. Après un chapitre introductif, on propose dans cette thèse la construction de deux familles de processus multifractals. Ces constructions reposent sur les travaux de Schmitt et de ses co-auteurs et sur ceux de Bacry et Muzy. Dans le chapitre 2, on construit des processus multifractals à partir de moyennes mobiles alpha-stables, tandis que le chapitre 3 est consacré à la construction des Marches Aléatoires Fractionnaires Multifractales d'indice de Hurst 0

Made available in DSpace on 2016-12-08T15:56:17Z (GMT). No. of bitstreams: 1 Eduardo do Nascimento.pdf: 17308021 bytes, checksum: 2ed27e1acd450db70b3f3070fc023b66 (MD5) Previous issue date: 2015-02-23 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work were studied nanodielectric epoxy coatings DGEBA/OTBG reinforced with alumina anoparticles of 10nm and zinc oxide nanopaticles of 90nm. The nanocomposites were processed by twin-screw extrusion with filler fractions of 1%phr, 3%phr, 6%phr and 15%phr. The particle dispersion was investigated with the FIB/FESEM method. Then, the images collected were treated for quantitative analysis of the dispersion. Two methods were used to quantify the dispersion. First, the index for nearest neighbor distance NND showed that nanocomposites reinforced with alumina have a greater state of agglomeration NND=0.45 compared to ZnO-nanocomposites reinforced with NND=0.65. This occured due to the higher surface area of the nanoparticles alumina. Second, a multifractal analysis of the particles-agglomerated distribution was performed. In this ase, the thesis presented suggests that the interphase percolation threshold should occur when the fractal imension of the agglomerations is similar to fractal dimension of the individual particles distribution. From the critical percolation threshold estimated by the dielectric behavior was possible to calculate the interphase thickness, correlating the state of dispersion and the dielectric breakdown of the nanocomposites. Was found a nterphase with 61nm for nanocomposites reinforced with ZnO and 12nm for nanocomposites reinforced with alumina. The dielectric spectra showed a quasi-DC conductivity behavior in nanocomposites with lower filler fraction due to the trapping of additional water molecules in the electrical diffuse double-layer in interfaces. As the filler loading is increased, the interfacial polarization is less pronunciated because the larger overlapping of interphase. When the percolation threshold is reached in nanocomposites with higher fraction of particles, the material exhibits conductivity-DC. The dielectric behavior corroborates the measured value of the dielectric breakdown. The nanocomposites reinforced with 1%phr of ZnO and alumina are increased by 50% and 34% the dielectric breakdown as compared to DGEBA/OTBG pristine matrix. Whereas in nanocomposites reinforced with 15%phr at interphase percolation threshold, the variation in dielectric breakdown was deleterious. It is shown that the addition of nanoparticles does not cause change in free volume of the polymer matrix, thus the free volume changes did not corroborate the nanodielectrics performance. The calorimetric aspects of the curing reaction and mechanical properties were also investigated. It was observed that the zinc oxide catalyzes the crosslinking reaction by decreasing the activation energy of 65kJ/mol in the neat epoxy matrix to 53kJ/mol in the nanocomposite. Thus, its mechanical behavior is unlike modified due to variation in crosslink density of the matrix. While the use of nanoparticles of Al2O3 leads to mechanical toughness of nanocomposite, the addition of ZnO nanoparticles leads to embrittlement, as seen through cracks induced by scratch test. The glass transition temperature and hardness were increased to 95°C and 177MPa the neat epoxy matrix to 104°C and 198MPa in the ZnO-nanocomposite, showing a direct relationship with the effect of adding the nanoparticles in the crosslinking reaction. / Neste trabalho estudou-se revestimentos epoxídicos nanodielétricos DGEBA/OTBG reforçado com nanopartículas de alumina 10nm e oxido de zinco 90nm. Os nanocompósitos foram processados por extrusão dupla-rosca com frações de reforços de 1%phr, 3%phr, 6%phr e 15%phr. A dispersão das partículas foi investigada com o método FIB/FESEM. Então, as imagens coletadas foram tratadas para as analises quantitativas da dispersão. Dois métodos foram empregados na quantificação da dispersão. Primeiro, o índice para o vizinho mais próximo NND mostrou que os nanocompósitos reforçados com alumina apresentam um maior estado de aglomeração NND=0,45 em comparação aos nanocompósitos reforçados com ZnO NND=0,65. Isto ocorreu devido a superior área superficial das nanopartículas de alumina. Segundo, foi realizada uma analise multifractal da distribuição de partículas-aglomerados. Neste caso, a tese apresentada propõe que o limiar de percolação da interfase deve ocorrer quando a dimensão fractal das aglomerações for da mesma ordem da dimensão fractal da distribuição de partículas individuais. A partir do limiar critico de percolação estimado pelo comportamento dielétrico foi possível calcular a espessura da interfase correlacionando o estado de dispersão e a rigidez dielétrica dos nanocompósitos. Encontrou-se uma