Struktur selbsterregter nichtlinearer Dichtewellen in staubigen Plasmen unter Schwerelosigkeit

In this thesis, dust-density waves are investigated in an extended three-dimensional dusty plasma under microgravity conditions in a radio-frequency discharge. The waves emerge spontaneously at low neutral gas pressures and high dust densities in the presence of streaming ions. They are often nonlinear and the wave field comprises a complicated structure, which is reflected, e.g., in the appearance of so-called topological defects, i.e., locations of splitting or merging wave fronts. The aim of the work is to examine this spatio-temporal structure of the waves in detail. For this, the established methods like, e.g., the Fourier analysis are only suitable to a limited extent due to the transient behavior of the waves. Thus, based on the Hilbert transform, a new workflow has been introduced, which allows to analyze the waves on different, individually chosen time scales. In particular, the method can be used to define instantaneous quantities such as the phase or the instantaneous frequency of the wave. With this, the wave field can be reconstructed at each recorded time step and the position and motion of the defects can be determined. Moreover, it is possible to investigate the global wave properties by averaging the instantaneous quantities temporally. In this work, the spatial frequency distribution of the waves is investigated primarily. It turns out that in most cases the frequency varies in space. Surprisingly, the observed decrease is not constant. Instead, so-called frequency clusters have been discovered. These are regions of almost constant frequency, which are separated by abrupt frequency jumps. Such an observation was not made so far in the field of dusty plasmas. It is not compatible with Huygens' linear wave theory, which involves a constant frequency but a varying wavelength. Furthermore, it has been found that both analyzed phenomena -defects and frequency clusters- cannot be observed independently. Rather, they are closely related to each other since the defects move almost exclusively along the cluster boundaries. A detailed analysis of the instantaneous frequency further reveals hints for an incomplete synchronization at the cluster boundaries as it is typical for isolated driven van-der-Pol oscillators. This result and a comparison with numerical studies suggest to model the waves as a system of coupled self-sustained van-der-Pol oscillators. For this purpose, complementary numerical investigations are realized within the scope of this work, which in particular include the behavior of the oscillators at the cluster boundaries. There, it is found that the oscillators affect each other in terms of so-called frequency pulling. The results of this thesis lead to a modified picture of dust-density waves, which treats the dust cloud as an ensemble of mutually interacting nonlinear self-sustained oscillators that represents the unstable saturated wave field.In dieser Dissertation werden Dichtewellen in einem ausgedehnten dreidimensionalen staubigen Plasma unter Schwerelosigkeit in einer Hochfrequenzentladung untersucht. Die Wellen treten bei niedrigen Neutralgasdrücken und hohen Staubdichten in Gegenwart strömender Ionen spontan auf. Sie sind zumeist nichtlinear und das Wellenfeld besitzt eine komplizierte Struktur, die sich beispielsweise durch das Auftreten von sogenannten topologischen Defekten, d.h. Orten, an denen sich Wellenfronten aufteilen oder vereinigen, bemerkbar macht. Ziel der Arbeit ist es, diese raum-zeitliche Struktur der Wellen detailliert zu untersuchen. Die etablierten Methoden, wie z.B. die Fourier-Analyse, sind dafür jedoch aufgrund des transienten Verhaltens der Wellen nur bedingt geeignet. Basierend auf der Hilbert-Transformation ist daher ein neues Verfahren eingeführt worden, welches eine Analyse auf unterschiedlichen, frei wählbaren Zeitskalen erlaubt. Mit dieser Methode lassen sich insbesondere instantane Größen, wie beispielsweise Phase und instantane Frequenz der Welle, definieren. So kann das Wellenfeld für jeden aufgenommenen Zeitschritt rekonstruiert und Position und Bewegung der Defekte bestimmt werden. Außerdem ist es durch eine zeitliche Mittelung der instantanen Größen möglich, die globalen Welleneigenschaften zu untersuchen. In dieser Arbeit wird primär die räumliche Frequenzverteilung der Wellen untersucht. Dabei zeigt sich, dass die Frequenz in den meisten Fällen räumlich variiert. Überraschenderweise erfolgt der beobachtete Abfall nicht stetig. Stattdessen sind sogenannte Frequenzcluster entdeckt worden. Dies sind Bereiche annähernd konstanter Frequenz, die durch abrupte Frequenzsprünge voneinander getrennt sind. Eine solche Beobachtung wurde in staubigen Plasmen zuvor nicht gemacht. Sie ist nicht mit dem linearen Wellenbild nach Huygens vereinbar, welches lediglich eine variable Wellenlänge zulässt. Weiterhin stellt sich heraus, dass die beiden analysierten Phänomene -- Defekte und Frequenzcluster -- nicht getrennt voneinander zu beobachten sind. Sie sind vielmehr eng miteinander verknüpft, da sich die Defekte fast ausschließlich entlang der Clustergrenzen bewegen. Eine detaillierte Analyse der instantanen Frequenz gibt ferner Hinweise auf eine unvollständige Synchronisation an den Clustergrenzen, wie es für isolierte, getriebene van-der-Pol Oszillatoren bekannt ist. Dieses Resultat und ein Vergleich der experimentellen Befunde mit numerischen Studien legen nahe, die Wellen als ein System gekoppelter selbsterregter van-der-Pol Oszillatoren zu modellieren. Dafür werden in dieser Arbeit komplementäre numerische Untersuchungen durchgeführt, die insbesondere das Verhalten der Oszillatoren an den Clustergrenzen umfassen. Es zeigt sich, dass sich die Oszillatoren dort gegenseitig durch sogenanntes Frequency-Pulling beeinflussen. Die Ergebnisse dieser Arbeit führen zu einem modifizierten Bild der Staubdichtewellen, welches die Staubwolke als ein Ensemble miteinander wechselwirkender nichtlinearer und selbsterregter Oszillatoren ansieht, das das instabile, gesättigte Wellenfeld repräsentiert

