Um dos mais investigados problemas na teoria qualitativa dos sistemas dinâmicos no plano é o XVI problema de Hilbert que investiga uma cota superior para o número de ciclos limites em sistemas diferenciais polinomiais e suas posições relativas. Por outro lado, os sistemas diferenciais suaves por partes tem despertado o interesse de muitos pesquisadores recentemente devido a sua estreita relação com outras áreas das ciências como física, biologia, economia e engenharias. Portanto é natural a busca pela extensão das técnicas e ferramentas da teoria qualitativa para essa classe de sistemas. Nessa dissertação apresentamos uma generalização da técnica do averaging para uma classe especial dos sistemas de Filippov, conhecida como sistemas diferenciais contínuos por partes, desenvolvida por Llibre-Novaes-Teixeira e, aplicamos essa técnica na investigação de uma classe particular de sistemas, que chamamos do tipo Kukles generalizado. / One of the most investigated problems in the qualitative theory of dynamical systems in the plane is the XVI Hilbert\'s problem which asks for the maximum number and position of limity cycles for all planar polynomial differential systems of degree n. On the other hand, recently piecewise continuous differential systems have attracting the interest of many researches specially because of their close relation with other sciences for instance physics, biology, economy and engineering. These relations motivate extensions of the qualitative tools for this class of systems. In this work we present a generalization of the averaging theory for a class of Filippov systems, namely piecewise continuous differential systems, developed by Llibre-Novaes-Teixeira and, we apply this theory to a particular class of differential systems, which we nominate generalized Kukles type.

Yes / Simulation of the suspended sediment concentration (SSC) has great significance in predicting the sediment transport rate, vegetation growth and the river ecosystem in the vegetated open channel flows. The present study focuses on investigating the vertical SSC profile in the vegetated open channel flows. To this end, a model of the dispersive flux is proposed in which the dispersive coefficient is expressed as partitioned linear profile above or below the half height of vegetation. The double-averaging method, i.e. time-spatial average, is applied to improve the prediction accuracy of the vertical SSC profile in the vegetated open channel flows. The analytical solution of SSC in both the submerged and the emergent vegetated open channel flows is obtained by solving the vertical double-averaging sediment advection-diffusion equation. The morphological coefficient, a key factor of the dispersive coefficient, is obtained by fitting the existing experimental data. The analytically predicted SSC agrees well with the experimental measurements, indicating that the proposed model can be used to accurately predict the SSC in the vegetated open channel flows. Results show that the dispersive term can be ignored in the region without vegetation, while the dispersive term has significant effect on the vertical SSC profile within the region of vegetation. The present study demonstrates that the dispersive coefficient is closely related to the vegetation density, the vegetation structure and the stem Reynolds number, but has little relation to the flow depth. With a few exceptions, the absolute value of the dispersive coefficient decreases with the increase of the vegetation density and increases with the increase of the stem Reynolds number in the submerged vegetated open channel flows. / Natural Science Foundation of China (Nos. 11872285 and 11672213), The UK Royal Society – International Exchanges Program (IES\R2\181122) and the Open Funding of State Key Laboratory of Water Resources and Hydropower Engineering Science (WRHES), Wuhan University (Project No: 2018HLG01).

Yes / Simulation of the suspended sediment concentration (SSC) has great significance in predicting the sediment transport rate, vegetation growth and the river ecosystem in the vegetated open channel flows. The present study focuses on investigating the vertical SSC profile in the vegetated open channel flows. To this end, a model of the dispersive flux is proposed in which the dispersive coefficient is expressed as partitioned linear profile above or below the half height of vegetation. The double-averaging method, i.e. time-spatial average, is applied to improve the prediction accuracy of the vertical SSC profile in the vegetated open channel flows. The analytical solution of SSC in both the submerged and the emergent vegetated open channel flows is obtained by solving the vertical double-averaging sediment advection-diffusion equation. The morphological coefficient, a key factor of the dispersive coefficient, is obtained by fitting the existing experimental data. The analytically predicted SSC agrees well with the experimental measurements, indicating that the proposed model can be used to accurately predict the SSC in the vegetated open channel flows. Results show that the dispersive term can be ignored in the region without vegetation, while the dispersive term has significant effect on the vertical SSC profile within the region of vegetation. The present study demonstrates that the dispersive coefficient is closely related to the vegetation density, the vegetation structure and the stem Reynolds number, but has little relation to the flow depth. With a few exceptions, the absolute value of the dispersive coefficient decreases with the increase of the vegetation density and increases with the increase of the stem Reynolds number in the submerged vegetated open channel flows. / the Natural Science Foundation of China (Nos. 11872285 and 11672213), The UK Royal Society – International Exchanges Program (IES\R2\181122) and the Open Funding of State Key Laboratory of Water Resources and Hydropower Engineering Science (WRHES), Wuhan University (Project No: 2018HLG01)