interfase com 61nm para os nanocompósitos reforçados com ZnO e com 12nm para os nanocompósitos reforçados com alumina. Os espectros dielétricos mostram um comportamento de condutividade quasi-DC nos nanocompósitos com pequena fração de reforços devido ao aprisionamento das adicionais moléculas de aguas nas duplascamadas elétricas nas interfaces. A medida que a fração e aumentada ocorrer uma polarização interfacial menos intensa dada a maior sobreposição das regiões de interfase. Quando o limiar de percolação e atingido nos nanocompósitos com grande fração de partículas, o material apresenta condutividade-DC. O comportamento dielétrico corrobora com o valor medido da rigidez dielétrica. Os nanocompósitos reforçados com 1%phr de alumina e ZnO aumentaram respectivamente em 50% e 34% o valor da rigidez dielétrica em comparação com a matriz DGEBA/OTBG não reforçada. Ao passo que, nos nanocompósitos com a interfase percolada reforçados com 15%phr a variação da rigidez dielétrica foi deletéria. Mostrou-se que a adição de nanopartículas não causa variação no volume livre da matriz polimérica, portanto, alterações do volume livre não corroboram para o desempenho de nanodielétricos. Os aspectos calorimétricos da reação de cura e as propriedades mecânicas também foram investigados. Observou-se que, o oxido de zinco catalisa a reação de reticulação diminuindo a energia de ativação de 65kJ/mol na matriz epoxídica não reforçada para 53kJ/mol no nanocompósito. Assim, o seu comportamento mecânico e diferentemente alterado devido a variação na densidade de reticulação da matriz. Enquanto, a utilização de nanopartículas de Al2O3 leva a tenacificação mecânica do nanocompósito, a adição de ZnO leva a fragilização, como verificado através de trincas induzidas com teste por risco realizado com um nanoindentador. A dureza e a temperatura de transição vítrea foram aumentadas de 177MPa e 95°C na matriz não reforçada para 198MPa e 104°C nos nanocompósitos, mostrando direta relação com o efeito da adição das nanopartículas na reação de reticulação.

Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems. In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time $t$, infinite sample size $S$, and infinite resolution (infinitesimal box-size $varepsilon$). Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension $D_mathrm{obs}(S,t,varepsilon)$. We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which $D_mathrm{obs}(S,t,varepsilon)$ is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on $S$, $t$, and $varepsilon$ in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields. In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension $D_mathrm{eff}$ on $t$ and $varepsilon$. We numerically explore this in weakly chaotic leaky systems. There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum maps. / Es ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet. In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit $t$, unendlicher Stichprobengröße $S$ und unendlicher Auflösung, also infinitesimaler Boxgröße $varepsilon$ auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden. Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension $D_mathrm{obs}(S,t,varepsilon)$ durch und zeigen, dass die drei Parameter $S$, $t$ und $varepsilon$ in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen $D_mathrm{obs}(S,t,varepsilon)$ den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von $S$, $t$ und $varepsilon$ anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht. Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an: Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen $varepsilon$-Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen. Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von $D_mathrm{eff}$ von $t$ und $varepsilon$ auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist. Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben.

Fractals have long been recognized to be a characteristic feature arising from chaotic dynamics; be it in the form of strange attractors, of fractal boundaries around basins of attraction, or of fractal and multifractal distributions of asymptotic measures in open systems. In this thesis we study fractal and multifractal measure distributions in leaky Hamiltonian systems. Leaky systems are created by introducing a fully or partially transparent hole in an otherwise closed system, allowing trajectories to escape or lose some of their intensity. This dynamics results in intricate (multi)fractal distributions of the surviving trajectories. These systems are suitable models for experimental setups such as optical microcavities or microwave resonators. In this thesis we perform an improved investigation of the fractality in these systems using the concept of effective dimensions. They are defined as the dimensions far from the usually considered asymptotics of infinite evolution time $t$, infinite sample size $S$, and infinite resolution (infinitesimal box-size $varepsilon$). Yet, as we show, effective dimensions can be considered as intrinsic to the dynamics of the system. We present a detailed discussion of the behaviour of the numerically observed dimension $D_mathrm{obs}(S,t,varepsilon)$. We show that the three parameters can be expressed in terms of limiting length