Studies in Nonlinear and Stochastic Phenomena and Quality Factor Enhancement in a Nanomechanical Resonator

University of Minnesota M.S.M.E. thesis.July 2019. Major: Mechanical Engineering. Advisor: Subramanian Ramakrishnan. 1 computer file (PDF); xiii, 117 pages.Nonlinear damping has recently been experimentally observed in carbon nanotube and graphene-based nanoelectromechanical (NEMS) resonators and shown to be an effective means to achieve higher quality (Q) factors. Moreover, it has been shown that white noise excitation can be exploited to shrink the resonance width of the frequency response characteristics of the resonator as a pathway to higher Q factors. Motivated thus, this thesis is a study of certain fundamental characteristics of the nonlinear dynamics of a nanoelectromechanical resonator in both the deterministic and stochastic regimes with a focus on the influence of those characteristics on the Q factor. Using a Duffing oscillator based model, this thesis: (1) derives an analytical expression between oscillation amplitude and frequency of a NEMS resonator using the harmonic balance method to study the frequency response characteristics and validates the results using numerical simulation, (2) studies the deterministic dynamics of a NEMS resonator deriving an analytical relationship between the phase angle and maximum oscillation of the resonator response, (3) derives an analytical expression between the resonance frequency and resonance amplitude, (4) studies the hysteresis characteristics both in the stochastic and deterministic regimes elucidating the effects of nonlinear damping and external excitation on the hysteresis region, (5) finds that stochastic excitation with increasing intensity can shrink the hysteresis width, (6) shows that increasing the magnitude of the linear damping coefficient results in the decrease of Q-factors, (7) shows that in the combined presence of both parametric and external excitation, increasing the ratio of pump frequency to external forcing frequency results in lower resonant frequency and lower resonance width, (8) observes that in the parametrically driven nanomechanical resonator, higher parametric oscillation amplitude increases the resonance amplitude with a small impact on the resonance frequency, (9) solves the stochastic model using the Euler-Maruyama method and generates frequency response curves where it is found that higher noise intensity of Levy stable stochastic process can increase the Q factor, (10) finds that the Q factor is increased by decreasing the nonlinear damping and external harmonic driving amplitude. In summary, this thesis presents a set of novel results on the nonlinear, stochastic dynamics of a NEMS resonator and discusses the implications of the results for achieving enhanced Q factors. The results are of interest both from a theoretical viewpoint as well as in sensing applications using a nanoresonator

