Columnar jointing is a fracture pattern, best known from locations such as the Giant's Causeway, or Fingal's Cave, in which cracks self-organize into a nearly hexagonal arrangement, leaving behind an ordered colonnade. In this thesis observations of columnar jointing are reported from both a controlled laboratory setting, and in cooled lava flows. Experiments were performed in slurries of corn starch and water, which form columnar joints when dried. This drying process is examined in detail, and it is shown how desiccation leads to the propagation of a sharp shrinkage front. In general, but with some significant exceptions, the size of columnar joints is inversely dependent on the speed of this shrinkage front during their formation. The exceptions, which include sudden jumps in column scale, show that hysteresis is also important in choosing the column scale. Novel observations of the 3D structure of joints in starch show that columnar joints do not settle down to a perfect hexagonal pattern, but rather mature into a continuously evolving dynamic pattern. This pattern is scale invariant, and the same statistical distribution of column shapes applies equally to joints in both starch and lava. Field work was performed to study columnar jointing in the basalts of the Columbia River Basalt Group and the island of Staffa, and the more heterogeneous lava flows of Southwestern British Columbia. The widths of columns and the heights of striae (chisel-like markings that record details of cooling) were examined in detail, and these length scales are shown to be inversely proportional to each other. An additional length scale, that of wavy columns, is also first reported here. Based on these measurements, empirical advective-diffusive models are developed to describe the transport of water in a drying starch-cake, and the transport of heat in a cooling lava flow. These models have only a single scaling parameter, the Péclet number, which relates the fracture front velocity times the column size to the (thermal or hydraulic) diffusivity. In both cases, the formation of columnar joints occurs at a Péclet number of about 0.2. This model explains the hundred-fold differences in scale between columnar joints in starches and lavas, and can be used as a tool for the interpretation of joint patterns in the field.

To describe learning, as an alternative to a neural network recently dynamical systems were introduced whose vector fields were plastic and self-organising. Such a system automatically modifies its velocity vector field in response to the external stimuli. In the simplest case under certain conditions its vector field develops into a gradient of a multi-dimensional probability density distribution of the stimuli. We illustrate with examples how such a system carries out categorisation, pattern recognition, memorisation and forgetting without any supervision.

In this thesis, low-order nonlinear models for the prediction of the nonlinear behaviour of thermoacoustic systems are developed. These models are based on thermoacoustic networks, in which linear acoustics is combined with a nonlinear heat release model. The acoustic networks considered in this thesis can take into account mean flow and non-trivial acoustic reflection coefficients, and are cast in state-space form to enable analysis both in the frequency and time domains. Starting from linear analysis, the stability of thermoacoustic networks is investigated, and the use of adjoint methods for understanding the role of the system's parameters on its stability is demonstrated. Then, a nonlinear analysis using various state-of-the-art methods is performed, to highlight the strengths and weaknesses of each method. Two novel frameworks that fill some gaps in the available methods are developed: the first, called Flame Double Input Describing Function (FDIDF), is an extension of the Flame Describing Function (FDF). The FDIDF approximates the flame nonlinear response when it is forced simultaneously with two frequencies, whereas the FDF is limited to one frequency. Although more expensive to obtain, the FDIDF contains more nonlinear information than the FDF, and can predict periodic and quasiperiodic oscillations. It is shown how, in some cases, it corrects the prediction of the FDF about the stability of thermoacoustic oscillations. The second framework developed is a weakly nonlinear formulation of the thermoacoustic equations in the Rijke tube, in which the acoustic response is not limited to a single-Galerkin mode, and is embedded in a state-space model. The weakly nonlinear analysis is strictly valid only close to the expansion point, but is much cheaper than any other available method. The above methods are applied to relatively simple thermoacoustic configurations, in which the nonlinear heat release model is that of a laminar conical flame or an electrical heater. However, in real gas turbines more complex flame shapes are found, for which no reliable low-order models exist. Two models are developed in this thesis for turbulent bluff-body stabilised flames: one for a perfectly premixed flame, in which the modelling is focused on the flame-flow interaction, accounting for the presence of recirculation zones and temperature gradients; the second for imperfectly premixed flames, in which equivalence ratio fluctuations, modelled as a passive scalar field, dominate the heat release response. The second model was shown to agree reasonably well with experimental data, and was applied in an industrial modelling project.

