A new type of cantilever beam gyroscope is introduced, modeled, and analyzed. The main structure includes a cantilever beam and a rigid body attached to the free end of the beam. The model accounts for the eccentricity, that is the offset of the center of mass of the rigid body relative to the beam's free end. The first and second moments of mass and the rotary inertia appear in the equations of motion and boundary conditions. The common mechanism of electrostatic actuation of microgyroscopes is used with the difference of computing the force at the center of mass resulting in the electrostatic force and moment in the boundary conditions. By using the extended Hamilton's principle, the method of assumed modes, and Lagrange's differential equations, the equations of motion, boundary conditions, and the discretized model are developed. The generalized model simplifies to other beam gyroscope models by setting the required parameters to zero. Considering the DC and AC components of the actuating and sensing methods, the response is resolved into the static and dynamic components. The static configuration is studied for an increasing DC voltage. For the uncoupled system of equations, the explicit equation relating the DC load and the static configuration is computed and solved for the static configuration of the beam-rigid body in each direction. Including the rotation rate, the stationary analysis is performed, the stationary pull-in voltage is identified, and it is shown that the angular rotation rate does not affect the static configuration. The modal frequencies of the beam-rigid body gyroscope are studied and the instability region due to the rotation rate is computed. It is shown that the gyroscope can operate in the frequency modulation mode and the amplitude modulation mode. To operate the beam-rigid body gyroscope in the frequency modulation mode, the closed-form relation of the observed modal frequency split and the input rotation rate is computed. The calibration curves are generated for a variety of DC loads. It is shown that the scale factor improves by matching the zero rotation rate natural frequencies. The method of multiple scales is used to study the reduced-order nonlinear dynamics of the oscillations around the static equilibrium. The modulation equations, the ``slow'' system, are derived and solved for the steady-state solutions. The computational shooting method is employed to evaluate the results of the perturbation method. The frequency response and force response plots are generated. For combinations of parameters resulting in a single-valued response, the two methods are in excellent agreement. The synchronization of the response occurs in the sense direction for initially mismatched natural frequencies. The global stability of the system is studied by drawing phase-plane diagrams and long-time integration of response trajectories. The separatrices are computed, the jump phenomena is numerically shown, and the dynamic pull-in of the response is demonstrated. The fold bifurcation points are identified and it is shown that the response jumps to the higher/lower branch beyond the bifurcation points in forward/backward sweep of the amplitude and the excitation frequency of AC voltage. The mechanical-thermal (thermomechanical) noise effect on the sense response is characterized by using a linear approximation of the system and the nonlinear "slow" system obtained by using the method of multiple scales. To perform linear analysis, the negligible effect of Coriolis force on the drive amplitude is discarded. The second-order drive resonator is solved for the drive amplitude and phase. Finding the sense response due to the thermal noise force and the Coriolis force and equating them computes the mechanical-thermal noise equivalent rotation rate in terms of system parameters and mode shapes. The noise force is included in the third-order equation of the perturbation and equation to account for that in the reduced-order nonlinear response. The numerical results of linear and reduced-order nonlinear thermal noise analyses agree. It is shown that higher quality factor, higher AC voltage, and operating at lower DC points result in better resolution of the microsensor.

La tesi, costituita da due parti e tre capitoli, si concentra sulle conseguenze macroeconomiche della globalizzazione, prendendo in considerazione vari schemi di un modello a generazioni sovrapposte, in un mercato finanziario integrato. Per ipotesi, gli agenti vivono per due periodi e sono divisi in due gruppi: i "vecchi" che posseggono il fattore capitale e i "giovani" che offrono lavoro e risparmio. Nella prima parte si suppone che i mercati siano perfetti. Dopo aver ricevuto il loro reddito, i giovani ottimizzano consumo e risparmio. Diverse ipotesi vengono avanzate sull'apertura dei mercati, ma le economie convergono sempre verso uno stato stazionario asintoticamente stabile. Nella seconda parte i mercati finanziari sono imperfetti e i prestiti sono razionati. I giovani agenti risparmiano tutto il loro reddito e consumano solo nel secondo periodo della loro vita. In queste nuove ipotesi si possono riscontrare dinamiche endogene di tipo periodico. / The essay, made by two parts and three chapters, focuses on macroeconomic effects of globalization, considering various schemes of a two-country OLG model with integrated financial market. For hypothesis, agents live for two periods and are divided in two groups: the "old" that own the capital factor and the "young" that supply labor and savings. In the first part markets are supposed to be perfect. After received their income, the young optimize their consumption and savings. Different hypotheses about the opening markets are considered, but the economies ever converge to an asymptotically stable steady state. In the second part the financial markets are imperfect and borrowing is constrained. The young agents save all their income and consume only in the second period of their life. In these new hypotheses endogenous, periodic dynamics may occur.