This study deals with a general method for the analysis of a semi-active control technique for a fast-shunt switching system. The benefit of the semi-active system is the reduction in power consumption, which is a significant disadvantage of a fully active system compared with a passive system. A semi-active system under consideration is a semi-actively shunted piezoelectric system, which converts the strain energy into electrical energy through strong electromechanical coupling achieved though the piezoelectric phenomenon. Our proposed semi-active approach combines a PZT-based energy harvesting with a fast switching system driven by a Pulse-Width Modulated (PWM) signal. The fast switching system enables continuous adaptation of vibration energy control/harvesting by varying the PWM duty cycle. This contrasts with a conventional capacitance switching system that can only change the capacitance at discrete values. The analysis of the current piezoelectric system combined with a fast-switching system poses a considerable challenge as it contains both continuous and discrete characteristics. The study proposes an enhanced averaging method for analyzing the piecewise linear system. The simulation of the averaged system is much faster than that of the time-varying system. Moreover, the analysis derives error bounds that characterize convergence in the time domain of the averaged system to the original system. The dissertation begins with the derivation of the equations governing the physics of a piezostructure combined with an electrical switching shunt network. The results of the averaging analysis and numerical simulation are presented in order to provide a basis for estimating the structural responses that range between open- and short-circuit conditions which constitutes two limiting conditions. An experimental study demonstrates that the capacitive shunt bimorph piezostructure coupled with a single switch can be adjusted continuously by varying the PWM duty cycle. And the behavior of such hybrid system can be well predicted by the averaging analysis. / Ph. D.

Nonlinear limit cycle oscillators are common in nature and man-made equipments, for example, they occur in electronics, robotics, lasers, chemical reactions, biological systems and economical models. Such oscillators demonstrate periodic behavior with fixed frequency and amplitude independently of the system’s initial conditions. The goal of the doctoral thesis is the development and application of phase reduction and averaging methods to analyze particular nonlinear problems in self-oscillatory systems. The phase reduction method allows us to reduce the dynamic of a weakly perturbed limit cycle oscillator to a single scalar equation that defines the dynamics of the phase. This method is usually applied to the systems described by ordinary differential equations. Here this method is extended for the systems with time delay. The phase reduction method is applied to analyze the delayed feedback control (DFC) algorithm. Such an approach allows us to obtain analytical results for slightly mismatched DFC scheme and to stabilize unstable periodic orbits with topological restriction. The averaging method is applied to self-oscillatory systems driven by high-frequency periodic force. The method allows to derive the equations for the slow motion, averaged over high-frequency oscillations. Using this method the mechanism of suppression of sustained neuronal spiking under high frequency electrical stimulation is investigated. / Netiesiniai ribinio ciklo osciliatoriai dažnai sutinkami gamtoje bei žmogaus sukonstruotose sistemose, pavyzdžiui elektronikoje, robototechnikoje, lazeriuose, cheminėse reakcijose, biologinėse sistemose bei ekonominiuose modeliuose. Tokie osciliatoriai vaizduoja periodinį elgesį su fiksuotu dažniu ir amplitude nepriklausomai nuo sistemos pradinių sąlygų. Šios disertacijos tikslas yra išplėtoti ir pritaikyti fazinės redukcijos ir vidurkinimo metodus konkretiems uždaviniams spręsti. Fazinės redukcijos metodas leidžia silpnai perturbuoto ribinio ciklo osciliatoriaus dinamiką redukuoti iki vieno skaliarinio kintamojo dinamikos. Darbe fazinės redukcijos metodas išplėtotas ribinio ciklo osciliatoriams su delsa. Plačiai išnagrinėtas fazinės redukcijos taikymas chaotinėms sistemoms, kurių nestabili periodinė orbita (NPO) yra stabilizuojama uždelstuoju grįžtamuoju ryšiu (UGR). Toks priėjimas leidžia gauti analizinius rezultatus silpnai išderintai URG valdomai sistemai bei stabilizuoti NPO su topologiniu ribojimu. Vidurkinimo metodas leidžia gauti sistemos dinamiką, suvidurkintą per aukšto dažnio periodą, kai sistema yra veikiama aukšto dažnio periodine jėga. Darbe yra išnagrinėtas neuronų savųjų osciliacijų nuslopinimo mechanizmas, stimuliuojant juos aukšto dažnio elektriniu signalu.