abstract: According to the CDC in 2010, there were 2.8 million emergency room visits costing $7.9 billion dollars for treatment of nonfatal falling injuries in emergency departments across the country. Falls are a recognized risk factor for unintentional injuries among older adults, accounting for a large proportion of fractures, emergency department visits, and urgent hospitalizations. The objective of this research was to identify and learn more about what factors affect balance using analysis techniques from nonlinear dynamics. Human balance and gait research traditionally uses linear or qualitative tests to assess and describe human motion; however, it is growing more apparent that human motion is neither a simple nor a linear task. In the 1990s Collins, first started applying stochastic processes to analyze human postural control system. Recently, Zakynthinaki et al. modeled human balance using the idea that humans will remain erect when perturbed until some boundary, or physical limit, is passed. This boundary is similar to the notion of basins of attraction in nonlinear dynamics and is referred to as the basin of stability. Human balance data was collected using dual force plates and Vicon marker position data for leans using only ankle movements and leans that were unrestricted. With this dataset, Zakynthinaki’s work was extended by comparing different algorithms used to create the critical curve (basin of stability boundary) that encloses the experimental data points as well as comparing the differences between the two leaning conditions. / Dissertation/Thesis / Masters Thesis Bioengineering 2016

abstract: This dissertation treats a number of related problems in control and data analysis of complex networks. First, in existing linear controllability frameworks, the ability to steer a network from any initiate state toward any desired state is measured by the minimum number of driver nodes. However, the associated optimal control energy can become unbearably large, preventing actual control from being realized. Here I develop a physical controllability framework and propose strategies to turn physically uncontrollable networks into physically controllable ones. I also discover that although full control can be guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and control energy to achieve actual control, and my work provides a framework to address this issue. Second, in spite of recent progresses in linear controllability, controlling nonlinear dynamical networks remains an outstanding problem. Here I develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another. I introduce the concept of attractor network and formulate a quantifiable framework: a network is more controllable if the attractor network is more strongly connected. I test the control framework using examples from various models and demonstrate the beneficial role of noise in facilitating control. Third, I analyze large data sets from a diverse online social networking (OSN) systems and find that the growth dynamics of meme popularity exhibit characteristically different behaviors: linear, “S”-shape and exponential growths. Inspired by cell population growth model in microbial ecology, I construct a base growth model for meme popularity in OSNs. Then I incorporate human interest dynamics into the base model and propose a hybrid model which contains a small number of free parameters. The model successfully predicts the various distinct meme growth dynamics. At last, I propose a nonlinear dynamics model to characterize the controlling of WNT signaling pathway in the differentiation of neural progenitor cells. The model is able to predict experiment results and shed light on the understanding of WNT regulation mechanisms. / Dissertation/Thesis / Doctoral Dissertation Electrical Engineering 2017

abstract: Synthetic biology is an emerging field which melds genetics, molecular biology, network theory, and mathematical systems to understand, build, and predict gene network behavior. As an engineering discipline, developing a mathematical understanding of the genetic circuits being studied is of fundamental importance. In this dissertation, mathematical concepts for understanding, predicting, and controlling gene transcriptional networks are presented and applied to two synthetic gene network contexts. First, this engineering approach is used to improve the function of the guide ribonucleic acid (gRNA)-targeted, dCas9-regulated transcriptional cascades through analysis and targeted modification of the RNA transcript. In so doing, a fluorescent guide RNA (fgRNA) is developed to more clearly observe gRNA dynamics and aid design. It is shown that through careful optimization, RNA Polymerase II (Pol II) driven gRNA transcripts can be strong enough to exhibit measurable cascading behavior, previously only shown in RNA Polymerase III (Pol III) circuits. Second, inherent gene expression noise is used to achieve precise fractional differentiation of a population. Mathematical methods are employed to predict and understand the observed behavior, and metrics for analyzing and quantifying similar differentiation kinetics are presented. Through careful mathematical analysis and simulation, coupled with experimental data, two methods for achieving ratio control are presented, with the optimal schema for any application being dependent on the noisiness of the system under study. Together, these studies push the boundaries of gene network control, with potential applications in stem cell differentiation, therapeutics, and bio-production. / Dissertation/Thesis / Doctoral Dissertation Biomedical Engineering 2018

