Made available in DSpace on 2014-06-11T19:26:55Z (GMT). No. of bitstreams: 0 Previous issue date: 2008-03-12Bitstream added on 2014-06-13T19:06:40Z : No. of bitstreams: 1 cardin_pt_me_sjrp.pdf: 780321 bytes, checksum: 2c76fcd2cf98ce623cf8bc779edb3379 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta dissertação estudamos critérios para determinar a existência, a não existência e a unicidade de ciclos limites de campos de vetores planares. Mais especificamente, estudamos equações de Lienard Äx + f(x; _ x) _ x + g(x) = 0; onde f e g satisfazem determinadas hip¶oteses. Em particular estudamos a equa»c~ao de van der Pol Äx + (x2 ¡ 1) _ x + x = 0; a qual é conhecida da teoria dos circuitos elétricos. Provamos a existência e a unicidade de ciclos limites para estas equações. Por fim estudamos a equação de van der Pol com o parâmetro 1 e o fenômeno canard que ocorre ao considerarmos um parâmetro adicional ®: As técnicas utilizadas s~ao as usuais de Análise Assintótica. / In this work we study the existence, the non existence and the uniqueness of limit cycles of planar vector felds. More specifically, we study Lienard equations Äx+f(x; _ x) _ x+g(x) = 0; where f and g satisfy some hypothesis. In particular we study the van der Pol equation Äx + (x2 ¡ 1) _ x + x = 0; which is knew of the circuit theory. We prove the existence and the uniqueness of limit cycles for these equations. In the last part we study the van der Pol equation with the parameter 1 and the canard phenomenon which appears when we consider an additional parameter ®: The techniques employed are the usual in the Asymptotic Analysis.

Contribution à l’étude du phénomène d’oscillation argumentaire. L’oscillateur argumentaire a un mouvement stable périodique, à une fréquence proche de sa fréquence fondamentale, lorsqu’il est soumis à une excitation provenant d’une source de type harmonique, à une fréquence qui est un multiple de ladite fréquence fondamentale, et agissant de manière telle que son interaction avec le système dépende des coordonnées d’espace du système. La présente thèse étudie quelques systèmes argumentaires et essaie de mettre en évidence des relations symboliques entre les paramètres de ces systèmes et leur comportement observé ou calculé. C’est la représentation de Van der Pol qui a été utilisée la plupart du temps pour représenter l’état du système, car elle est bien adaptée à la méthode de centrage, où l'on cherche une solution sous forme d’un signal de type sinusoïdal, d’amplitude et de phase lentement variables. L’originalité de la présente thèse vis-à-vis des publications antérieures est dans la modélisation, plus proche des systèmes physiques réels, dans les développements symboliques qui donnent des représentations inédites, dans le mode de réalisation des expériences, qui utilisent toutes une visualisation de Van der Pol en temps réel, et dans l’objet de l’expérience de la poutre excitée axialement de manière argumentaire. Au cours de cette thèse, des systèmes simples à un DDL ont été modélisés, construits et expérimentés. Des relations symboliques, notamment concernant les probabilités de capture par des attracteurs, ainsi que des critères de stabilité et une solution symbolique approchée, ont été mis en évidence. Un système continu constitué d’une poutre élancée excitée axialement a ensuite été modélisé à l’aide de deux modèles et expérimenté ; toujours dans le domaine symbolique, des propriétés ont été étudiées, notamment concernant des combinaisons de plages de paramètres permettant au phénomène argumentaire d’exister / Contribution to the study of the argumental oscillation phenomenon. The argumental oscillator has a stable periodic motion at a frequency close to its fundamental frequency when it is subjected to an excitation from a harmonic source at a frequency which is a multiple of said fundamental frequency, and acting in such a way that its interaction with the system depends on the space coordinates of the system. This thesis studies some argumental systems and tries to demonstrate symbolic relations between the parameters of these systems and their observed or calculated behavior. The Van der Pol representation was used most of the time to represent the state of the system, as it is well adapted to the averaging method, where a solution is sought as a signal of sinusoidal type, with slowly varying amplitude and phase. The originality of this thesis with respect to previous publications is in the modeling, closer to real physical systems, in the symbolic developments that give new representations, in the embodiment of the experiments, which all use a real-time Van der Pol visualization, and in the object of the experiment of the beam axially excited in an argumental way. During this thesis, simple systems with one DDL have been modeled, built and tested. Symbolic relationships have been highlighted, in particular with regard to the probabilities of capture by attractors, as well as stability criteria and an approximate symbolic solution. A continuous system consisting of an axially excited slender beam was then modeled using two models, and tested; still in the symbolic domain, properties have been studied, especially concerning combinations of parameter ranges allowing the argumental phenomenon to occur

