Neste trabalho foi realizado um mapeamento de ferramentas analíticas para representação e tratamento computacional de Sistemas Fuzzy. Este mapeamento evidenciou a existência de relações e a adequação do uso da Teoria Fuzzy na construção de modelos para a solução de problemas envolvendo Sistemas Dinâmicos e, em particular, de Sistemas Dinâmicos Caóticos. Propôs-se uma diferenciação entre dois tipos de Sistemas Dinâmicos Fuzzy (SDF\'s) - os Sistemas Dinâmicos Fuzzy Intrínsecos (SDFI\'s) e os Sistemas Dinâmicos Fuzzy Extrínsecos (SDFE\'s). Foram utilizados SDFE\'s na modelagem Fuzzy para a solução de problemas de fixação de trajetórias em sistemas caóticos. Foram desenvolvidos Estudos de Caso que permitiram verificar, por meio de simulações em bilhares e testes em circuito de Chua (este último implementado em protótipo físico) a adequação do uso desta técnica na solução destes problemas. No desenvolvimento da Tese foram obtidas as seguintes realizações relacionadas ao circuito de Chua (CCH): Tratamento computacional sobre circuito real; Utilização de um conversor analógico-digital de oito bits seguido por filtro passa-baixas para compensar a baixa resolução na leitura dos sinais; Utilização de circuito girador para implementar o indutor do CCH; Proposta e realização de um circuito inédito para definir o resistor de controle, via porta paralela do PC, de modo simples e de fácil reprodução. Com relação aos bilhares foram obtidas as seguintes realizações: Definição de um novo tipo de bilhar (Bilhar Newtoniano de Garms & Andrade); Desenvolvimento da análise detalhada das equações de simulação dos bilhares empregados nos Estudos de Caso. Ao rediscutir-se a Lógica Seqüencial Fuzzy definiu-se e desenvolveu-se, por meio da aplicação de realimentação em Circuitos Seqüenciais Fuzzy nos Sistemas Dinâmicos, um Astável Fuzzy (oscilações não-periódicas), o qual exemplifica um SDFI. Finalmente realizaram-se algumas interpretações da Física pela Teoria Fuzzy, utilizando-se o conceito de SDFI\'s. / A mapping of analytical tools for representation and computational treatment of Fuzzy Systems was made in this thesis. This mapping evidenced the existence of relations and the adequacy of the use of the Fuzzy Theory in the models construction for the problems solution involving dynamic systems e, in particular, of chaotic systems. A differentiation was considered about two types of Fuzzy Dynamic Systems - the Intrinsic Fuzzy Dynamic Systems (IFDS) and the Extrinsic Fuzzy Dynamic Systems (EFDS). The EFDS in the Fuzzy modeling is used for the problems solution of trajectories setting in chaotic systems. Case Studies had been developed that allow to verify, by means of simulation in billiards and tests in circuit of Chua (implemented in physical prototype), the adequacy of the use of this strategy in the solution of these problems. Related with Chua circuit, can be mentioned the following achievements: Computational treatment on real circuit; Use of a 8-bits AD converter followed by lowpass filter to compensate this low resolution signals reading; Use of gyrator circuit to implement the inductor used in this circuit; Proposal and accomplishment of a circuit to define the control resistor, via PC parallel port, of simple reproduction. Related with billiards, the following achievements are mentioned: Definition of a new billiard - Garms & Andrade Newtonian Billiard; Development of detailed equations of the simulations billiards analysis. When discussing again the Sequential Logic Fuzzy, is defined and developed, by means of the application of feedback in Sequential Circuits Fuzzy in the Dynamic Systems, an Astable Fuzzy (non-periodic oscillations), which exemplifies an IFDS. Finally, some interpretations of the Physics for the Fuzzy Theory are also presented with the use of the IFDS concept.

