Difference equations arise in many fields of mathematics, both as discrete analogs of continuous behavior (analysis, numerical approximations) and as independent models for discrete behavior (population dynamics, economics, biology, ecology, etc.). In recent years, many models - especially in mathematical biology - are based on higher order nonlinear difference equations. As a result, there has been much focus on the existence of periodic solutions of certain classes of these equations and the asymptotic behavior of these periodic solutions. In this dissertation, we study the existence and global attractivity of both periodic and quasiperiodic solutions of two different higher order nonlinear difference equations. Both equations arise in biological applications.

In this work, some new results to exploit the recurrence properties of quasiperiodic dynamical systems are presented by means of a two dimensional visualization technique, Recurrence Plots(RPs). Quasiperiodicity is the simplest form of dynamics exhibiting nontrivial recurrences, which are common in many nonlinear systems. The concept of recurrence was introduced to study the restricted three body problem and it is very useful for the characterization of nonlinear systems. I have analyzed in detail the recurrence patterns of systems with quasiperiodic dynamics both analytically and numerically. Based on a theoretical analysis, I have proposed a new procedure to distinguish quasiperiodic dynamics from chaos. This algorithm is particular useful in the analysis of short time series. Furthermore, this approach demonstrates to be efficient in recognizing regular and chaotic trajectories of dynamical systems with mixed phase space. Regarding the application to real situations, I have shown the capability and validity of this method by analyzing time series from fluid experiments. / In dieser Arbeit stelle ich neue Resultate vor, welche zeigen, wie man Rekurrenzeigenschaften quasiperiodischer, dynamischer Systeme für eine Datenanalyse ausnutzen kann. Die vorgestellten Algorithmen basieren auf einer zweidimensionalen Darstellungsmethode, den Rekurrenz-Darstellungen. Quasiperiodizität ist die einfachste Dynamik, die nicht-triviale Rekurrenzen zeigt und tritt häufig in nichtlinearen Systemen auf. Nicht-triviale Rekurrenzen wurden im Zusammenhang mit dem eingeschränkten Dreikörper-problem eingeführt. In dieser Arbeit, habe ich mehrere Systeme mit quasiperiodischem Verhalten analytisch untersucht. Die erhaltenen Ergebnisse helfen die Wiederkehreigenschaften dieser Systeme im Detail zu verstehen. Basierend auf den analytischen Resultaten, schlage ich einen neuen Algorithmus vor, mit dessen Hilfe selbst in kurzen Zeitreihen zwischen chaotischem und quasiperiodischem Verhalten unterschieden werden kann. Die vorgeschlagene Methode ist besonders effizient zur Unterscheidung regulärer und chaotischer Trajektorien mischender dynamischer Systeme.Die praktische Anwendbarkeit der vorgeschlagenen Analyseverfahren auf Messdaten, habe ich gezeigt, indem ich erfolgreich Zeitreihen aus fluid-dynamischen Experimenten untersucht habe.

Understanding the phenomenon of wave scattering by random media is a ubiquitous problem that has instigated extensive research in the field. This thesis focuses on wave scattering by quasiperiodic media as an alternative approach to provide insight into the effects of structural aperiodicity on the propagation of the waves. Quasiperiodic structures are aperiodic yet ordered so have attributes that make them beneficial to explore. Quasiperiodic lattices are also used to model the atomic structures of quasicrystals; materials that have been found to have a multitude of applications due to their unusual characteristics. The research in this thesis is motivated by both the mathematical and physical benefits of quasiperiodic structures and aims to bring together the two important and distinct fields of research: waves in heterogeneous media and quasiperiodic lattices. A review of the past literature in the area has highlighted research that would be beneficial to the applied mathematics community. Thus, particular attention is paid towards developing rigorous mathematical algorithms for the construction of several quasiperiodic lattices of interest and further investigation is made into the development of periodic structures that can be used to model quasiperiodic media. By employing established methods in multiple scattering new techniques are developed to predict and approximate wave propagation through finite and infinite arrays of isotropic scatterers with quasiperiodic distributions. Recursive formulae are derived that can be used to calculate rapidly the propagation through one- and two-dimensional arrays with a one-dimensional Fibonacci chain distribution. These formulae are applied, in addition to existing tools for two-dimensional multiple scattering, to form comparisons between the propagation in one- and two-dimensional quasiperiodic structures and their periodic approximations. The quasiperiodic distributions under consideration are governed by the Fibonacci, the square Fibonacci and the Penrose lattices. Finally, novel formulae are derived that allow the calculation of Bloch-type waves, and their properties, in infinite periodic structures that can approximate the properties of waves in large, or infinite, quasiperiodic media.

