Time varying flow structures are involved in a large percentage of fluid flows although there is still much unknown regarding their behavior. With the development of high spatiotemporal resolution measurement systems it is becoming more feasible to measure these complex flow structures, which in turn will lead to a better understanding of their impact. One method that has been developed for studying these flow structures is finite time Lyapunov exponents (FTLEs). These exponents can reveal regions in the fluid, referred to as Lagragnian coherent structures (LCSs), where fluid elements diverge or attract. Better knowledge of how these time varying structures behave can greatly impact a wide range of applications, from aircraft design and performance, to an improved understanding of mixing and transport in the human body. This work provides the development of new methodologies for measuring and studying three-dimensional time varying structures. Provided herein is a method to improve replacement of erroneous measurements in particle image velocimetry data, which leads to increased accuracy in the data. Also, a method for directly measuring the finite time Lyapunov exponents from particle images is developed, as well as an experimental demonstration in a three-dimensional flow field. This method takes advantage of the information inherently contained in these images to improve accuracy and reduce computational requirements. Lastly, this work provides an in depth look at the flow field for developing wall jets across a wide range of Reynolds numbers investigating the mechanisms that contribute to their development. / Ph. D.

Many chemical reactions can be described as the crossing of an energetic barrier. This process is mediated by an invariant object in phase space. One can construct a normally hyperbolic invariant manifold (NHIM) of the reactive dynamical system which is an invariant sphere that can be considered as the geometric representation of the transition state itself. The NHIM has invariant cylinders (reaction channels) attached to it. This invariant geometric structure survives as long as the invariant sphere is normally hyperbolic. We applied this theory to the hydrogen exchange reaction in three degrees of freedom in order to figure out the reason of the transition state theory (TST) failure. Energies high above the reaction threshold, the dynamics within the transition state becomes partially chaotic. We have found that the invariant sphere first ceases to be normally hyperbolic at fairly low energies. Surprisingly normal hyperbolicity is then restored and the invariant sphere remains normally hyperbolic even at very high energies. This observation shows two different energy values for the breakdown of the TST and the breakdown of the NHIM. This leads to seek another phase space object that is related to the breakdown of the TST. Using theory of the dividing surface including reactive islands (RIs), we can investigate such an object. We found out that the first nonreactive trajectory has been found at the same energy values for both collinear and full systems, and coincides with the first bifurcation of periodic orbit dividing surface (PODS) at the collinear configuration. The bifurcation creates the unstable periodic orbit (UPO). Indeed, the new PODS (UPO) is the reason for the TST failure. The manifolds (stable and centre-stable) of the UPO clarify these expectations by intersecting the dividing surface at the boundary of the reactive island (on the collinear and the three (full) systems, respectively).

Friction compensation is challenging but imperative for control engineers. For high-performance engineering systems, a friction-model-based controller is typically required to accommodate the nonlinearities arisen from the friction model employed. It is well known that the parameters of the friction model used in the friction compensation are nearly impossible to be accurately identified. Therefore, the objective of this research is to study the effect of these parametric uncertainties on the stability of a set-point position control system. With the above goal in mind, a variety of aspects are investigated in this work. Firstly, several common friction features and friction models are surveyed to provide background knowledge which helps select the friction model with the highest accuracy for our study. Secondly, an experimental setup is proposed and fabricated to validate the levels of accuracy given by the candidate friction models. The comparisons between the numerical and experimental results confirm that the LuGre friction model is the best approximation of the observed friction behaviours among all models selected. Moreover, a series of profound discussions addressing the relation between the candidate models’ structures and their numerical friction feature predictions are provided and followed by a summary table that recapitulates the properties of the candidate friction models. Last but not least, the state space models of the proposed setup formulated by the improved version of the LuGre model and the two controllers of interest, namely input-output linearization controller and nominal characteristic trajectory following (NCTF) controller, are derived for the stability analysis under the parametric uncertainties. Two parameters in the friction model used, σ_0 and σ_1, are perturbed for the stability analysis in which the results applying the concept of Lyapunov exponents (LEs) clearly illustrate the significant effect of the varying σ_0 and σ_1 values on the system stability. The effect of parametric uncertainties can depend quite significantly on the incorporated controller, and the stability results obtained here are applicable to the design and analysis of other systems that are inherently similar to our setup. The stability analysis conducted is this work is recommended for other control systems to avoid unwanted qualitative behaviours under parametric perturbations. / October 2016