Realizamos uma montagem experimental da experiência da torneira gotejante, com desenvolvimento dos sistemas de aquisição de dados e de controle do experimento e os respectivos programas de aquisição e analise de dados. O sistema apresentou uma grande variedade de comportamentos, tais como crises, intermitências, bifurcações (tangentes de Hopf), orbitas semi-periódicas e ciclos-limite. Utilizamos técnicas de dinâmica simbólica com a determinação de orbitas periódicas, e representações na forma de grafos das gramaticas simbólicas associadas a diversos atratores. Foram feitas estimativas e representações gráficas de complexidade de conjunto. Foram comparados os resultados obtidos com diversos tipos de bico de torneira. / We built a dripping faucet experiment. We have developed the data acquisition and control systems, as well as the acquisition and data analysis softwares. The dynamical system showed a great variety of non-linear behavior such as crises, intermittencies, Hopf and tangent bifurcations, quasi-periodic orbits and circle limits. We have employed symbolic dynamics tools for periodic orbits extraction and grammar approximations. The grammars associated to some attractors were represented as directed graphs, with approximations and pictoral representations of their set complexity. The results obtained with different types of neeple faucet were compared.

Hydraulic structures such as dykes and groynes are commonly used to help control river flows and reduce flood risk. The present research aims to develop an idealized model of the hydrodynamics in the vicinity of a large river bend, and the advection and mixing processes where groynes are located. In this study a curvilinear model of shallow water equations is applied to investigate chaotic advection of particles in a river bend similar in dimensions to a typical bend in the River Danube, Hungary. First, a curvilinear grid generator is developed based on Poisson-type elliptic partial differential equations. The grid generator is verified for benchmark tests concerning a circular domain and for distorted grids in a rectangular domain. It is found that multi-grid (MG) and conjugate gradient (CG) methods performed better computationally than successive over-relaxation (SOR) in generating the curvilinear grids. The open channel hydrodynamics are modelled using the shallow water equations (SWEs) derived by depth-averaging the continuity and Navier-Stokes momentum equations. Both Cartesian and curvilinear forms of the shallow water equations are presented. Both sets of equations are discretized spatially using finite differences and the solution marched forward in time using fourth-order Runge-Kutta scheme. The shallow water solvers are verified and validated for uniform flow in the rectangular channel, wind-induced set up in rectangular and circular basins, flow past a sidewall expansion, and Shallow flow in a rectangular channel with single groyne. A Lagrangian particle tracking model is used to predict the trajectories of tracer particles, and bilinear interpolation is used to provide a representation of the continuous flow field from discrete results. The particle tracking model is verified for trajectories in the flow field of a single free vortex and in the alternating flow field of a pair of blinking vortices. Excellent agreement is obtained with analytical solutions, previously published results in the literature. The combined shallow flow and Lagrangian particle tracking model is then used to simulate particle advection in the flow past a side-wall cavity containing a groyne and reasonable agreement is obtained with published experimental and alternative numerical data. Finally, the combined model is applied to simulate the shallow flow hydrodynamics, advection and mixing processes in the vicinity of groynes in river bend, the dimensions representative of a typical bend in the Danube River, Hungary.

Realizamos uma montagem experimental da experiência da torneira gotejante, com desenvolvimento dos sistemas de aquisição de dados e de controle do experimento e os respectivos programas de aquisição e analise de dados. O sistema apresentou uma grande variedade de comportamentos, tais como crises, intermitências, bifurcações (tangentes de Hopf), orbitas semi-periódicas e ciclos-limite. Utilizamos técnicas de dinâmica simbólica com a determinação de orbitas periódicas, e representações na forma de grafos das gramaticas simbólicas associadas a diversos atratores. Foram feitas estimativas e representações gráficas de complexidade de conjunto. Foram comparados os resultados obtidos com diversos tipos de bico de torneira. / We built a dripping faucet experiment. We have developed the data acquisition and control systems, as well as the acquisition and data analysis softwares. The dynamical system showed a great variety of non-linear behavior such as crises, intermittencies, Hopf and tangent bifurcations, quasi-periodic orbits and circle limits. We have employed symbolic dynamics tools for periodic orbits extraction and grammar approximations. The grammars associated to some attractors were represented as directed graphs, with approximations and pictoral representations of their set complexity. The results obtained with different types of neeple faucet were compared.