Periodic driving is a ubiquitous tool for controlling experimental quantum systems. When the drive fields are of comparable, incommensurate frequencies, new theoretical tools are required to treat the resulting quasiperiodic time dependence. Similarly, new and surprising phenomena of topological origin may emerge in this regime, including the quantized pumping of energy from one drive field to another. This dissertation will describe how to exploit this energy pumping to coherently translate––or boost––quantum states of a cavity in the Fock basis. This protocol enables the preparation of highly excited Fock states for use in quantum metrology––one need only boost low occupation Fock states. Energy pumping, and hence boosting, may be achieved nonadiabatically as a robust edge effect associated to an anomalous localized topological phase (ALTP) of fermions on a wire, called the quasiperiodic Floquet-Thouless energy pump (QP pump). We present a simple coupled-layer model for the QP pump, and describe the broader topological classification which characterizes its robust properties. Finally, we argue that energy pumping by the edge modes is robust to the introduction of weak interactions between fermions, making the QP pump a stable, interacting, non-equilibrium phase of matter.

In recent years, the study of metamaterials has been a subject of much interest, with acoustic metamaterials being applied to a wide range of applications. This utility is in part due to the incorporation of various elements in their design. The addition of local resonators provides greater versatility in controlling vibrations. Nonlinear elements introduce features such as discrete breathers and frequency shift. Electromechanical metamaterials have been established to have great potential for use in simultaneous energy harvesting in addition to vibration control. Furthermore, metamaterials with quasiperiodic patterning have been shown to possess useful properties such as edge-localized modes. However, no works investigate the interaction between all these elements, especially in the nonlinear regime. In this work, we investigate a unique metamaterial with local resonators, nonlinearity, electromechanical elements, and quasiperiodicity. The proposed metamaterial is examined using both analytical and numerical techniques in order to firmly establish the effects of each element. First, a nonlinear metamaterial with electromechanical local resonators is studied using the perturbation method of multiple scales, wavepacket excitation and direct integration, and specto-spatial processing techniques. The effect of the electromechanical local resonators is established for both the linear and nonlinear regimes, notably including the addition of new bandgaps and pass bands. The influence of electrical parameters on the system dynamics is explored through parametric analysis, demonstrating their use in tuning the system response. It is also shown that nonlinear phenomena such as localized solitons and frequency shift are present in the voltage response of the electromechanical metamaterial. Next, a nonlinear metamaterial with local resonators and quasiperiodicity is investigated using the method of multiple scales as well as numerical solution of the method of harmonic balance. Topological features stemming from quasiperiodicity are observed in the linear and nonlinear regimes. The presence of local resonators is shown to result in an additional, topologically trivial bandgap. The influence of quasiperiodic parameters and the source of quasiperiodicity on the system's band structure and mode shapes are established in both the linear and nonlinear regimes. Nonlinearity is also shown to affect topological features such as edge modes, resulting in amplitude dependence that can affect the localization of these modes in the nonlinear regime. Finally, a metamaterial with nonlinearity, electromechanical local resonators, and quasiperiodic patterning is modeled and investigated. Multiple configurations are examined, including different shunt circuits coupled to the electromechanical resonators and different sources of quasiperiodic patterning. It is shown that electromechanical local resonators produce two topologically trivial bandgaps, compared to the single trivial bandgap of the purely mechanical resonator. The influence of mechanical, electrical, and quasiperiodic parameters is explored to establish the effects of these parameters on bandgap formation in the linear regime. The behavior of the metamaterial in the nonlinear regime was found to be consistent with a purely mechanical system, with no adverse effects from the presence of electromechanical elements. The impact of nonlinear and quasiperiodic phenomena on energy harvesting is also investigated. Through exploration of this unique metamaterial, it is shown that beneficial features from all elements can be present at once, resulting in a versatile metamaterial with great potential for numerous applications. / Doctor of Philosophy / In recent years, the study of metamaterials has been a subject of much interest. Despite their name, metamaterials are not homogenous materials, but engineered structures designed to possess properties not found in naturally occurring materials. Many elements can be incorporated into metamaterial design, each with its own benefits. These can range from nonlinear springs, which allow the metamaterial to