Neste trabalho nós estudamos os expoentes de Lyapunov de aplicações f : Td → Td homotópicas a uma aplicação Anosov linear e a continuidade absoluta de folheações. Nós mostramos para algumas classes de homotopia de aplicações que a soma dos expoentes de Lyapunov está limitado pela soma dos expoentes de Lyapunov da aplicação Anosov linear. Além disso, admitindo uma propriedade conhecida como densidade uniformemente limitada (UBD) nas folheações, mostramos uma igualdade entre a soma dos expoentes de Lyapunov de f e do Anosov linear. Também construímos um conjunto C1 aberto de difeomorfismos parcialmente hiperbólicos do toro T4, preservando volume, com folheação central bidimensional não compacta e não absolutamente contínua. Ainda construímos um exemplo parcialmente hiperbólico com folhas centrais bidimensionais, não compactas onde a desintegração do volume ao longo da folheação central não é nem Lebesgue nem atômica. / In this work we study the Lyapunov exponents of maps f : Td → Td homotopic to a linear Anosov map. We proof for some homotopic classes of maps which the sum of Lyapunov exponents is bounded by the sum of the Lyapunov exponents of the linear Anosov map. Moreover, by assuming a property known as uniformly bounded density (UBD) in the foliations, we show an equality between the sum of the Lyapunov exponents of f and the linear Anosov. We also construct an C1 open set of volume preserving partially hyperbolic diffeomorphisms with non compact two dimensional center foliation and non absolutely continuous. We still build an example of partially hyperbolic diffeomorphism with non compact bidimensional center leaves where the disintegration of volume along the center foliation is neither Lebesgue nor atomic.

Nas últimas décadas, diversos sistemas de comunicação baseados em caos foram propostos. Dentre eles, vários utilizam uma função para codificar uma mensagem em um sinal caótico, que é caracterizado como um sinal de banda larga. Dado que o canal de transmissão é limitado em banda por natureza, é necessário determinar e controlar o espectro do sinal caótico transmitido por esse sistema. Nesse sentido, um sistema de comunicação em banda limitada, baseado em sincronismo caótico, foi proposto recentemente utilizando-se filtros digitais para controlar a largura de banda dos sinais transmitidos. Esses filtros, inseridos no sistema de comunicação, modificam o sistema original gerador do sinal caótico, tornando-se necessário analisar como essa inserção afeta o sincronismo caótico. Nessa tese, apresenta-se uma análise desse sistema de comunicação digital de tempo discreto, baseado em sincronismo caótico, considerando-se um canal com ruído aditivo branco gaussiano. As condições necessárias para a sincronização desse sistema são obtidas analiticamente, por meio de um teorema, para um mapa gerador de caos qualquer. O desempenho desse sistema é avaliado em termos da taxa de erro de bit, e, para melhorar seu desempenho, propõe-se filtrar o ruído fora da banda do sinal na entrada do receptor. Apesar das condições de sincronismo terem sido determinadas, a inserção dos filtros também pode modificar a natureza caótica dos sinais, e não há garantia que os sinais transmitidos sejam caóticos. Para analisar a natureza caótica dos sinais transmitidos pelo sistema de comunicação, o maior expoente de Lyapunov é obtido numericamente em função dos coeficientes dos filtros, dos parâmetros do mapa e da função de codificação da mensagem. / In recent decades, several chaos-based communication systems have been proposed. Many of them use a function to encode a message into a chaotic signal, which is characterized as wideband. Since every transmission channel is bandlimited in nature, it is necessary to determine and to control the spectrum of the chaotic signal transmitted by this system. This way, a bandlimited chaos-based communication system was recently proposed using digital filters and chaotic synchronization. These filters, inserted in the communication system, modify the original chaotic generator system, becoming necessary to study how their insertion affect chaotic synchronization. In this work, we present an analysis of this discrete-time chaos-based digital communication system considering an additive white Gaussian noise channel. The synchronization conditions of this system is analytically obtained, through a theorem, for a generic chaos generator map. The system performance is evaluated in terms of bit error rate, and, to obtain a performance improvement, it is also proposed to filter the out-of-band noise in the receiver. Although the conditions for chaotic synchronization have been determined, the filters insertion can also modify the chaotic nature of the signals, and there is no guarantee that the transmitted signals remain chaotic. To analyze the chaotic nature of the communication system transmitted signals, the largest Lyapunov exponent is numerically accessed as a function of the filters coefficients, the parameters of the map and the message coding function.