Neste trabalho foi realizado um mapeamento de ferramentas analíticas para representação e tratamento computacional de Sistemas Fuzzy. Este mapeamento evidenciou a existência de relações e a adequação do uso da Teoria Fuzzy na construção de modelos para a solução de problemas envolvendo Sistemas Dinâmicos e, em particular, de Sistemas Dinâmicos Caóticos. Propôs-se uma diferenciação entre dois tipos de Sistemas Dinâmicos Fuzzy (SDF\'s) - os Sistemas Dinâmicos Fuzzy Intrínsecos (SDFI\'s) e os Sistemas Dinâmicos Fuzzy Extrínsecos (SDFE\'s). Foram utilizados SDFE\'s na modelagem Fuzzy para a solução de problemas de fixação de trajetórias em sistemas caóticos. Foram desenvolvidos Estudos de Caso que permitiram verificar, por meio de simulações em bilhares e testes em circuito de Chua (este último implementado em protótipo físico) a adequação do uso desta técnica na solução destes problemas. No desenvolvimento da Tese foram obtidas as seguintes realizações relacionadas ao circuito de Chua (CCH): Tratamento computacional sobre circuito real; Utilização de um conversor analógico-digital de oito bits seguido por filtro passa-baixas para compensar a baixa resolução na leitura dos sinais; Utilização de circuito girador para implementar o indutor do CCH; Proposta e realização de um circuito inédito para definir o resistor de controle, via porta paralela do PC, de modo simples e de fácil reprodução. Com relação aos bilhares foram obtidas as seguintes realizações: Definição de um novo tipo de bilhar (Bilhar Newtoniano de Garms & Andrade); Desenvolvimento da análise detalhada das equações de simulação dos bilhares empregados nos Estudos de Caso. Ao rediscutir-se a Lógica Seqüencial Fuzzy definiu-se e desenvolveu-se, por meio da aplicação de realimentação em Circuitos Seqüenciais Fuzzy nos Sistemas Dinâmicos, um Astável Fuzzy (oscilações não-periódicas), o qual exemplifica um SDFI. Finalmente realizaram-se algumas interpretações da Física pela Teoria Fuzzy, utilizando-se o conceito de SDFI\'s. / A mapping of analytical tools for representation and computational treatment of Fuzzy Systems was made in this thesis. This mapping evidenced the existence of relations and the adequacy of the use of the Fuzzy Theory in the models construction for the problems solution involving dynamic systems e, in particular, of chaotic systems. A differentiation was considered about two types of Fuzzy Dynamic Systems - the Intrinsic Fuzzy Dynamic Systems (IFDS) and the Extrinsic Fuzzy Dynamic Systems (EFDS). The EFDS in the Fuzzy modeling is used for the problems solution of trajectories setting in chaotic systems. Case Studies had been developed that allow to verify, by means of simulation in billiards and tests in circuit of Chua (implemented in physical prototype), the adequacy of the use of this strategy in the solution of these problems. Related with Chua circuit, can be mentioned the following achievements: Computational treatment on real circuit; Use of a 8-bits AD converter followed by lowpass filter to compensate this low resolution signals reading; Use of gyrator circuit to implement the inductor used in this circuit; Proposal and accomplishment of a circuit to define the control resistor, via PC parallel port, of simple reproduction. Related with billiards, the following achievements are mentioned: Definition of a new billiard - Garms & Andrade Newtonian Billiard; Development of detailed equations of the simulations billiards analysis. When discussing again the Sequential Logic Fuzzy, is defined and developed, by means of the application of feedback in Sequential Circuits Fuzzy in the Dynamic Systems, an Astable Fuzzy (non-periodic oscillations), which exemplifies an IFDS. Finally, some interpretations of the Physics for the Fuzzy Theory are also presented with the use of the IFDS concept.

