Dissertations / Theses on the topic 'Model of coupled harmonic oscillators'

This diploma thesis deals with numerical simulations of the optical response of plasmonic infrared antennas placed on silicon substrates with thin film of silicon dioxide and subsequently with fitting of scattering spectra by model of coupled harmonic oscillators. In this work, we study an influence of length of antennas on the strength of coupling of localized surface plasmons in the antennas with phonons in silicon dioxide film. Also, the influence of silicon dioxide film thickness on this coupling is investigated.

In this thesis, the synchronization in the arrays of identical and non-identical coupled harmonic oscillators is studied. Both linear and nonlinear coupling is considered. The study consists of two main parts. The first part concentrates on theoretical analysis and the second part contains the simulation results. The first part begins with introducing the harmonic oscillators and the basics of synchronization. Then some theoretical aspects of synchronization of linearly and nonlinearly coupled harmonic oscillators are presented. The theoretical results say that linearly coupled identical harmonic oscillators synchronize for any frequency of oscillation. For nonlinearly coupled identical harmonic oscillators, synchronization is shown to occur at large enough frequency values. In the second part, the simulator and simulation results are presented. A GUI is designed in MATLAB to run the simulations. In the simulations, synchronization of coupled harmonic oscillators are studied according to different coupling strength values, different frequency values, different coupling graph types (e.g. all-to-all, ring, tree) and different coupling function types (e.g. linear, saturation, cubic). The simulation results do not only support the theoretical part of the thesis but also give some idea about the part of the synchronization of coupled harmonic oscillators uncovered by theory.

When a quantum system interacts with many other quantum mechanical objects, the behaviour of the system is strongly affected; this is referred to as an open quantum system (OQS). Since the inception of quantum theory the development of OQSs has been synonymous with realistic descriptions of quantum mechanical models. With recent activity in the advancement of quantum technologies, there has been vested interest in manipulating OQSs. Therefore understanding and controlling environmental effects, by structuring environments, has become an important field. The method of choice for tackling OQSs is the master equation approach, which requires approximations and doesn't allow direct assessment of the environment. This thesis tackles the issues of OQSs with an unorthodox method; we employ a series of coupled quantum harmonic oscillators to simulate an OQS. This permits the use of the covariance matrix technique which allows us to avoid approximations and analyse the environment modes. We investigate the Markov approximation and Rotating-Wave approximation (RWA), which are commonly used in the field. By considering four OQS models, we study an entanglement-based non-Markovian behaviour (NMB) quantifier (ENMBQ). The relevance of detuning, coupling strength and bath structures in determining the amount of NMB is noted. A brief study of the factors that affect a fidelity-based NMB quantifier is also conducted. We also analyse the effect on the ENMBQ if the terms excluded by the RWA are included in the models. Finally, an examination of the applicability of the RWA in the presence of strong coupling is undertaken in a three oscillator model. The fidelity-based analysis utilised could allow one to ascertain when and if the RWA can be applied to a model of interest, including OQSs. The knowledge within, and the methodology used throughout this thesis, could arm researchers with insights to control the flow of quantum information in their systems.

Thesis advisor: Jan R. Engelbrecht
Thesis advisor: Renato E. Mirollo
Many phenomena in nature that involve ordering in time can be understood as collective behavior of coupled oscillators. One paradigm for studying a population of self-sustained oscillators is the Kuramoto model, where each oscillator is described by a phase variable, and interacts with other oscillators through trigonometric functions of phase differences. This dissertation studies $N$ identical Kuramoto oscillators in a general form \[ \dot{\theta}_{j}=A+B\cos\theta_{j}+C\sin\theta_{j}\qquad j=1,\dots,N, \] where coefficients $A$, $B$, and $C$ are symmetric functions of all oscillators $(\theta_{1},\dots,\theta_{N})$. Dynamics of this model live in group orbits of M\"obius transformations, which are low-dimensional manifolds in the full state space. When the system is a phase model (invariant under a global phase shift), trajectories in a group orbit can be identified as flows in the unit disk with an intrinsic hyperbolic metric. A simple criterion for such system to be a gradient flow is found, which leads to new classes of models that can be described by potential or Hamiltonian functions while exhibiting a large number of constants of motions. A generalization to extended phase models with non-identical couplings gives rise to richer structures of fixed points and bifurcations. When the coupling weights sum to zero, the system is simultaneously gradient and Hamiltonian. The flows mimic field lines of a two-dimensional electrostatic system consisting of equal amounts of positive and negative charges. Bifurcations on a partially synchronized subspace are discussed as well
Thesis (PhD) — Boston College, 2017
Submitted to: Boston College. Graduate School of Arts and Sciences
Discipline: Physics