behave differently as its deformation increases, to electromechanical components, which convert the motion of the metamaterial into electrical voltage. While these elements have been examined individually and in certain combinations, no works examine the combination of elements proposed in this dissertation. In this work, we investigate the impact of nonlinearity, electromechanical components, and two other beneficial elements on the system's vibration response. Combinations of these elements are examined using various analysis techniques, which are used to establish the effects of each element individually as well as their interaction when combined. Multiple variations are examined for each element, such as different types of nonlinearity or different circuits attached to the electromechanical elements. This allows us to confirm the presence of valuable features exclusive to the elements incorporated into the metamaterial. Through exploration of multiple combinations of these metamaterial elements, it is shown that beneficial features from all elements can be present at once, resulting in a versatile metamaterial with great potential for numerous applications.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Desde a descoberta do estado quasicristalino por Daniel Shechtman et al. em 1984 e da fabricação por Roberto Merlin et al. de uma superrede artificial de GaAs/ AlAs em 1985 com características da sequência de Fibonacci, um grande número de trabalhos teóricos e experimentais tem relatado uma variedade de propriedades interessantes no comportamento de sistemas aperiódicos. Do ponto de vista teórico, é bem sabido que a cadeia de Fibonacci em uma dimensão se constitui em um protótipo de sucesso para a descrição do estado quasicristalino de um sólido. Dependendo da regra de inflação, diferentes tipos de estruturas aperiódicas podem ser obtidas. Esta diversidade originou as chamadas regras metálicas e devido à possibilidade de tratamento analítico rigoroso este modelo tem sido amplamente estudado. Neste trabalho, propriedades de localização em uma dimensão são analisadas considerando-se um conjunto de regras metálicas e o modelo de ligações fortes de banda única. Considerando-se o Hamiltoniano de ligações fortes com um orbital por sítio obtemos um conjunto de transformações relativas aos parâmetros de dizimação, o que nos permitiu calcular as densidades de estados (DOS) para todas as configurações estudadas. O estudo detalhado da densidade de estados integrada (IDOS) para estes casos, mostra o surgimento de plateaux na curva do número de ocupação explicitando o aparecimento da chamada escada do diabo" e também o caráter fractal destas estruturas. Estudando o comportamento da variação da energia em função da variação da energia de hopping, construímos padrões do tipo borboletas de Hofstadter, que simulam o efeito de um campo magnético atuando sobre o sistema. A natureza eletrônica dos auto estados é analisada a partir do expoente de Lyapunov (γ), que está relacionado com a evolução da função de onda eletrônica ao longo da cadeia unidimensional. O expoente de Lyapunov está relacionado com o inverso do comprimento de localização (ξ= 1 /γ), sendo nulo para os estados estendidos e positivo para estados localizados. Isto define claramente as posições dos principais gaps de energia do sistema. Desta forma, foi possível analisar o comportamento autossimilar de cadeias com diferentes regras de formação. Analisando-se o espectro de energia em função do número de geração de cadeias que seguem as regras de ouro e prata foi feito, obtemos conjuntos do tipo-Cantor, que nos permitiu estudar o perfil do calor específico de uma cadeia e Fibonacci unidimensional para diversas gerações / Since the discovery of a quasicrystalline state by Daniel Shechtman et al. in 1984 and the growth of artificial GaAs/AlAs superlattices on nonperiodic Fibonacci sequence by Roberto Merlin et al., a number of theoretical and experimental works have reported a variety of interesting physical properties of aperiodic systems. Theoretically, it is well known that in one dimension, the Fibonacci chain is a successful prototype to describe a quasicrystalline state. Depending on the in ation rule, different kinds of aperiodic structures can be obtained. This diversity originates the called metallic means, and due to the possibility of analytical and rigorous mathematical treatments the Fibonacci model has been applied by several authors. In this work, electronic localization properties are studied, taking into account a set of metallic means in one dimension. Considering a single band tight-binding Hamiltonian, a set of decimation transformations is obtained allowing the calculation of the Density of States (DOS) for all configurations. The detailed study of the Integrated Density of States (IDOS), shows the appearance of plateaux in the occupation number curve exhibiting the so-called "devil's staircase"indicating the fractal nature of the structures. Studying the behavior of the energy as a function of the hopping we derive Hofstadter butter y type patters, which simulate the effect of a magnetic field acting on the system. The electronic nature of the eigenstate is analyzed by looking at the Lyapunov exponent which is related to the evolution of the electronic wave unction along the one dimensional chain. Since it is zero for an extended state and positive for a localized one, defining the main gaps positions, it is related to the inverse of the localization length. Through a careful analysis of the Lyapunov curves it was also possible to obtain the perfect self-similarity structures for all chains. In particular,for the chains that follow the golden and silver rules, the study of the energy behavior was done by analyzing the energy spectrum as a function of the generation number of each one of the chains. The results yield Cantor-like sets, which allowed us to calculate the specific heat profile for several generations of the one-dimensional Fibonacci chain.