We consider a type of forced "Van Der Pol" oscillator where the forced function is periodic and oscillatory around the t-axis. This problem derived from an electrical model. The important issues here is that this circuits presents the spiking phenomena over a one time period and it has important applications in signal processing and digital communication. The three most important problems that we addressed here in this thesis are to compute the number of spikes a solution completes in one time period (it can be used to transform the analog signal into digital information), how the dynamics of the number of spikes change with respect to the parameters amplitude (k) and frequency (w), and when the coupled circuits synchronize (i.e., the driver and the respond are on synchronous). Sophisticated mathematical and numerical analysis has been developed that enable us to give a complete study of the problems above described.

Learning is a mechanism intrinsic to all sentient biological systems. Despite the diverse range of paradigms that exist, it appears that an artificial system has yet to be developed that can emulate learning with a comparable degree of accuracy or efficiency to the human brain. With the development of new approaches comes the opportunity to reduce this disparity in performance. A model presented by Janson and Marsden [arXiv:1107.0674 (2011)] (Memory foam model) redefines the critical features that an intelligent system should demonstrate. Rather than focussing on the topological constraints of the rigid neuron structure, the emphasis is placed on the on-line, unsupervised, classification, retention and recognition of stimuli. In contrast to traditional AI approaches, the system s memory is not plagued by spurious attractors or the curse of dimensionality. The ability to continuously learn, whilst simultaneously recognising aspects of a stimuli ensures that this model more closely embodies the operations occurring in the brain than many other AI approaches. Here we consider the pertinent deficiencies of classical artificial learning models before introducing and developing this memory foam self-shaping system. As this model is relatively new, its limitations are not yet apparent. These must be established by testing the model in various complex environments. Here we consider its ability to learn and recognize the RGB colours composing cartoons as observed via a web-camera. The self-shaping vector field of the system is shown to adjust its composition to reflect the distribution of three-dimensional inputs. The model builds a memory of its experiences and is shown to recognize unfamiliar colours by locating the most appropriate class with which to associate a stimuli. In addition, we discuss a method to map a three-dimensional RGB input onto a line spectrum of colours. The corresponding reduction of the models dimensions is shown to dramatically improve computational speed, however, the model is then restricted to a much smaller set of representable colours. This models prototype offers a gradient description of recognition, it is evident that a more complex, non-linear alternative may be used to better characterize the classes of the system. It is postulated that non-linear attractors may be utilized to convey the concept of hierarchy that relates the different classes of the system. We relate the dynamics of the van der Pol oscillator to this plastic self-shaping system, first demonstrating the recognition of stimuli with limit cycle trajectories. The location and frequency of each cycle is dependent on the topology of the systems energy potential. For a one-dimensional stimuli the dynamics are restricted to the cycle, the extension of the model to an N dimensional stimuli is approached via the coupling of N oscillators. Here we study systems of up to three mutually coupled oscillators and relate limit cycles, fixed points and quasi-periodic orbits to the recognition of stimuli.

No description available.

In this thesis, the stabilization and tracking problem of the Van der Pol oscillator is studied by using advanced control techniques. First, the linear state feedback and linear adaptive state feedback controllers for the stabilization problem are designed. Then, non-linear state feedback and output feedback controllers are proposed for the tracking problem with known parameters. Finally, a dynamic output feedback controller based on adaptive backstepping technique is introduced for the tracking problem when all parameters of the Van der Pol system are unknown.