A prominent feature of the dynamics of large neuronal networks are the synchrony-driven collective oscillations generated by the interplay between synaptic coupling and synaptic delays. This thesis investigates the emergence of delay-induced oscillations in networks of heterogeneous spiking neurons. Building on recent theoretical advances in exact mean field reductions for neuronal networks, this work explores the dynamics and bifurcations of an exact firing rate model with various forms of synaptic delays. In parallel, the results obtained using the novel firing rate model are compared with extensive numerical simulations of large networks of spiking neurons, which confirm the existence of numerous synchrony-based oscillatory states. Some of these states are novel and display complex forms of partial synchronization and collective chaos. Given the well-known limitation of traditional firing rate models to describe synchrony-based oscillations, previous studies greatly overlooked many of the oscillatory states found here. Therefore, this thesis provides a unique exploration of the oscillatory scenarios found in neuronal networks due to the presence of delays, and may substantially extend the mathematical tools available for modeling the plethora of oscillations detected in electrical recordings of brain activity.
Una característica fonamental de la dinàmica d'una xarxa neuronal és l'emergència d'oscil·lacions degudes a sincronització. L'origen d'aquestes oscil·lacions és molt sovint degut les interaccions sinàptiques i als seus retards temporals inherents. Aquesta tesi analitza la emergència d'oscil·lacions produïdes per retards sinàptics en xarxes neuronals heterogènies. A partir de troballes recents en teories de camp mig per xarxes neuronals, aquest treball explora la dinàmica i les bifurcacions d'un model de {\it rate} amb diferents tipus de retards sinàptics. En paral·lel els resultats obtinguts mitjançant el nou model de rate són comparats amb simulacions numèriques de grans xarxes neuronals. Aquestes simulacions confirmen l'existència de nombrosos estats oscil·latoris produïts per sincronització. Alguns d'aquests estats són nous I mostren formes complexes de sincronització parcial i de caos col·lectiu. Gran part d'aquestes oscil·lacions han estat àmpliament ignorades a la literatura, degut a la limitació dels models tradicionals de rate per descriure estats amb un alt nivell de sincronització. Així doncs aquesta tesi ofereix una exploració única dels possibles escenaris oscil·latoris en xarxes neuronals amb retards sinàptics, i amplia significativament les eines matemàtiques disponibles per a la modelització de la gran diversitat d'oscil·lacions neuronals presents en les mesures elèctriques de l'activitat cerebral.

On s’intéresse dans cette thèse à des systèmes couplés de type champ moyen en étudiant l’existence de l’état de synchronisation qui se caractérise par une distance uniformément bornée dans le temps entre chaque paire de composantes d’une solution. L’étude se base sur une méthode perturbative. Néanmoins les résultats obtenus ne sont pas évidents dans le cas non-perturbé. En outre dans le cas où le système couplé est périodique et grâce au Théorème du point fixe on montre l’existence d’une solution périodique sur le tore. L’étude de stabilité et de stabilité exponentielle est établie dans le cas linéaire et appliquée à ce type de systèmes couplés
We study in this thesis a class of a perturbed interconnected mean-field system, also known as a coupled systems. Under some assumptions we prove the existence of an invariant open set by the flow of the perturbed system ; in other word, we prove that the distance between the components of an orbit is uniformly bounded, this property is also called synchronization. We use the perturbation method to obtain the result. However the result is not trivial for the not perturbed system. We use the fixed point theorem to prove the existence of a periodic orbit in the torus. We study in addition the stability and the exponential stability of such systems by studying the stability of a linear systems