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Desde a descoberta do estado quasicristalino por Daniel Shechtman et al. em 1984 e da fabricação por Roberto Merlin et al. de uma superrede artificial de GaAs/ AlAs em 1985 com características da sequência de Fibonacci, um grande número de trabalhos teóricos e experimentais tem relatado uma variedade de propriedades interessantes no comportamento de sistemas aperiódicos. Do ponto de vista teórico, é bem sabido que a cadeia de Fibonacci em uma dimensão se constitui em um protótipo de sucesso para a descrição do estado quasicristalino de um sólido. Dependendo da regra de inflação, diferentes tipos de estruturas aperiódicas podem ser obtidas. Esta diversidade originou as chamadas regras metálicas e devido à possibilidade de tratamento analítico rigoroso este modelo tem sido amplamente estudado. Neste trabalho, propriedades de localização em uma dimensão são analisadas considerando-se um conjunto de regras metálicas e o modelo de ligações fortes de banda única. Considerando-se o Hamiltoniano de ligações fortes com um orbital por sítio obtemos um conjunto de transformações relativas aos parâmetros de dizimação, o que nos permitiu calcular as densidades de estados (DOS) para todas as configurações estudadas. O estudo detalhado da densidade de estados integrada (IDOS) para estes casos, mostra o surgimento de plateaux na curva do número de ocupação explicitando o aparecimento da chamada escada do diabo" e também o caráter fractal destas estruturas. Estudando o comportamento da variação da energia em função da variação da energia de hopping, construímos padrões do tipo borboletas de Hofstadter, que simulam o efeito de um campo magnético atuando sobre o sistema. A natureza eletrônica dos auto estados é analisada a partir do expoente de Lyapunov (γ), que está relacionado com a evolução da função de onda eletrônica ao longo da cadeia unidimensional. O expoente de Lyapunov está relacionado com o inverso do comprimento de localização (ξ= 1 /γ), sendo nulo para os estados estendidos e positivo para estados localizados. Isto define claramente as posições dos principais gaps de energia do sistema. Desta forma, foi possível analisar o comportamento autossimilar de cadeias com diferentes regras de formação. Analisando-se o espectro de energia em função do número de geração de cadeias que seguem as regras de ouro e prata foi feito, obtemos conjuntos do tipo-Cantor, que nos permitiu estudar o perfil do calor específico de uma cadeia e Fibonacci unidimensional para diversas gerações / Since the discovery of a quasicrystalline state by Daniel Shechtman et al. in 1984 and the growth of artificial GaAs/AlAs superlattices on nonperiodic Fibonacci sequence by Roberto Merlin et al., a number of theoretical and experimental works have reported a variety of interesting physical properties of aperiodic systems. Theoretically, it is well known that in one dimension, the Fibonacci chain is a successful prototype to describe a quasicrystalline state. Depending on the in ation rule, different kinds of aperiodic structures can be obtained. This diversity originates the called metallic means, and due to the possibility of analytical and rigorous mathematical treatments the Fibonacci model has been applied by several authors. In this work, electronic localization properties are studied, taking into account a set of metallic means in one dimension. Considering a single band tight-binding Hamiltonian, a set of decimation transformations is obtained allowing the calculation of the Density of States (DOS) for all configurations. The detailed study of the Integrated Density of States (IDOS), shows the appearance of plateaux in the occupation number curve exhibiting the so-called "devil's staircase"indicating the fractal nature of the structures. Studying the behavior of the energy as a function of the hopping we derive Hofstadter butter y type patters, which simulate the effect of a magnetic field acting on the system. The electronic nature of the eigenstate is analyzed by looking at the Lyapunov exponent which is related to the evolution of the electronic wave unction along the one dimensional chain. Since it is zero for an extended state and positive for a localized one, defining the main gaps positions, it is related to the inverse of the localization length. Through a careful analysis of the Lyapunov curves it was also possible to obtain the perfect self-similarity structures for all chains. In particular,for the chains that follow the golden and silver rules, the study of the energy behavior was done by analyzing the energy spectrum as a function of the generation number of each one of the chains. The results yield Cantor-like sets, which allowed us to calculate the specific heat profile for several generations of the one-dimensional Fibonacci chain.