Nas últimas décadas, diversos sistemas de comunicação baseados em caos foram propostos. Dentre eles, vários utilizam uma função para codificar uma mensagem em um sinal caótico, que é caracterizado como um sinal de banda larga. Dado que o canal de transmissão é limitado em banda por natureza, é necessário determinar e controlar o espectro do sinal caótico transmitido por esse sistema. Nesse sentido, um sistema de comunicação em banda limitada, baseado em sincronismo caótico, foi proposto recentemente utilizando-se filtros digitais para controlar a largura de banda dos sinais transmitidos. Esses filtros, inseridos no sistema de comunicação, modificam o sistema original gerador do sinal caótico, tornando-se necessário analisar como essa inserção afeta o sincronismo caótico. Nessa tese, apresenta-se uma análise desse sistema de comunicação digital de tempo discreto, baseado em sincronismo caótico, considerando-se um canal com ruído aditivo branco gaussiano. As condições necessárias para a sincronização desse sistema são obtidas analiticamente, por meio de um teorema, para um mapa gerador de caos qualquer. O desempenho desse sistema é avaliado em termos da taxa de erro de bit, e, para melhorar seu desempenho, propõe-se filtrar o ruído fora da banda do sinal na entrada do receptor. Apesar das condições de sincronismo terem sido determinadas, a inserção dos filtros também pode modificar a natureza caótica dos sinais, e não há garantia que os sinais transmitidos sejam caóticos. Para analisar a natureza caótica dos sinais transmitidos pelo sistema de comunicação, o maior expoente de Lyapunov é obtido numericamente em função dos coeficientes dos filtros, dos parâmetros do mapa e da função de codificação da mensagem. / In recent decades, several chaos-based communication systems have been proposed. Many of them use a function to encode a message into a chaotic signal, which is characterized as wideband. Since every transmission channel is bandlimited in nature, it is necessary to determine and to control the spectrum of the chaotic signal transmitted by this system. This way, a bandlimited chaos-based communication system was recently proposed using digital filters and chaotic synchronization. These filters, inserted in the communication system, modify the original chaotic generator system, becoming necessary to study how their insertion affect chaotic synchronization. In this work, we present an analysis of this discrete-time chaos-based digital communication system considering an additive white Gaussian noise channel. The synchronization conditions of this system is analytically obtained, through a theorem, for a generic chaos generator map. The system performance is evaluated in terms of bit error rate, and, to obtain a performance improvement, it is also proposed to filter the out-of-band noise in the receiver. Although the conditions for chaotic synchronization have been determined, the filters insertion can also modify the chaotic nature of the signals, and there is no guarantee that the transmitted signals remain chaotic. To analyze the chaotic nature of the communication system transmitted signals, the largest Lyapunov exponent is numerically accessed as a function of the filters coefficients, the parameters of the map and the message coding function.