Le cycle de division cellulaire et l'horloge circadienne sont deux processus fondamentaux de la régulation cellulaire qui génèrent une expression rythmique des gènes et des protéines. Dans les cellules mammifères, les mécanismes qui sous-tendent les interactions entre le cycle cellulaire et l'horloge restent très mal connus. Dans cette thèse, nous étudions ces deux oscillateurs biologiques, à la fois individuellement et en tant que système couplé, pour comprendre et reproduire leurs principales propriétés dynamiques, détecter les composants essentiels du cycle cellulaire et de l'horloge, et identifier les mécanismes de couplage. Chaque oscillateur biologique est modélisé par un système d'équations différentielles ordinaires non linéaires et ses paramètres sont calibrés par rapport à des données expérimentales: le modèle du cycle cellulaire se base sur les modifications post-traductionnelles du complexe Cdk1-CycB et mène à un oscillateur de relaxation dont la dynamique et la période sont contrôlés par les facteurs de croissance; le modèle de l'horloge circadienne reproduit l'oscillation antiphasique BMAL1/PER:CRY et l'adaptation de la durée des états d'activation et répression par rapport à deux signaux d’entrée hormonaux déphasés. Pour analyser les interactions entre les deux oscillateurs nous étudions la synchronisation des deux rythmes pour des régimes de couplage uni- ou bi-directionnels. Les simulations numériques reproduisent les ratios entre les périodes de l'horloge et du cycle cellulaire, tels que 1:1, 3:2 et 5:4. Notre étude suggère des mécanismes pour le ralentissement du cycle cellulaire avec des implications pour la conception de nouvelles chronothérapies
The cell division cycle and the circadian clock are two fundamental processes of cellular control that generate cyclic patterns of gene activation and protein expression, which tend to be synchronous in healthy cells. In mammalian cells, the mechanisms that govern the interactions between cell cycle and clock are still not well identified. In this thesis we analyze these two biological oscillators, both separately and as a coupled system, to understand and reproduce their main dynamical properties, uncover essential cell cycle and clock components, and identify coupling mechanisms. Each biological oscillator is first modeled by a system of non-linear ordinary differential equations and its parameters calibrated against experimental data: the cell cycle model is based on post-translational modifications of the mitosis promoting factor and results in a relaxation oscillator whose dynamics and period are controlled by growth factor; the circadian clock model is transcription-based, recovers antiphasic BMAL1/PER:CRY oscillation and relates clock phases to metabolic states. This model shows how the relative duration of activating and repressing molecular clock states is adjusted in response to two out-of-phase hormonal inputs. Finally, we explore the interactions between the two oscillators by investigating the control of synchronization under uni- or bi-directional coupling schemes. Simulations of experimental protocols replicate the oscillators’ period-lock response and recover observed clock to cell cycle period ratios such as 1:1, 3:2 and 5:4. Our analysis suggests mechanisms for slowing down the cell cycle with implications for the design of new chronotherapies

碩士
國立成功大學
物理學系
102
Peres-Horodecki-Simon criterion and logarithmic negativity are very powerful tools to determine the separability and to measure the entanglement of Gaussian states. In this thesis, we set up several models, all of which comprise a number of coupled oscillators, and by facilitating the separability criterion and measure we're able to calculate the entanglement between each pair of oscillators at any time analytically, which reveals several interesting phenomena, including entanglement sudden death and revival of entanglement. Also, we compare the entanglement between center of mass coordinates and that of their member oscillators, and thereby understand the role of it in a composite system. Lastly, we'll make an attempt at appreciating the effects of particle numbers on entanglement. We hope these analytically solvable models can help us understand more about the entanglement of interacting systems and of large systems.

Frequency synthesizers are essential components for modern wireless and wireline communication systems as they provide the local oscillator signal required to transmit and receive data at very high rates. They are also vital for computing devices and microcontrollers as they generate the clocks required to run all the digital circuitry responsible for the high speed computations. Data rates and clocking speeds are continuously increasing to accommodate for the ever growing demand on data and computational power. This places stringent requirements on the performance metrics of frequency synthesizers. They are required to run at higher speeds, cover a wide range of frequencies, provide a low jitter/phase noise output and consume minimum power and area. In this work, we present new techniques and architectures for implementing high speed frequency synthesizers which fulfill the aforementioned requirements. We propose a new architecture and design approach for the realization of wideband millimeter-wave frequency synthesizers. This architecture uses two-step multi-order harmonic generation of a low frequency phase-locked signal to generate wideband mm-wave frequencies. A prototype of the proposed system is designed and fabricated in 90nm Complementary Metal Oxide Semiconductor (CMOS) technology. Measurement results demonstrated that a very wide tuning range of 5 to 32 GHz can be achieved, which is costly to implement using conventional techniques. Moreover the power consumption per octave resembles that of state-of-the art reports. Next, we propose the N-Push cyclic coupled ring oscillator (CCRO) architecture to implement two high performance oscillators: (1) a wideband N-Push/M-Push CCRO operating from 3.16-12.8GHz implemented by two harmonic generation operations using the availability of different phases from the CCRO, and (2) a 13-25GHz millimeter-wave N-Push CCRO with a low phase noise performance of -118dBc/Hz at 10MHz. The proposed oscillators achieve low phase noise with higher FOM than state of the art work. Finally, we present some improvement techniques applied to the performance of phase locked loops (PLLs). We present an adaptive low pass filtering technique which can reduce the reference spur of integer-N charge-pump based PLLs by around 20dB while maintaining the settling time of the original PLL. Another PLL is presented, which features very low power consumption targeting the Medical Implantable Communication Standard. It operates at 402-405 MHz while consuming 600microW from a 1V supply.