The thesis is based on using dynamical systems theories and techniques to study the qualitative dynamics of herpes simplex virus type 2 (HSV-2), a sexually-transmitted disease of major public health significance. A deterministic model for the interaction of the virus with the immune system in the body of an infected individual (in vivo) is designed first of all. It is shown, using Lyapunov function and LaSalle's Invariance Principle, that the virus-free equilibrium of the model is globally-asymptotically stable whenever a certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present equilibrium when the threshold quantity exceeds unity. Using persistence theory, it is shown that the virus will always be present in vivo whenever the reproduction threshold exceeds unity. The analyses (theoretical and numerical) of this model show that a future HSV-2 vaccine that enhances cell-mediated immune response will be effective in curtailling HSV-2 burden in vivo. A new single-group model for the spread of HSV-2 in a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less than unity. The model has a unique endemic equilibrium, which is shown to be globally-stable for a special case, when the reproduction number exceeds unity. The model is extended to incorporate an imperfect vaccine with some therapeutic benefits. Using centre manifold theory, it is shown that the resulting vaccination model undergoes a vaccine-induced backward bifurcation (the epidemiological importance of the phenomenon of backward bifurcation is that the classical requirement of having the reproduction threshold less than unity is, although necessary, no longer sufficient for disease elimination. In such a case, disease elimination depends upon the initial sizes of the sub-populations of the model). Furthermore, it is shown that the use of such an imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity). The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies. Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it is shown that the risk-structured model undergoes backward bifurcation under certain conditions (the backward bifurcation property can be removed if the susceptible population is not stratified according to the risk of acquiring infection). Thus, one of the main findings of this thesis is that risk structure can induce the phenomenon of backward bifurcation in the transmission dynamics of HSV-2 in a population.

Η τοπολογική ταξινόμηση και μελέτη διανυσματικών πεδίων αποτελεί το κύριο θέμα αυτής της διατριβής. Στο Κεφάλαιο 1 δίνονται οι απαραίτητοι ορισμοί, καθώς και τα αποτελέσματα επί της ταξινόμησης διανυσματικών πεδίων σε μονοδιάστατες και δισδιάτατες πολλαπλότητες. Στο Κεφάλαιο 2 τεχνικές της Θεωρίας Κόμβων χρησιμοποιούνται προκειμένου να μελετηθεί η τοπολογική δομή ορισμένων παράξενων ελκυστών που εμφανίζονται στη διεθνή βιβλιογραφία. Στο Κεφάλαιο 3 αναπτύσσεται μία μέθοδος η οποία επιτρέπει την ολική τοπολογική ταξινόμηση διανυσματικών πεδίων σε ευκλείδειους χώρους οποιασδήποτε διάστασης. Η μέθοδος αυτή έπειτα εφαρμόζεται στην ταξινόμηση διανυσματικών πεδίων του R^2 και του R^3. Στο Κεφάλαιο 4 μελετάται ένα διανυσματικό πεδίο του R^3 αμετάβλητο από την D_2 ομάδα. Δίνεται η ολική του μελέτη, για διάφορες τιμές των παραμέτρων, και το μερικό του διάγραμμα διακλάδωσης. Αποδεικνύεται η ύπαρξη χάους και συνδέεται με τις συμμετρικές ιδιότητες του συστήματος, ενώ η μελέτη ολοκληρώνεται με τη συμπεριφορά του συστήματος στο άπειρο. / The topological classification and study of vector fields is the subject of this thesis. In Chapter 1 the necessary definitions are given, along with the known results on the classification of vector fields on 1-dimensional and 2-dimensional manifolds. In Chapter 2 methods of Knot Theory are used for the clarification of the topological study of some strange attractors found in the bibliography. In Chapter 3 a technique is developed, which can be used to classify globally vector fields defined on Euclidean spaces of any dimension. This technique is then used to classify some vector fields of R^2 and R^3. In the final Chapter 4 a vector field of R^3 is studied which is invariant under the D_2 symmetry group. We present its global phase portrait, for various parameter values, and its partial bifurcation diagram. The existence of chaos is proven and its connection to the symmetry properties of the attractor is discussed. We end its study presenting its behavior at infinity.