Numerical Verification of an Analytical Model for Phase Noise in MEMS Oscillators.

A new analytical formulation for phase noise in MEMS oscillators was recently presented encompassing the role of essential nonlinearities in the electrical and mechanical domains. In this paper, we validate the effectiveness of the proposed analytical formulation with respect to the unified theory developed by Demir et al. describing phase noise in oscillators. In particular, it is shown that, over a range of the second-order mechanical nonlinear stiffness of the MEMS resonator, both models exhibit an excellent match in the phase diffusion coefficient calculation for a square-wave MEMS oscillator.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/TUFFC.2016.257536

Analytical Method for Computation of Phase-Detector Characteristic

Discovery of undesirable hidden oscillations, which cannot be found by simulation, in models of phase-locked loop (PLL) showed the importance of development and application of analytical methods for the analysis of such models. Approaches to a rigorous nonlinear analysis of analog PLL with multiplier phase detector (classical PLL) and linear filter are discussed. An effective analytical method for computation of multiplier/mixer phase-detector characteristics is proposed. For various waveforms of high-frequency signals, new classes of phase-detector characteristics are obtained, and dynamical model of PLL is constructed

NEAR-GRAZING AND NOISE-INFLUENCED DYNAMICS OF ELASTIC CANTILEVERS WITH NONLINEAR TIP INTERACTION FORCES

Within this dissertation work, numerical, analytical, and experimental studies are conducted with macro-scale and micro-scale elastic structures in the presence of nonlinear force interactions. The specific physical systems explored within this work are an atomic force microscope (AFM) micro-cantilever and a macro-scale cantilever experiencing similar tip interaction forces as the AFM cantilever operated in tapping mode. The tip sample forces in an AFM operation are highly nonlinear, with long-range attractive forces and short-range repulsive forces. In the macro-scale case, magnetic attractive forces and repulsive forces, which arise due to impacts with a compliant surface are used to generate similar nonlinear tip interaction forces. For elastic structures subjected to off-resonance base excitations, bifurcations close to grazing events are studied in detail, and the observed nonlinear phenomena are found to be common across the considered length scales. The dynamics of the considered systems are studied with a reduced-order computational model based on Galerkin projection with a single mode approximation. Along with studies on the bifurcation behavior, the effects of added Gaussian white noise on the system dynamics are also examined. Non-smooth system dynamics is studied by constructing local maps near the discontinuity. Period-doubling events are examined by using Poincaré maps and discontinuity mapping analysis. An important component of this dissertation research is the investigations into the effects of noise on the dynamics of these structures. Experimental and numerical efforts are used to examine the stochastic dynamics of the cantilever structures when a random component is added to the harmonic input. The noise effects are studied when the excitation frequency is close to a system resonance as well as when it is off-resonance. An analytical-numerical method with moment evolution equations is used to study the effects of noise. The effects of noise on contact and adhesion phenomena are explored. Through this dissertation work, the importance of considering noise-influenced dynamics in micro-scale applications such as AFM operations is illustrated. In addition, this work helps shed light on universality of nonlinear phenomenon across different length scales

Characterization of causes of signal phase and frequency instability Final report

Characteristic instabilities in phase and frequency errors of reference oscillator

Dynamics of coupled mechanical oscillators with friction

Complex phenomena such as stick-slip vibrations, chaos and self-organized dynamics are frequently encountered in several mechanical systems with friction. Some applications include control of robot manipulators, distribution of earthquakes, suspension dynamics in vehicles, among others. These systems are strongly nonlinear. Spring-mass oscillators with friction have emerged as a simple jet effective model capturing the dynamics of much more complex system. In this dissertation, we study stability and dynamics of single and coupled mechanical oscillators with friction, mathematically described by differential equations with discontinuous right-hand sides. One particular problem in discontinuous systems is the computation of the basins of attraction of their stable equilibria or other attractors; for example, they provide important information about complex behavior caused by friction or damping, useful in the design of mechanical devices. To cope with this problem, we implemented an algorithm for the computation of basins of attraction in discontinuous systems based on the Simple Cell Mapping method, which has been evaluated via a set of representative applications. In the second part of the thesis, a piecewise smooth analysis of two coupled oscillators was carried out, finding out some conditions that guarantee the stability of the sliding dynamics in the presence of one or more intersecting surfaces. Finally, the dynamics of a network of $N$ mechanical oscillators was studied from the point of view of synchronization, where the goal was to steer the positions and velocities of each oscillator in the network towards a common behavior. In particular, an extensive numerical analysis for studying synchronization in chaotic friction oscillators was performed, characterizing the influence of dynamic coupling and providing an estimation of the synchronization region in terms of the coupling parameters. Initially, we considered the simple case of two coupled oscillators, then we extended the analysis to the case of larger networks of coupled systems with different network topologies. Moreover, preliminary analytical results of the convergence on a network of $N$ friction oscillators based on contraction analysis are investigated. The results were also validated through a representative example