Netiesiniai ribinio ciklo osciliatoriai dažnai sutinkami gamtoje bei žmogaus sukonstruotose sistemose, pavyzdžiui elektronikoje, robototechnikoje, lazeriuose, cheminėse reakcijose, biologinėse sistemose bei ekonominiuose modeliuose. Tokie osciliatoriai vaizduoja periodinį elgesį su fiksuotu dažniu ir amplitude nepriklausomai nuo sistemos pradinių sąlygų. Šios disertacijos tikslas yra išplėtoti ir pritaikyti fazinės redukcijos ir vidurkinimo metodus konkretiems uždaviniams spręsti. Fazinės redukcijos metodas leidžia silpnai perturbuoto ribinio ciklo osciliatoriaus dinamiką redukuoti iki vieno skaliarinio kintamojo dinamikos. Darbe fazinės redukcijos metodas išplėtotas ribinio ciklo osciliatoriams su delsa. Plačiai išnagrinėtas fazinės redukcijos taikymas chaotinėms sistemoms, kurių nestabili periodinė orbita (NPO) yra stabilizuojama uždelstuoju grįžtamuoju ryšiu (UGR). Toks priėjimas leidžia gauti analizinius rezultatus silpnai išderintai URG valdomai sistemai bei stabilizuoti NPO su topologiniu ribojimu. Vidurkinimo metodas leidžia gauti sistemos dinamiką, suvidurkintą per aukšto dažnio periodą, kai sistema yra veikiama aukšto dažnio periodine jėga. Darbe yra išnagrinėtas neuronų savųjų osciliacijų nuslopinimo mechanizmas, stimuliuojant juos aukšto dažnio elektriniu signalu. / Nonlinear limit cycle oscillators are common in nature and man-made equipments, for example, they occur in electronics, robotics, lasers, chemical reactions, biological systems and economical models. Such oscillators demonstrate periodic behavior with fixed frequency and amplitude independently of the system’s initial conditions. The goal of the doctoral thesis is the development and application of phase reduction and averaging methods to analyze particular nonlinear problems in self-oscillatory systems. The phase reduction method allows us to reduce the dynamic of a weakly perturbed limit cycle oscillator to a single scalar equation that defines the dynamics of the phase. This method is usually applied to the systems described by ordinary differential equations. Here this method is extended for the systems with time delay. The phase reduction method is applied to analyze the delayed feedback control (DFC) algorithm. Such an approach allows us to obtain analytical results for slightly mismatched DFC scheme and to stabilize unstable periodic orbits with topological restriction. The averaging method is applied to self-oscillatory systems driven by high-frequency periodic force. The method allows to derive the equations for the slow motion, averaged over high-frequency oscillations. Using this method the mechanism of suppression of sustained neuronal spiking under high frequency electrical stimulation is investigated.

Um dos mais investigados problemas na teoria qualitativa dos sistemas dinâmicos no plano é o XVI problema de Hilbert que trata dos ciclos limites. Mais precisamente, a segunda parte do referido problema questiona sobre o número máximo de ciclos limites de um sistema diferencial polinomial plano de grau n. Por ciclo limite entendemos uma órbita fechada isolada no conjunto de todas as órbitas periódicas de um sistema diferencial plano.Uma maneira clássica de obter um ciclo limite é perturbando um sistema com uma singularidade do tipo centro. Nesta dissertação apresentamos dois métodos utilizados para a análise do número de ciclos limites que bifurcam de um centro, a saber o método das integrais abelianas e o método do averaging / One of the most investigated problems in the qualitative theory of dynamical systems in the plane is the XVI Hilberts problem which deals with limit cycles. More precisely, the second part of the problem asks about the maximum number of limit cycles of a polynomial differential system of degree n. A limit cycle is a single closed orbit on the set of all periodic orbits of a differential planar system. A classic way to obtain a limit cycle is perturbing a system with a singularity of center type.In this work we discuss about two methods used to investigate the number of limit cycles which bifurcate from a center; they are known as Abelian integrals and averaging theory