Made available in DSpace on 2017-07-10T17:11:52Z (GMT). No. of bitstreams: 1 Texto completo - Rosiney.pdf: 7631119 bytes, checksum: 43c0461bb49060121b74d945a88d53d4 (MD5) Previous issue date: 2012-11-26 / The present work proposes a study of the dynamics of a gyroscope using simulated data of an analytical model by comparing with experimental data. Classical mechanical modeling approaches are used to identify the equilibrium points, stability and verification of the regions where the motion equations of the gyroscope can present regular or chaotic behavior. The Lyapunov exponents are identified through the standard method, Eckmann-Ruelle Method, Wolf method with time series and the 0-1 test. The results achieved illustrate the main advantages and drawbacks of each method and allow to observe qualitatively and quantitatively information about the motion of the gyroscope used. / Este trabalho propõe um estudo da dinâmica de um giroscópio usando dados de simulação de um modelo analítico comparando com dados experimentais. Verifica-se a modelagem usando mecânica clássica, estudo de pontos de equilíbrio, estabilidade e verificação de regiões onde o movimento do giroscópio pode ficar regular ou caótico. Os expoentes de Lyapunov são identificados usando o método padrão, método de Eckmann-Ruelle, método deWolf com séries temporais e o teste 0-1. Os resultados alcançados nesta dissertação permitiram comparar as principais vantagens e desvantagens de cada um dos métodos e extrair informações qualitativas e quantitativas sobre o movimento do giroscópio em estudo.

Estudos sobre o comportamento dinâmico de estruturas não lineares são até os dias de hoje motivo de extensas pesquisas em todo o mundo. Desde o início do desenvolvimento da teoria das oscilações não lineares buscou-se compreender os mecanismos básicos, como perturbações que provocassem respostas complexas nas estruturas flexíveis. Este trabalho apresenta um estudo teórico e experimental do comportamento dinâmico de uma semi-asa de um avião acoplada a uma turbina com a hélice desbalanceada, esse sistema é representado através de uma viga engastada excitada por uma fonte não ideal localizada na extremidade oposta ao engaste. Entende-se como sistema não ideal aquele que considera que a excitação é influenciada pela própria resposta do sistema. Para sistemas dinâmicos não ideais, deve-se adicionar uma equação que descreva como a fonte não ideal interage com o sistema. Considera-se na equação do sistema apenas o primeiro modo de vibrar. Os resultados de simulação numérica apresentados são obtidos utilizando o software Matlab® 8.0 e o parâmetro de controle a ser analisado é o torque do motor. Os resultados que mostram o comportamento dinâmico do sistema são o histórico no tempo, plano de fase, FFT e para identificar o comportamento caótico calculam-se os expoentes de Lyapunov. O gráfico que mostra a presença do efeito Sommerfeld (salto) no sistema também é apresentado. Na parte experimental, apresenta-se todo o procedimento experimental, assim como os resultados: Histórico no tempo, plano de fase reconstruído, FFT, expoentes de Lyapunov e as análises que ilustram a presença do efeito Sommerfeld no experimento. / Studies about the dynamic behaviour of nonlinear structures have been to this date subject of extensive research all around the world. Since the beginning of the development of the nonlinear oscillation theory one has tried to understand the basic mechanisms, like disruptions that would cause complex answers on flexible structures. This paper presents a theoretical and practical study of the dynamic behaviour of a semi-wing of an airplane installed on a turbine with unbalanced propellers; this system is represented through a cantilever beam excited by a non-ideal source located at the end opposite to the coupling. As a non-ideal system we mean the one that considers that the excitement is influenced by the system\'s response itself. For non-ideal dynamic systems, one must add an equation that describes how the non-ideal source interacts with the system. Only the first vibrating mode is considered in the system\'s equation. The numeric simulation results shown are obtained by using the Matlab® 8.0 software and the control parameter to be analyzed is the motor torque. The results that show the dynamic behaviour of the system are time history, phase plan, FFT and to identify the chaotic behaviour the Lyapunov\'s indexes are calculated. The graphic that shows the presence of the Sommerfeld effect (jump) in the system is also presented. In the experimental part, all the practical procedure is presented, as well as experimental results, like, for example: Time history, phase plan reconstruction, FFT, Lyapunov exponents and the analyses that illustrate the presence of the Sommerfeld effect on the experiment.