Made available in DSpace on 2014-12-17T15:14:50Z (GMT). No. of bitstreams: 1 RebenRMG.pdf: 3800723 bytes, checksum: a25f5d04ee938499344a1aa36b4e8be8 (MD5) Previous issue date: 2008-12-29 / Conselho Nacional de Desenvolvimento Cient?fico e Tecnol?gico / There is nowadays a growing demand for located cooling and stabilization in optical and electronic devices, haul of portable systems of cooling that they allow a larger independence in several activities. The modules of thermoelectrical cooling are bombs of heat that use efect Peltier, that consists of the production of a temperature gradient when an electric current is applied to a thermoelectrical pair formed by two diferent drivers. That efect is part of a class of thermoelectrical efcts that it is typical of junctions among electric drivers. The modules are manufactured with semiconductors. The used is the bismuth telluride Bi2Te3, arranged in a periodic sequence. In this sense the idea appeared of doing an analysis of a system that obeys the sequence of Fibonacci. The sequence of Fibonacci has connections with the golden proportion, could be found in the reproductive study of the bees, in the behavior of the light and of the atoms, as well as in the growth of plants and in the study of galaxies, among many other applications. An apparatus unidimensional was set up with the objective of investigating the thermal behavior of a module that obeys it a rule of growth of the type Fibonacci. The results demonstrate that the modules that possess periodic arrangement are more eficient / H? atualmente uma demanda crescente por resfriamento localizado e estabiliza??o de temperatura em dispositivos ?pticos e eletr?nicos, alem de sistemas de refrigera??o port?teis que permitem uma maior independ?ncia em diversas atividades. Os m?dulos de refrigera??o termoel?trica s?o bombas de calor que utilizam efeito Peltier, que consiste na produ??o de um gradiente de temperatura quando uma corrente el?trica ? aplicada a um par termoel?trico formado por dois condutores diferentes. Esse efeito faz parte de uma classe de efeitos termoel?tricos que ? t?pico de jun??es entre condutores el?tricos. Os m?dulos s?o fabricados com semicondutores. O mais utilizado ? o telureto de bismuto Bi2Te3, arranjados em uma seq??ncia peri?dica. Neste sentido surgiu a id?ia de fazer uma an?lise de um sistema que obedece a seq??ncia de Fibonacci. A seq??ncia de Fibonacci tem conex ?es com a propor??o ?urea, podendo ser encontrada no estudo reprodutivo das abelhas, no comportamento da luz e dos ?tomos, como tamb?m no crescimento de plantas e no estudo de gal?xias, dentre muitas outras aplica??es. Foi montado um aparato unidimensional com o objetivo de investigar o comportamento t?rmico de um m?dulo que obedece a uma regra de crescimento do tipo Fibonacci. Os resultados demonstram que os m?dulos que possuem arranjo peri?dico s?o mais eficientes iv

Electromagnetic wave phenomena represent a crucial factor during the forma- tion of the Earth's magnetosphere, as they are responsible for the energy trans- fer in the collisionless plasma medium. Multipoint observations of such wave phenomena are particularly useful to distinguish between spatial and temporal intensity variations. Moreover, an approximate event spatial extent and prop- agation directions can be determined. The thesis is focused on the statistical study of conjugate observations of wave phenomena called quasiperiodic (QP) emissions observed by the Van Allen Probes spacecraft and ground-based Kan- nuslehto station. Altogether, 26 simultaneously observed events were analyzed. This approach is unique in the sense that most such analyses up to date were only case studies. The thesis further presents the analysis of the dependence of properties of another wave phenomena, called magnetospheric line radiation (MLR), on the geomagnetic activity indices and solar wind parameters. Geomag- netic activity effects on the event characteristics are revealed. Finally, the effect of interplanetary shocks on the overall very low frequency (VLF) wave intensity measured by the DEMETER spacecraft is studied.