L'objectif est d'élaborer un nouveau procédé de fabrication de mousses métalliques par voie de fonderie en modélisant l'infiltration et la solidification d'un métal liquide dans un milieu poreux. La modélisation est faite en deux étapes.Tout d'abord, à l'échelle locale un brin de la mousse métallique est considéré comme un tube capillaire et l'infiltration et solidification d'un métal liquide dans un moule cylindrique est étudiée. Deuxièmement,le modèle macroscopique de la solidification diffusive d'un métal liquide dans un milieu poreux est obtenu par prise de moyenne volumique. Le modèle local est codée dans un outil CFD opensource et trois études paramétriques ont été faites permettant la détermination des relations de la longueur et le temps d'infiltration en fonction de paramètres de fonctionnement. La modélisation de la solidification d’un métal liquide dans un milieu poreux est simplifié en considérant que le moule est complètement saturé par un métal liquide au repos,par suite la solidification se produit par diffusion pure (pas de convection). L'équilibre thermique local (LTE) est considéré entre les phases solide et liquide du métal tandis qu'un non équilibre thermique local (LTNE) est retenue entre la phase métallique et le moule. Les problèmes de fermeture associés ainsi que le problème macroscopique ont été résolus numériquement. / The objective of this work is to elaborate a new manufacturing process of metal foams via casting by modelling the infiltration and solidification of liquid metal inside a porous medium.However, due to the complexity of this problem the study is divided into two steps. First, at local scale one strut of the metal foam is considered as a capillary tube and the infiltration and solidification of liquid metal inside a cylindrical mould is studied. Second, a macroscopic model of diffusive solidification is derived using the volume average method. The local model is coded in an open source CFD tool and three parametric studies were done where the relations between the infiltration length and time as function of the operating parameters are determined. The modelling of the solidification of liquid metal inside a porous medium is simplified by considering that the mould is fully saturated by liquid metal at rest, solidification occurs by pure diffusion. Local thermal equilibrium (LTE) is considered between the solid and liquid phases of the metal while local thermal non equilibrium (LTNE) is retained between the metallic mixture and the mould. The associated closure problems as well as the macroscopic problem were numerically solved.

Made available in DSpace on 2014-06-11T19:26:56Z (GMT). No. of bitstreams: 0 Previous issue date: 2009-06-03Bitstream added on 2014-06-13T18:47:52Z : No. of bitstreams: 1 silvajunior_jb_me_sjrp.pdf: 533913 bytes, checksum: 2ffa5488599336df8a97baf938757756 (MD5) / Neste trabalho estudamos o Método Averaging. Este método é uma ferramenta extremamente útil para quantificar o número de ciclos limites que podem bifurcar de uma singularidade do tipo centro de um sistema de equações diferenciais. A parte inicial do trabalho apresenta a Teoria de Aproximação Assintótica e um primeiro contato com o Averaging. Posteriormente apresentamos uma versão do Averaging via a Teoria do Grau de Brouwer. Finalmente fizemos algumas aplicações do método apresentando uma cota superior para o número de ciclos limites que podem bifurcar a partir das órbitas periódicas de centros de um sistema de equações diferenciais. Além disso, mostramos através de exemplos concretos que esta cota superior pode ser realizada. / In this work we study the Averaging Method. This method is a useful tool in order to give the maximum number of limit cycles that can bifurcate from a center type singularity of a di®erential equation system. In the first part of the work we present the Asymptotic Approximation Theory and a first view of the averaging. After that, we present a version of the averaging via Brouwer Degree Theory. Finally we give some applications of this method presenting an upper bound for the number of limit cycles that can bifurcate from a center type singularity of a di®erential equation system. Moreover, we show by presenting concrete examples that this upper bound can be realized.

Orientador: Claudio Aguinaldo Buzzi / Banca: Claudio Gomes Pessoa / Banca: Luis Fernando de Osório Mello / Resumo: Nosso objetivo neste trabalho é estudar o método do averaging através do grau topológico de Brouwer e utilizá-lo para investigar o número de ciclos limites que bifurcam de uma singularidade do tipo centro quando perturbamos um sistema de equações diferenciais através de um pequeno parâmetro ε. Começaremos apresentando o método do averaging que aaprece na literatura clássica e algumas aplicações deste. Depois faremos uma breve discussão sobre o grau topológico de Brouwer, seguido do teorema do averaging que faz menção a este conceito. Finalmente, exibiremos algumas aplicações inéditas do método. / Abstract: The aim of this is to study the averaging method using the Brouwer topological degree in order to investigative the number of limit cycles that can bifurcate from a center type singularity when a differential systemas is perturbed by a small parameter ε. To this respect, initially, we present "classical" averaging method and some of its applications. So we introduce the Brouwer topological degree, followed by the averaging theorem. Finally, we show some original applications of the averaging method. / Mestre