Neste trabalho investiga-se o emprego de técnicas de estimação em sistemas de modulação digital que utilizam sinais caóticos. Inicialmente, aspectos básicos das teorias de sistemas não-lineares e de modulações digitais são revisitados seguidos de técnicas recentemente propostas de modulações digitais caóticas com receptores por correlação coerente, não-coerente e diferencial: o CSK (Chaos Shift Keying), o DCSK (Differential Chaos Shift Keying) e algumas de suas variantes, em especial o FM-DCSK (Frequency Modulated DCSK). Nessa descrição, utiliza-se a notação de equivalente passa-baixas de tempo discreto para facilitar a comparação com modulações digitais convencionais. Deduz-se o limite de Cramér-Rao para a estimação da condição inicial de órbitas caóticas em função de propriedades estatísticas do mapa que as gerou e descrevem-se dois estimadores para elas: o MLE (Maximum Likelihood Estimator) que se aplica a mapas com densidade invariante uniforme e o algoritmo de Viterbi para o qual se apresenta uma generalização a fim de aplicá-lo a uma classe maior de mapas. Por apresentar ganho de estimação maior na faixa de relação sinal-ruído de interesse, este último é utilizado em propostas de sistemas de modulação digital que utilizam estimação de órbitas para detectar o símbolo enviado: o ML-CSK (Maximum Likelihood CSK) modificado para poder usar mapas com densidade invariante não-uniforme, empregando um ou dois mapas e o ML-DCSK (Maximum Likelihood DCSK). Por simulação, avaliou-se o desempenho em termos de taxa de erro desses sistemas sob ruído branco aditivo gaussiano. / In this work, we investigate the use of estimation techniques to digital modulation systems that use chaotic signals. Initially, basic aspects of nonlinear systems and digital modulation theory are reviewed followed by currently proposed techniques of chaotic digital modulation with coherent, noncoherent and differential correlation receivers: CSK (Chaos Shift Keying), DCSK (Differential Chaos Shift Keying) and some of its variants in special FM-DCSK (Frequency Modulated DCSK). These systems are described using a discrete-time lowpass equivalent model to facilitate comparison with conventional digital modulation systems. We derive Cramér-Rao lower bounds for the estimation of the initial condition of chaotic orbits as a function of the statistical properties of the chaos generating map and describe two chaotic orbits estimators: the MLE (Maximum Likelihood Estimator) that applies only to maps with uniform invariant density and the Viterbi algorithm for which a generalization is presented that allows its application to a broader class of maps. Because of the larger estimation gains attained in the signal-to-noise ratio range of interest, the latter is used in proposed digital modulation systems that use orbit estimation to detect the transmitted symbol: ML-CSK (Maximum Likelihood CSK) modified to allow maps with nonuniform invariant density using one map or two maps and ML-DCSK (Maximum Likelihood DCSK). The performance of these systems in terms of symbol error rate is accessed via simulation under additive white gaussian noise perturbations.

Zhao, Jin. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 79-83). / Abstracts in English and Chinese. / Abstract --- p.i / Acknowledgement --- p.iii / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Synchronization of Master-slave Systems --- p.1 / Chapter 1.2 --- Output Regulation --- p.2 / Chapter 1.3 --- Typical Nonlinear Systems --- p.4 / Chapter 1.4 --- Organization --- p.4 / Chapter 2 --- Synchronization of Chua's Circuit and Van der Pol Oscillator via Inter- nal Model Approach --- p.6 / Chapter 2.1 --- Introduction --- p.6 / Chapter 2.2 --- Problem Formulation --- p.8 / Chapter 2.3 --- Preliminaries --- p.10 / Chapter 2.4 --- Solvability of the Problem --- p.13 / Chapter 2.4.1 --- The solution of the regulator equations --- p.14 / Chapter 2.4.2 --- Steady-state generator --- p.15 / Chapter 2.4.3 --- Internal model --- p.19 / Chapter 2.4.4 --- Stabilization --- p.20 / Chapter 2.4.5 --- Simulation --- p.22 / Chapter 2.5 --- Conclusions --- p.27 / Chapter 3 --- Robust Output Regulation of Output Feedback Systems with Nonlinear Exosystems --- p.28 / Chapter 3.1 --- Introduction --- p.28 / Chapter 3.2 --- Assumptions and Preliminaries --- p.29 / Chapter 3.3 --- Solvability of the Synchronization Problem --- p.33 / Chapter 3.4 --- Comparing Two Approaches for Output Regulation --- p.42 / Chapter 3.4.1 --- Differences between the two approaches for the output regulation problem --- p.42 / Chapter 3.4.2 --- Solvability of the regulator equations --- p.43 / Chapter 3.4.3 --- Solvability of stabilization --- p.47 / Chapter 3.5 --- Conclusions --- p.49 / Chapter 4 --- Applications of Robust Regional Synchronization via Output Regulation Techniques --- p.50 / Chapter 4.1 --- Problem Formulation --- p.50 / Chapter 4.2 --- Duffing Oscillator Synchronizes with Chua's Circuit --- p.51 / Chapter 4.2.1 --- Transfer the synchronization problem into the stabilization problem --- p.53 / Chapter 4.2.2 --- Boundedness of Chua's circuit --- p.57 / Chapter 4.2.3 --- Stabilization --- p.59 / Chapter 4.2.4 --- Simulation Results --- p.64 / Chapter 4.3 --- The Chaotic SMIB Power System Synchronizes with Van der Pol Oscillator --- p.64 / Chapter 4.3.1 --- Transfer the synchronization problem into the stabilization problem --- p.68 / Chapter 4.3.2 --- Stabilization --- p.71 / Chapter 4.3.3 --- Simulation Results --- p.74 / Chapter 4.4 --- Conclusions --- p.76 / Chapter 5 --- Conclusions --- p.77 / Bibliography --- p.79