Deterministic and stochastic dynamics of multi-variable neuron models : resonance, filtered fluctuations and sodium-current inactivation

Neurons are the basic elements of the networks that constitute the computational units of the brain. They dynamically transform input information into sequences of electrical pulses. To conceive the complex function of the brain, it is crucial to understand this transformation and identify simple neuron models which accurately reproduce the known features of biological neurons. This thesis addresses three different features of neurons. We start by exploring the effect of subthreshold resonance on the response of a periodically forced neuron using a simple threshold model. The response is studied in terms of an implicit one-dimensional time map that corresponds to the Poincar´e map of the forced system. Qualitatively distinct responses are found, including mode locking and chaos. We analytically find the stability regions of mode-locking solutions, and identify the transition to chaos through period-adding bifurcations. We show that the response becomes chaotic when the forcing frequency is close to the resonant frequency. Then we will consider an experimentally verified model with realistic spikegenerating mechanism and study the effect of filtered synaptic fluctuations on the firing-rate response of the neuron. Using a population density method as well as an efficient numerical method, we find the steady-state firing rate in two limits of fast and slow synaptic inputs and present the linear response theory for the firing rate of the model in response to both time-dependent mean inputs and time-dependent noise intensity. Finally, a novel model is introduced that incorporates threshold variability of neurons. We determine the modulation of the input-output properties of the model due to oscillatory inputs and in the presence of filtered synaptic fluctuations.EThOS - Electronic Theses Online ServiceUniversity of WarwickOverseas Research Students Awards Scheme (ORSAS)GBUnited Kingdo

Asteroseismology of 20,000 Kepler Red Giants

Asteroseismology is the study of stellar pulsations. Over the past decade, asteroseismology, in particular, the study of the solar-like oscillations, has entered its golden age and has caught attention across various fields, including exoplanet science and Galactic archaeology. With high-quality data from the CoRoT, Kepler, and TESS space missions, asteroseismology has been demonstrated to be an essential tool to characterise stars and probe stellar internal structure. In this thesis, I apply asteroseismic analyses to Kepler red giants showing a rich spectrum of solar-like oscillations, which are stochastically excited and intrinsically damped by surface turbulence. First of all, I give a brief introduction to some critical phases of stellar evolution and the fundamentals of asteroseismology. Then, I present in Chapter 2 the results on asteroseismology of 1523 red giants that were misclassified by the Kepler Input Catalog. A dedicated method was used to successfully discriminate for each star the real oscillation power excess from its alias. In Chapter 3, I present a catalog of global seismic parameters, masses, and radii for 16,000 red giants, which has been the largest and most homogeneous catalog. The results showed that oscillation amplitude and granulation power depend on mass and metallicity. Next, I show in Chapter 4 the results on asteroseismology of Long Period Variables (LPVs), which has provided strong evidence to address three long-standing open questions in LPVs. Oscillations and granulation are very valuable signals to characterise stars, but they introduce challenges to confirm and characterise small transiting exoplanets using radial velocity (RV) measurements. Chapter 5 shows predictions of RV jitter induced by oscillations and granulation for dwarfs and giants in terms of stellar parameters. The conclusions and some suggestions for future work are given in Chapter 6

Probabilistic and thermodynamic aspects of dynamical systems

The probabilistic approach to dynamical systems giving rise to irreversible behavior at the macroscopic, mesoscopic, and microscopic levels of description is outlined. Signatures of the complexity of the underlying dynamics on the spectral properties of the Liouville, Frobenius-Perron, and Fokker-Planck operators are identified. Entropy and entropy production-like quantities are introduced and the connection between their properties in nonequilibrium steady states and the characteristics of the dynamics in phase space are explored.info:eu-repo/semantics/publishe