scales that define the parameter ranges in which $D_mathrm{obs}(S,t,varepsilon)$ is an effective dimension of the system. We provide dynamical and statistical arguments for the dependence of these scales on $S$, $t$, and $varepsilon$ in strongly chaotic systems and show that the knowledge of the scales allows us to define meaningful effective dimensions. We apply our results to three main fields. In the context of numerical algorithms to calculate dimensions, we show that our findings help to numerically find the range of box sizes leading to accurate results. We further show that they allow us to minimize the computational cost by providing estimates of the required sample-size and iteration time needed. A second application field of our results is systems exhibiting non-trivial dependencies of the effective dimension $D_mathrm{eff}$ on $t$ and $varepsilon$. We numerically explore this in weakly chaotic leaky systems. There, our findings provide insight into the dynamics of the systems, since deviations from our predictions based on strongly chaotic systems at a given parameter range are a sign that the stickiness inherent to such systems needs to be taken into account in that range. Lastly, we show that in quantum analogues of chaotic maps with a partial leak, a related effective dimension can be used to explain the numerically observed deviation from the predictions provided by the fractal Weyl law for systems with fully absorbing leaks. Here, we provide an analytical description of the expected scaling based on the classical dynamics of the system and compare it with numerical results obtained in the studied quantum maps. / Es ist seit langem bekannt, dass Fraktale eine charakteristische Begleiterscheinung chaotischer Dynamik sind. Sie treten in Form von seltsamen Attraktoren, von fraktalen Begrenzungen der Einzugsbereiche von Attraktoren oder von fraktalen und multifraktalen Verteilungen asymptotischer Maße in offenen Systemen auf. In dieser Arbeit betrachten wir fraktal und multifraktal verteilte Maße in geöffneten hamiltonschen Systemen. Geöffnete Systeme werden dadurch erzeugt, dass man ein völlig oder teilweise transparentes Loch im Phasenraum definiert, durch das Trajektorien entkommen können oder in dem sie einen Teil ihrer Intensität verlieren. Die Dynamik in solchen Systemen erzeugt komplexe (multi)fraktale Verteilungen der verbleibenden Trajektorien, beziehungsweise ihrer Intensitäten. Diese Systeme sind zur Modellierung experimenteller Aufbauten, wie zum Beispiel optischer Mikrokavitäten oder Mikrowellenresonatoren, geeignet. In dieser Arbeit führen wir eine verbesserte Untersuchung der Fraktalität in derartigen Systemen durch, die auf dem Konzept der effektiven Dimensionen beruht. Diese sind als die Dimensionen definiert, die weit weg von den üblicherweise betrachteten Limites unendlicher Iterationszeit $t$, unendlicher Stichprobengröße $S$ und unendlicher Auflösung, also infinitesimaler Boxgröße $varepsilon$ auftreten. Dennoch können effektive Dimensionen, wie wir zeigen, als der Dynamik des Systems inhärent angesehen werden. Wir führen eine detaillierte Diskussion der numerisch beobachteten Dimension $D_mathrm{obs}(S,t,varepsilon)$ durch und zeigen, dass die drei Parameter $S$, $t$ und $varepsilon$ in Form grenzwertiger Längenskalen ausgedrückt werden können, die die Parameterbereiche definieren, in denen $D_mathrm{obs}(S,t,varepsilon)$ den Wert einer effektiven Dimension des Systems annimmt. Wir beschreiben das Verhalten dieser Längenskalen in stark chaotischen Systemen als Funktionen von $S$, $t$ und $varepsilon$ anhand statistischer Überlegungen und anhand von auf der Dynamik basierenden Aussagen. Weiterhin zeigen wir, dass das Wissen um diese Längenskalen die Definition aussagekräftiger effektiver Dimensionen ermöglicht. Wir wenden unsere Ergebnisse hauptsächlich in drei Bereichen an: Im Kontext numerischer Algorithmen zur Dimensionsberechnung zeigen wir, dass unsere Ergebnisse es erlauben, diejenigen $varepsilon$-Bereiche zu finden, die zu korrekten Ergebnissen führen. Weiterhin zeigen wir, dass sie es uns erlauben, den Rechenaufwand zu minimieren, indem sie uns eine Abschätzung der benötigten Stichprobengröße und Iterationszeit ermöglichen. Ein zweiter Anwendungsbereich sind Systeme, die sich durch eine nichttriviale Abhängigkeit von $D_mathrm{eff}$ von $t$ und $varepsilon$ auszeichnen. Hier ermöglichen unsere Ergebnisse ein besseres Verständnis der Systeme, da Abweichungen von den Vorhersagen basierend auf der Annahme von starker Chaotizität ein Anzeichen dafür sind, dass im entsprechenden Parameterbereich die Eigenschaft dieser Systeme, dass Bereiche in ihrem Phasenraum Trajektorien für eine begrenzte Zeit einfangen können, relevant ist. Zuletzt zeigen wir, dass in quantenmechanischen Analoga chaotischer Abbildungen mit partiellen Öffnungen eine verwandte effektive Dimension genutzt werden kann, um die numerisch beobachteten Abweichungen vom fraktalen weyl'schen Gesetz für völlig transparente Öffnungen zu erklären. In diesem Zusammenhang zeigen wir eine analytische Beschreibung des erwarteten Skalierungsverhaltens auf, die auf der klassischen Dynamik des Systems basiert, und vergleichen sie mit numerischen Erkenntnissen, die wir über die Quantenabbildungen gewonnen haben.