I describe the miniaturization of a linear-type ion trap mass spectrometer for possible applications in portable chemical analysis, and demonstrate the advantages of using lithographically patterned electrode plates in realizing an ion trap with dimension r0 less than 1 mm. The focus of the work was to demonstrate the viability and feasibility of the patterned electrode approach to trap miniaturization, and also to discover potential obstacles to its use. Planar ceramic substrates were patterned with metal electrodes using photolithography. Plates that were originally used in a linear trap with a half-spacing (r0) of 2.19 mm were positioned much closer together such that r0 = 0.95 mm. A capacitive voltage divider provided different radiofrequency (RF) amplitudes to each electrode, and the capacitor values were adjusted to provide the correct electric field at this closer spacing. Electron ionization mass spectra of toluene and dichloromethane demonstrate instrument performance with better than unit mass resolution. Compared with the larger plate spacing, the signal intensity is reduced, corresponding to the reduced trapping capacity of the smaller device, but the mass resolution of the larger device is retained. A further miniaturized linear ion trap with a half-spacing of 362 µm was designed and tested. A series of obstacles and troubleshooting on ion source, analytical method, and electronics were present. These experiments show promise for further miniaturization using patterned ceramic plates and provide a guide for the ion trap miniaturization. The feasibility of a wire linear ion trap was also demonstrated. Unit mass resolution was obtained, indicating a promise for further optimization and miniaturization of the wire linear ion trap. In addition to the practical experiments on the miniaturized linear ion traps, I theoretically studied ion motion in the toroidal ion trap using SIMION simulations, which show classical chaotic behavior of single ions. The chaotic motion is a result of the non-linear components of the electric fields as established by the trap electrodes, and not by Coulombic interaction from other ions. The chaotic behavior was observed specifically in the ejection direction of ions located in non-linear resonance bands within and adjacent to the region of stable trapping. The non-linear bands crossing through the stability regions correspond to hexapole resonance conditions, while the chaotic ejection observed immediately adjacent to the stable trapping region corresponds to a "fuzzy" ejection boundary. Fractal-like patterns were obtained in a series of zoomed-in regions of the stability diagram.

The concern of this thesis was to investigate the nonlinear dynamics inherent in oscillating waterfalls.

In this thesis we apply chaotic dynamic data analysis to the area of discrete time signal processing. A newly developed Hidden Filter Hidden Markov Model is introduced in detection of chaotic signals. Numerical experiments have verified that this novel nonlinear model outperforms linear AR model in detecting chaotic signals buried by noise having similar power spectra. A simple Histogram Model is proposed which can also be used to do detection on the data sets with chaotic behavior. Receiver Operating Characteristics for a variety of noise levels and model classes are reported.