In dieser Arbeit werden zwei nichtlineare Phänomene untersucht, Multistabilität durch verzögerte Rückkopplung und Synchronisation von quasiperiodischen Oszillationen. Dies geschieht mit Hilfe von Halbleiterlasern und auf dem selben Chip wie der Laser integrierter ultrakurzer optischer Rückkopplung. Verzögerte Rückkopplung ist unter anderem die Ursache für das Phänomen der Faltung von Lasermoden, und damit für das Auftreten von mehreren möglichen Laserzuständen für die selben Parameter. Ein tristabiles Regime von Dauerstrichzuständen kann im Experiment für mehrere breite Parameterbereiche der Rückkopplung beobachtet werden. Sehr nahe der Laserschwelle wird einer der Laserzustände durch den stabilen ``aus''''-Zustand ersetzt. Theoretische Betrachtungen im Rahmen des paradigmatischen Lang-Kobayashi Models verzögerter Rückkopplung ermöglichen eine in sich konsistente Interpretation der experimentellen Ergebnisse. Neben der Beeinflussung des stationären Verhaltens eines Halbleiterlasers kann verzögerte Rückkopplung Instabilitäten in der Laseremission hervorrufen. Abhängig von Rückkoppelstärke und -phase werden zwei verschiedene Intensitätspulsationen des emittierten Lichtes beobachtet. Synchronisationsprozesse solcher Pulsationen wurden von mir in einem System von zwei verschiedenen gekoppelten Multisektionslasern untersucht. Periodische Selbstpulsationen von Laser 1 werden hierfür in Laser 2 injiziert, welcher sich in einem Regime quasiperiodischer Intentensitätspulsationen mit zwei fundamentalen Frequenzen befindet. Das Experiment zeigt eine neue Art von Übergang zu synchronem Verhalten, welche kürzlich mit Hilfe von gekoppelten generischen Phasen- und van der Pol Oszillatormodellen aufgedeckt wurde. Desweiteren konnten bislang unerforschte Prozesse des Kohärenzübertrags auch zu nichtsynchronisierten Oszillationen beobachtet werden. / In this work two nonlinear phenomena are investigated, multistability due to delayed feedback and synchronization of quasiperiodic oscillations. The experimental devices are semiconductor lasers with ultra-short optical feedback, which is integrated on the same chip as the laser. Delayed feedback causes the folding of lasing modes, leading to hysteresis effects and even the coexistence of several laser states for the same parameters. A regime of tristability of continuous-wave (cw) states is found for multiple ranges of applied currents. Very close to threshold, one of the lasing states may be replaced by the stable ``off''''-state. Theoretical investigations in the framework of the paradigmatic Lang-Kobayashi model provide a consistent understanding of the experimental findings. Besides modifying the stationary behavior of a semiconductor laser, delayed feedback can cause instabilities of the laser output. Depending on strength and phase of the feedback, two types of self-sustaining pulsations of the emitted light intensity are found in our devices. Synchronization processes of such pulsations are studied in a system of two coupled multisection lasers. Periodic self-pulsations of laser 1 are injected into laser 2, which is operating in a regime with two-frequency quasiperiodic self-pulsations. The experimental system demonstrates the new type of transitions to synchrony between three frequencies which has been recently revealed using generic coupled phase and van der Pol oscillator models. Moreover, carefully determining the coherence of the noisy oscillations, so far unexplored processes of coherence transfer to nonsynchronized oscillations are revealed.