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2006-03-03Bitstream added on 2014-06-13T20:27:28Z : No. of bitstreams: 1 bronzi_ma_me_sjrp.pdf: 904425 bytes, checksum: 2344eb35a112034c2f1741b2e229f1ec (MD5) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos intersecções homoclínicas de variedades estável e instável de pontos peródicos. Toda intersecção homoclínica produz um comportamento curioso na dinâmiôa. Nosso modelo de tal fenômeno é a famosa ferradura de Smale, a qual é um conjunto hiperbólico para um difeomorfismo. Além disso, estudamos dinâmica não hiperbólica cuja perda de hiperbolicidade é divido à tangências homoclínicas. Elas tem um papel central na teoria de sistemas dinâmicos. O desdobramento de uma tangência homoclínica produz dinâmicas muito interessantes. Neste trabalho estudamos a criação de cascatas de bifurcações de duplicação de período e um esquema de renormalização para uma tangência homoclínica. / We study homoclinic intersection of stable and unstable manifolds of periodic points. Every homoclinic intersection produce a intricate behavior of the dynamics. Our model of such phenomena is the so called Smalesþs horseshoe, which is a hyperbolic set for a di eomorphism. We also study non hyperbolic dynamics whose lack of hyperbolicity is due to homoclinic tangencies. They play a central role in the theory of dynamical systems. The unfolding of a homoclinic tangency produce many interesting dynamics. In this work we study creation of cascade of period doubling bifurcations and a renormalization scheme for a homoclinic tangency.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-07-25Bitstream added on 2014-06-13T20:35:11Z : No. of bitstreams: 1 cauz_lo_me_sjrp.pdf: 1991139 bytes, checksum: c18750cde05438df23eec43208d0eb54 (MD5) / Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) / Neste trabalho apresentamos um estudo da dinâmica de um sistema vibrante não ideal, composto por um motor e uma mola, conhecido como vibrador centrífugo. O objetivo deste estudo é mostrar a diferença de comportamento do sistema, quando consideramos molas duras (coeficiente de elasticidade cúbica positivo) ou molas suaves (coeficiente de elasticidade cúbica negativo). Para mola dura foi analisada a estabilidade dos pontos de equilíbrio, e mostrada por meio da teoria de variedade central e do teorema de Bezout a existência da bifurcação de Hopf. Para mola suave, þe mostrada a existência de uma órbita heteroclínica conectando dois pontos de sela. Usando o método clássico de Melnikov, é discutida a existência ou não do comportamento caótico para um determinado nível de energia e para certos valores do coeficiente de amortecimento. Toda a análise é acompanhada de simulações numéricas para a confirmação dos resultados. / In this work we present a study of the dynamics of a non-ideal vibrating system, composed by a motor and a spring, which is known as centrifugal vibrator. The purpose of this study is to show the difference of behavior of the system when we consider hard springs (positive coefficient of cubical elasticity) or soft springs (negative coefficient of cubical elasticity). For hard spring the stability of the fixed point was analyzed, and by means of the Central Manifolds Theory and the Bezout theorem the existence of the Hopf Bifurcation is shown. For soft spring, it is shown the existence of a heteroclinic orbit connecting two saddle points. Using the classical Melnikov method it is discussed the existence, or not, of the chaotic behavior for some energy level and certain values of the damping coefficient. All the analysis is followed by numerical simulations to confirm the results.