The complexity of the interface between the muscular system and the nervous system is still elusive. We investigated how the neuromuscular system functions and how it is influenced by various perturbations. Postural stability was selected as the model system, because this system provides complex output, which could indicate underlying mechanisms and feedback loops of the neuromuscular system. We hypothesized that aging, physical pain, and mental and physical perturbations affect balancing strategy, and based on these observations, we constructed a model that simulates many aspects of the neuromuscular system. Our results show that aging changes the control strategy of balancing from more chaotic to more repetitive. The chaotic elements ensure quick reactions and strong capacity to compensate for the perturbations; this adeptly reactive state changes into a less reactive, slower, probably less mechanically costly balancing strategy. Mental tasks during balancing also decreased the chaotic elements in balancing strategy, especially if the subject experienced chronic pain. Additional motoric tasks, such as tying knots while balancing, were correlated with age but unaffected by chronic pain. Our model competently predicted the experimental findings, and we proceeded to use the model with an external data set from Physionet to predict the balancing strategy of Parkinson’s patients. Our neurological model, comprised of RLC circuits, provides a mechanistic explanation for the neuromuscular system adaptations.

Estudos sobre o comportamento dinâmico de estruturas não lineares são até os dias de hoje motivo de extensas pesquisas em todo o mundo. Desde o início do desenvolvimento da teoria das oscilações não lineares buscou-se compreender os mecanismos básicos, como perturbações que provocassem respostas complexas nas estruturas flexíveis. Este trabalho apresenta um estudo teórico e experimental do comportamento dinâmico de uma semi-asa de um avião acoplada a uma turbina com a hélice desbalanceada, esse sistema é representado através de uma viga engastada excitada por uma fonte não ideal localizada na extremidade oposta ao engaste. Entende-se como sistema não ideal aquele que considera que a excitação é influenciada pela própria resposta do sistema. Para sistemas dinâmicos não ideais, deve-se adicionar uma equação que descreva como a fonte não ideal interage com o sistema. Considera-se na equação do sistema apenas o primeiro modo de vibrar. Os resultados de simulação numérica apresentados são obtidos utilizando o software Matlab® 8.0 e o parâmetro de controle a ser analisado é o torque do motor. Os resultados que mostram o comportamento dinâmico do sistema são o histórico no tempo, plano de fase, FFT e para identificar o comportamento caótico calculam-se os expoentes de Lyapunov. O gráfico que mostra a presença do efeito Sommerfeld (salto) no sistema também é apresentado. Na parte experimental, apresenta-se todo o procedimento experimental, assim como os resultados: Histórico no tempo, plano de fase reconstruído, FFT, expoentes de Lyapunov e as análises que ilustram a presença do efeito Sommerfeld no experimento. / Studies about the dynamic behaviour of nonlinear structures have been to this date subject of extensive research all around the world. Since the beginning of the development of the nonlinear oscillation theory one has tried to understand the basic mechanisms, like disruptions that would cause complex answers on flexible structures. This paper presents a theoretical and practical study of the dynamic behaviour of a semi-wing of an airplane installed on a turbine with unbalanced propellers; this system is represented through a cantilever beam excited by a non-ideal source located at the end opposite to the coupling. As a non-ideal system we mean the one that considers that the excitement is influenced by the system\'s response itself. For non-ideal dynamic systems, one must add an equation that describes how the non-ideal source interacts with the system. Only the first vibrating mode is considered in the system\'s equation. The numeric simulation results shown are obtained by using the Matlab® 8.0 software and the control parameter to be analyzed is the motor torque. The results that show the dynamic behaviour of the system are time history, phase plan, FFT and to identify the chaotic behaviour the Lyapunov\'s indexes are calculated. The graphic that shows the presence of the Sommerfeld effect (jump) in the system is also presented. In the experimental part, all the practical procedure is presented, as well as experimental results, like, for example: Time history, phase plan reconstruction, FFT, Lyapunov exponents and the analyses that illustrate the presence of the Sommerfeld effect on the experiment.