碩士
國立成功大學
物理學系碩博士班
96
Synchronization is a natural phenomenon which is an active research topic. Many models were published to approximate these phenomena. In one of these models, all basic entities are considered as identical oscillators with pulse-coupled interaction among them. We use a model to approximate these synchronous phenomena. And we discuss the conditions of two, three and N oscillators. Finally, we find some compatible domains of parameters to make all oscillators be synchronous for all initial conditions.

The field of coupled resonant systems is a rich research area with enumerable real-world applications, including the fields of neural computing and pattern recognition, energy harvesting, and even modeling the behavior of certain types of biological systems. This work is primarily focused on the study of the behaviors of two subsets of this field: large networks of globally coupled resonators (which, in this work, refers to passive, damped resonant elements which require external stimulus) and smaller networks of oscillators (referring to active devices capable of self-sustained motion), which are coupled through a network of light-sensitive resistive elements. In the case of the former, we begin by developing an analytical and experimental framework to examine the behaviors of this system under various conditions, such as different coupling modalities and element-level parametric mistunings. Once a proper understanding of the dynamics of these systems has been established, we go on to develop the system into a single-input, single-output, multi-analyte volatile organic compound sensor. For the study of oscillator networks, we begin by building a device which utilizes a network of Colpitts oscillators, coupled through a series of color-filtered CdSe photocells. We then establish that through the analysis of particular emergent behaviors (most notably, frequency locking within the network), this type of system may show promise as a threshold color sensor. By exploiting these behaviors, this type of system may find applications in neuromorphic computing (particularly in optical pattern recognition).

An innovative mathematical architecture for modelling neuronal electrical activity is presented, called the cognitive rhythm generator (CRG), wherein the proposed architecture is a hybrid model comprised of three interconnected stages, namely: (1) a bank of neuronal modes; (2) a ring device (limit-cycle oscillator); and (3) a static output nonlinearity (mapper). Coupled CRG networks are employed to emulate and elucidate the dynamics of biological neural networks, including the recurrent networks in the hippocampus. Several species of ring devices are described and investigated, including the clock, labile clock, hourglass and multistable ring systems, and their applications to neuronal modelling explored. Complexity measures such as the maximum Lyapunov exponent, correlation dimension and detrended fluctuation analysis are applied to compare model and biological records and validate the CRG methodology. The basis of neural coding is also examined in mathematical detail, with particular regard to its description by Volterra-Wiener kernel formalism, from which the neuronal modes are derived. Applications to theta-gamma coding are discussed. Further on in the thesis, a CRG epileptiform network model of spontaneous seizure-like events (SLEs) is developed and used as a platform to test neuromodulation approaches for seizure abatement. (Neuromodulation mentioned here refers to methods involving electrical stimulation of neural tissue for therapeutic benefit). Spontaneous SLE transitions in the epileptiform network are shown to be related to the mechanism of intermittency, as determined by examining the state space dynamics of the model. The onset of SLEs is associated with increased network excitability and decreased stability, consistent with experimental results from the low-magnesium/high-potassium in vitro model of epilepsy. Lastly, a novel strategy for therapeutic neuromodulation is presented wherein a coupled CRG network (called the “therapeutic network”) is interfaced with the epileptiform network model, forming a closed loop for responsive, biomimetic neuromodulation of the epileptiform network. Relevance to clinical applications and future work is discussed.