Made available in DSpace on 2016-12-12T20:15:51Z (GMT). No. of bitstreams: 1 Anderson Hoff.pdf: 11193047 bytes, checksum: f07cba24b1a4da1b53270bac747a0252 (MD5) Previous issue date: 2014-02-25 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Estruturas de bifurcação delimitam regiões periódicas imersas em áreas de caos em planos de parâmetros de sistemas dinâmicos. Neste trabalho são estudadas as estruturas de bifurcação de sistemas dinâmicos contínuos quadridimensionais, um circuito de Chua e um acoplamento de dois osciladores de FitzHugh-Nagumo. Os resultados numéricos foram obtidos através do cálculo dos expoentes de Lyapunov, através de integração numérica dos sistemas, e das curvas de bifurcação, por continuação numérica através do MatCont. Investigou-se as bifurcações que formam o endoesqueleto de camarões em planos de parâmetros no circuito de Chua, além de estruturas espirais, caos transiente e bacias de atração caóticas e periódicas. Análise semelhante foi realizada no acoplamento de dois osciladores de FitzHugh-Nagumo, identificando estruturas periódicas imersas em regiões caóticas, estruturas de línguas de Arnold imersas em regiões de comportamento quase-periódico, com períodos organizados e conectadas com regiões periódicas, e a sensibilidade do sistema às condições iniciais.

Made available in DSpace on 2016-12-12T20:15:52Z (GMT). No. of bitstreams: 1 Rodrigo Antonio da Silva.pdf: 16150641 bytes, checksum: 4f03858f903f15428bb43b27fd6e1fe8 (MD5) Previous issue date: 2015-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we investigate parameter planes constructed for a set of three autonomous, ten-parameter, first-order nonlinear ordinary differential equations, which models a tri-trophic food web system. By using Lyapunov exponents, bifurcation diagrams, and trajectories in the phase-space, to numerically characterize the dynamics of the model in a parameter plane, we show that it presents typical periodic structures embedded in a chaotic region, forming spiral structures that coils up around a focal point while period-adding bifurcation take place. / Nesse trabalho investigamos planos de parâmetros construídos para um conjunto de três equações diferenciais ordinárias, autônomas, não lineares de primeira ordem com dez parâmetros que modela uma cadeia alimentar tritrófica. Usamos expoentes de Lyapunov, diagramas de bifurcação, e curvas no espaço de fase para caracterizar numericamente a dinâmica do modelo em um plano de parâmetro e, mostramos que este apresenta estruturas periódicas típicas em meio à regiões caóticas, formando espirais que se enrolam ao redor de um ponto focal ao passo que surgem bifurcações de adição de período.

Submitted by Helmut Patrocinio (hell.kenn@gmail.com) on 2017-12-01T23:43:39Z No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) / Approved for entry into archive by Ismael Pereira (ismael@neuro.ufrn.br) on 2017-12-04T12:33:01Z (GMT) No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) / Made available in DSpace on 2017-12-04T12:33:37Z (GMT). No. of bitstreams: 2 license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Heloisa_Onias_Dissertacao_2012.pdf: 9805428 bytes, checksum: 00e0f3bac6584320107351966c70da69 (MD5) Previous issue date: 2012-08 / O circuito RLD, formado por um resistor, um indutor e um diodo em s?rie, apresenta uma din?mica muito rica quando for?ado por uma tens?o externa harm?nica e vem sendo estudado h? d?cadas. Contudo, ainda existem t?picos em din?mica n?o-linear sendo estudados com variantes deste circuito. Varreduras nos par?metros de controle podem fazer com que esse sistema oscile eletronicamente entre regi?es peri?dicas e regi?es ca?ticas. O diodo ? o elemento n?o linear respons?vel pelo surgimento do caos. Utilizando um modelo de capacit?ncia n?o linear para descrever o comportamento do diodo, podemos escrever as equa??es para esse sistema e estudar a sua din?mica numericamente. Nosso principal objetivo foi o estudo de expoentes cr?ticos complexos em bifurca??es din?micas. Para isso, realizamos um estudo num?rico do circuito RLD for?ado senoidalmente utilizando como par?metros de controle a frequ?ncia e a amplitude da tens?o de entrada. Constru?mos, a partir das s?ries temporais da corrente total e da tens?o no diodo, diagramas de bifurca??o com diferentes cortes estrobosc?picos, que apresentam cascata de dobramento de per?odo, janelas peri?dicas e