Phase and amplitude dynamics of quantum self-oscillators

Self-oscillators form a special class of oscillators, generating and maintaining a periodic motion while having some (or complete) independence of the frequency spectrum of oscillations from the spectrum of their power source. Pendulum clocks, brain neurons, fireflies, and cardiac pacemaker cells, are all examples of self-oscillators. Self-oscillations are not limited to the regime of classical physics, but are seen in the quantum regime as well. In both regimes, self-oscillators may demonstrate two intriguing phenomena: (1) Synchronization, a phenomenon in which self-oscillators adjust their rhythm due to weak coupling to a drive or to another self-oscillating systems; (2) Amplitude death, a phenomenon in which two or more coupled self-oscillators approach a stable rest-state. In the work presented in this thesis, we have mostly investigated these phenomena in quantum self-oscillators. Chapter 2 tries to answer the question ``Are there quantum effects in the synchronization phenomenon, which cannot be modeled classically?" Using a quantum model of a self-oscillator with nonlinearity in its energy spectrum, we have answered this question in the affirmative. We have demonstrated that the anharmonic, discrete energy spectrum of the oscillator leads to multiple resonances in both phase locking and frequency entrainment. Coupling two quantum anharmonic self-oscillators, we show in Ch. 3 that genuine quantum effects are also expected in the amplitude death phenomenon. This is apparent in the multiple resonances of the mean phonon number of the oscillators, reflecting their quantized nature. Chapter 4 is concerned with the investigation of the synchronization phenomenon in an experimental system, an optomechanical cell coupled to a drive. In the classical parameter regime, we derive analytical Adler equations describing the synchronization of the optomechanical cell to two different drives: (1) an optical drive and (2) a mechanical drive. We demonstrate numerically that synchronization should also be observed in the quantum parameter regime. In Ch. 5 we describe our work in the field of Cooper pair splitters, a device consisting of two quantum dots side-coupled to a conventional superconductor. In this work, we go beyond the standard approximation of assuming the quantum dots to have a large charging energy. We derive a low-energy Hamiltonian describing the system, and suggest a scheme for the generation of a spin triplet state shared between the quantum dots, therefore extending the capabilities of the Cooper pair splitter to create entangled nonlocal electron pairs

Various Dynamical Regimes, and Transitions from Homogeneous to Inhomogeneous Steady States in Nonlinear Systems with Delays and Diverse Couplings

This dissertation focuses on the effects of distributed delays modeled by \u27weak generic kernels\u27 on the collective behavior of coupled nonlinear systems. These distributed delays are introduced into several well-known periodic oscillators such as coupled Landau-Stuart and Van der Pol systems, as well as coupled chaotic Van der Pol-Rayleigh and Sprott systems, for a variety of couplings including diffusive, cyclic, or dynamic ones. The resulting system is then closed via the \u27linear chain trick\u27 and the linear stability analysis of the system and conditions for Hopf bifurcations that initiate oscillations are investigated. A variety of dynamical regimes and transitions between them result. As an example, in certain cases the delay produces transitions from amplitude death (AD) or oscillation death (OD) regimes to Hopf bifurcation-induced periodic behavior, where typically we observe the delayed limit cycle shrinking or growing as the delay is varied towards or away from the bifurcation point respectively. The conditions for transition between AD parameter regimes and OD parameter regimes are investigated for systems in which OD is possible. Depending on the coupling, these transitions are mediated by pitchfork or transcritical bifurcations. The systems are then investigated numerically, comparing with the predictions from the linear stability analysis and previous work. In several cases the various transitions among AD, OD and periodic domains that we observe are more intricate than the simple AD states, and the rough boundaries of the parameter regimes where they occur, which have been predicted by linear stability analysis and also experimentally verified in earlier work. The final chapter extends these studies by including the effects of periodically amplitude modulated distributed delays in both position and velocity. The existence of quasiperiodic solutions motivates the derivation of a second slow flow, together with a comparison of results and predictions from the second slow flow and the numerical results, as well as using the second slow flow to approximate the radii of the toroidal attractor. Finally, the effects of varying the delay parameter are briefly considered