A generalized low order model (LOM) for the fluctuating lift coefficient caused by vortex shedding from a tandem cylinder pair is proposed to expand upon models from previous authors. This model could provide a reduced computational time method for collecting qualitative and quantitive data from a tandem shedding pair. A delay coupled system with sufficient bifurcation characteristics is developed to account for the different flow regimes (extended-body, reattachment, and co-shedding) which occur as cylinder spacing is varied. Coefficient and parameter fitting is performed to fit experimental data. Finally, results and physical interpretations of the interactions in the model are discussed. It was found that many aspects of the flow at varying L/D ratios could be modeled by the LOM, including vortex suppression in the forward cylinder at the critical spacing, and amplitude growth in the rear cylinder in the co-shedding regime.

Le travail présenté dans ce mémoire traite de l'étude d'oscillateurs et d'Oscillateurs Contrôlés en Tension (OCT) différentiels couplés appliqués à la commande d'un réseau d’antennes linéaire. Après avoir rappelé les concepts d'antennes réseaux et d'oscillateurs, une synthèse de la théorie élaborée par R. York et donnant les équations dynamiques modélisant deux oscillateurs de Van der Pol couplés par un circuit résonnant a été présentée. Après avoir montré la limitation de cette approche concernant la prédiction de l'amplitude des oscillateurs, une nouvelle formulation des équations non linéaires décrivant les états de synchronisation a été proposée. Néanmoins, compte tenu du caractère trigonométrique et fortement non linéaire de ces équations, une nouvelle écriture facilitant la résolution numérique a été proposée. Ceci a permis l'élaboration d'un outil de Conception Assistée par Ordinateur (CAO) fournissant une cartographie de la zone de synchronisation de deux oscillateurs de Van der Pol couplés. Celle-ci permet de déterminer rapidement les fréquences d'oscillation libres nécessaires à l'obtention du déphasage souhaité. Pour ce faire, une procédure de modélisation de deux oscillateurs et OCTs différentiels couplés, par deux oscillateurs de Van der Pol couplés par une résistance a été élaborée. Les résultats fournis par l'outil de CAO proposé ont ensuite été comparés avec les résultats de simulations de deux oscillateurs et OCTs différentiels couplés obtenus avec le logiciel ADS d'Agilent. Une très bonne concordance des résultats a alors été obtenue montrant ainsi l'utilité et la précision de l'outil présenté. / The work presented in this thesis deals with the study of coupled differential oscillators and Voltage Controlled Oscillators (VCO) used to control antenna arrays. After reminding the concept of antenna arrays and oscillators, an overview of R. York's theory giving the dynamics for two Van der Pol oscillators coupled through a resonant network was presented. Then, showing the limitation of this approach regarding the prediction of the oscillators' amplitudes, a new formulation of the nonlinear equations describing the oscillators' locked states was proposed. Nevertheless, due to the trigonometric and strongly non-linear aspect of these equations, mathematical manipulations were applied in order to obtain a new system easier to solve numerically. This has allowed to the elaboration of a Computer Aided Design (CAD) tool, which provides a cartography giving the frequency locking region of two coupled differential Van der Pol oscillators. This cartography can help the designer to rapidly find the free-running frequencies of the two outermost differential oscillators or VCOs of the array required to achieve the desired phase shift. To do so, a modeling procedure of two coupled differential oscillators and VCOs as two coupled differential Van der Pol oscillators, with a resistive coupling network was performed. Then, in order to validate the results provided by our CAD tool, we compared them to the simulation results of two coupled differential oscillators and VCOs obtained with Agilent’s ADS software. Good agreements between the simulations of the circuits, the models and the theoretical results from our CAD tool were found.