transi??o intermitente. Tamb?m realizamos estudos num?ricos do comportamento da m?dia na regi?o de transi??o caos-peri?dico na busca de encontrar um expoente cr?tico caracter?stico e oscilas??es na m?dia, elementos que j? foram observados no mapa log?stico. N?o foram poss?veis observar numericamente as oscila??es, mas observamos um decaimento exponencial com expoente cr?tico de aproximadamente 0,5. Montamos um sistema de controle, aquisi??o e tratamento de dados experimentais no qual ? poss?vel a realiza??o remota de experimentos simult?neos com dois circuitos diferentes. Obtivemos diagramas de bifurca??es experimentais nos quais observamos que o sistema apresentahisterese e alta sensibilidade ?s condi??es do experimento como, por exemplo, o passo de varredura do par?metro de controle. / The RLD circuit, formed by a resistor, an inductor and a diode in series, displays a very rich dynamics when forced by an external harmonic voltage, and it has being studied for decades. However, there are some topics in nonlinear dynamics that are still studied with variants of this circuit nowadays. Changes in the control parameters may cause electronic oscillations between regular and chaotic regions.The diode is the nonlinear element responsible for the appearance of chaos. Using a nonlinear capacitance model to describe the behavior of the diode, we can write the equations for this system and study its dynamics numerically. Our main objective was the study of critical exponents in complex dynamic bifurcations. For that, we did a numerical study of the RLD circuit forced sinusoidally using as control parameters the amplitude of the input voltage and the frequency. We made, from the time series obtained, bifurcation diagrams with different stroboscopic cuts, which have cascade of period-doubling, periodic windows and intermittent transition. We also did numerical studies of the average behavior in the periodic-chaos transition region searching for characteristic critical exponent and oscilas??es on average, elements that have been observed in the logistic map. It was not possible to observe the oscillations numerically, but we observed an exponential decay with critical exponent of approximately 0.5. We set up a system able to control, acquire and process experimental data making it possible to perform remote simultaneous experiments with two different circuits. We have obtained experimental diagrams bifurcations in which we observe that the system has hysteresis and high sensitivity to the conditions of the experiment such as the step of scanning the control parameter.

[pt] Nesta dissertação estuda-se a flambagem e o comportamento pós-crítico inicial de pórticos planos através da formulação de elementos finitos com graus de liberdade adicionais para posterior implementação no programa de análise FTOOL. Realizaram-se análises linearizadas para a determinação das cargas críticas clássicas e modos de flambagem de colunas com diferentes condições de contorno. Em um segundo momento, realizaram-se testes numéricos no sentido de prever a estabilidade do caminho pós-crítico (sensibilidade a imperfeições) de algumas estruturas cujos resultados analíticos são conhecidos. Finalmente, criaram- se alguns exemplos no FTOOL para comprovar a sua eficácia na obtenção de cargas críticas e avaliação do comportamento pós-crítico inicial de pórticos planos. Utilizou-se o pórtico de Roorda como exemplo para a detecção da sensibilidade a imperfeições, devido à sua simplicidade e ao conhecimento dos seus resultados analíticos. / [en] The buckling and post-buckling behavior of portal frames are studied by the formulation of finite elements with additional degrees of freedom in order to implement a new routine for obtaining critical loads in FTOOL, a structural analysis educational interactive system. Linearized analyses were performed in order to obtain critical loads and buckling modes for columns with different boundary conditions. Later, numerical tests were done in order to predict the stability of the post- critical path (sensitivity to imperfections) of some structures which analytical results are known. Finally, some examples were modeled in FTOOL to verify its accuracy in obtaining critical loads of portal frames. Roorda s frame was used as an example for the detection of the sensitivity to imperfections, based on its simplicity and knowledge of its analytical results.