Submitted by CAMILO CARRANZA (carranzacamilo@gmail.com) on 2017-07-25T19:58:22Z No. of bitstreams: 1 Camilo_PhD_Thesis.pdf: 11035322 bytes, checksum: efe400c07b13cabff41e927078789c59 (MD5) / Approved for entry into archive by Luiz Galeffi (luizgaleffi@gmail.com) on 2017-07-26T18:31:30Z (GMT) No. of bitstreams: 1 carranzalopez_jc_dr_ilha.pdf: 11035322 bytes, checksum: efe400c07b13cabff41e927078789c59 (MD5) / Made available in DSpace on 2017-07-26T18:31:30Z (GMT). No. of bitstreams: 1 carranzalopez_jc_dr_ilha.pdf: 11035322 bytes, checksum: efe400c07b13cabff41e927078789c59 (MD5) Previous issue date: 2017-06-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Nesta tese são investigadas, teórica e experimentalmente, a sincronização em fase e a sincronização em anti-fase de dois metrônomos oscilando sobre uma base móvel, a partir de um modelo aqui proposto. Uma descrição do funcionamento do mecanismo de escapamento dos metrônomos é feita, junto a um estudo da relação entre este e o oscilador de van der Pol. Também uma aproximação experimental do valor do amortecimento do metrônomo é fornecida. A frequência instantânea das respostas, numérica e experimental, do sistema é usada na analise. A diferença de outros trabalhos prévios, os dados experimentais têm sido adquiridos usando vídeos dos experimentos e extraídos com ajuda do software Tracker. Para investigar a relação entre as condições iniciais do sistema e seu estado final de sincronização, foram usados mapas bidimensionais chamados ‘basins of attraction’. A relação entre o modelo proposto e um modelo prévio também é mostrada. Encontrou-se que os parâmetros relevantes em relação a ambos os tipos de sincronização são a razão entre a massa do metrônomo e a massa da base, e o amortecimento do sistema. Tem-se encontrado, tanto experimental quanto teoricamente, que a frequência de oscilação dos metrônomos aumenta quando o sistema sincroniza-se em fase, e se mantém a mesma de um metrônomo isolado quando o sistema sincroniza-se em anti-fase. A partir de simulações numéricas encontrou-se que, em geral, incrementos no amortecimento do sistema levam ao sistema se sincronizar mais em fase do que em anti-fase. Adicionalmente se encontrou que, para dado valor de amortecimento, diminuir a massa da base leva a uma situação em que a sincronização em anti-fase é mais comum do que a sincronização em fase. / This thesis concerns a theoretical and experimental investigation into the synchronization of two coupled metronomes. A simplified model is proposed to study in-phase and anti-phase synchronization of two metronomes oscillating on a mobile base. A description of the escapement mechanism driving metronomes is given and its relationship with the van der Pol oscillator is discussed. Also an experimental value for the damping in the metronome is determined. The instantaneous frequency of the responses from both numerical and experimental data is used in the analysis. Unlike previous studies, measurements are made using videos and the time domain responses of the metronomes extracted by means of tracker software. Basins of attraction are used to investigate the relationship between initial conditions, parameters and both final synchronization states. The relationship between the model and a previous pendulum model is also shown. The key parameters concerning both kind of synchronization have been found to be the mass ratio between the metronome mass and the base mass, and the damping in the system. It has been shown, both theoretically and experimentally, that the frequency of oscillation of the metronomes increases when the system reaches in-phase synchronization, and is the same as an isolated metronome when the system synchronizes in anti-phase. From numerical simulations, it has been found that, in general, increasing damping leads the system to synchronize more in-phase than in anti-phase. It has also been found that, for a given damping value, decreasing the mass of the base results in the situation where anti-phase synchronization is more common than in-phase synchronization.

在這篇論文中，我們探討了在任何正參數之下，范德波爾方程的極限環結果。藉由改良後的同倫擾動方法，我們求得了一些極限環的近似結果。 相對於傳統的擾動方法，這種同倫方法在方程中並不受限於小的參數。除此之外，我們也設計了一個演算法來計算極限環的近似振幅及頻率。 / In this thesis, we study the limit cycle of van der Pol equation for parameter ε>0. We give some approximate results to the limit cycle by using the modified homotopy perturbation technique. In constract to the traditional perturbation methods, this homotopy method does not require a small parameter in the equation. Besides, we also devise a new algorithm to find the approximate amplitude and frequency of the limit cycle.