The objective of this dissertation is to investigate several dynamical approaches aiming to improve the sensitivity and performance of microelectromechanical systems (MEMS) resonant sensors. Resonant sensors rely on tracking shifts in the dynamic features of microstructures during sensing, such as their resonance frequency. We aim here to demonstrate analytically and experimentally several new concepts aiming to sharpen their response, enhance the signal to noise ratio, and demonstrate smart functionalities combined into a single resonator. The dissertation starts with enhancing the excitations of the higher order modes of vibrations of clamped-clamped microbeam resonators. The concept is based on using partial electrodes with shapes that induce strong excitation of the mode of interest. Using a half electrode, the second mode is excited with a high amplitude of vibration. Also, using a two-third electrode configuration is shown to amplify the third mode resonance amplitude compared with the full electrode under the same electrical loading conditions. Then, we demonstrate the effectiveness of higher order mode excitation and metal organic frameworks (MOFs) functionalization for improving the sensitivity and selectivity of resonant gas sensors. Also, using a single mode only, we show the possibility of realizing a smart switch triggered upon exceeding a threshold mass when operating the resonator near the dynamic pull-in instability. The second part of the dissertation deals with the dynamics of the microbeam under a two-source harmonic excitation. We experimentally demonstrate resonances of an additive and subtractive type. It is shown that by properly tuning the frequency and amplitude of the excitation force, the frequency bandwidth of the resonator is controlled. Finally, we employ the multimode excitation of a single resonator to demonstrate smart functionalities. By monitoring the frequency shifts of two modes, we experimentally demonstrate the effectiveness of this technique to measure the environmental temperature and gas concentration. Also, we present a hybrid sensor and switch device, which is capable of accurately measuring gas concentration and perform switching when the concentration exceeds a specific (safe) threshold. In contrast to the single mode operation, we show that monitoring the third mode enhances sensitivity, improves accuracy, and lowers the sensor sensitivity to noise.

This work presents a combination of analytical and numerical approaches to gain an insight of the dynamics of marine risers and micro machined resonators. Despite their scale difference, we show that both systems share similarities in terms of initial static deformation, quadratic and cubic nonlinearities, and internal resonances. First, we utilize the state space method to study the eigenvalue problem of vertical riser. An orthonormalization step is introduced to recover the numerical scheme during numerical integration and we investigate the effect of applied tension, apparent weight, and higher-order modes on the accuracy of the scheme. We show that the method is advantageous to find eigenvalues and mode shapes of vertical risers in comparison to other methods. The work is extended to study the eigenvalue problem of inclined risers considering the influence of static deflection, self-weight and mid-plane stretching. The linear dynamics is solved using Galerkin method. The results demonstrate that under the influence of tension and configuration angle, the frequencies exhibit commensurate ratio with respect to the first natural frequency leading to the possible activation of internal resonances. Next, we study the nonlinear interactions of inclined risers considering two-to-one and three-to-one internal resonances under single and multifrequency excitations. The multiple times scale method is applied to study the nonlinear interaction and results are compared to those from a Galerkin solution showing good agreement. Time histories and perturbation’s response curves, in addition to the dynamical solution obtained by Galerkin and stability analysis using Floquet theory are utilized to examine the system. These results feature nonlinear energy exchange, saddle node jumps, and Hopf bifurcations leading to complex dynamic motion that can endanger the riser structure. Finally, the analysis using pertubation is extended to investigate the two-to-one internal resonance in micromachined arch beams between its first two symmetric modes. The response is analyzed using the perturbation method considering the nonlinear interaction and two simultaneous excitations at higher AC voltages. Good agreement is found among the results of pertubations, Galerkin and experimental data from fabricated Silicon arch beam. Different types of bifurcations are observed which can lead to quasi-periodic and potentially chaotic motions.

No description available.

Thulium (Tm) based high repetition rate compact optical frequency comb sources operating in the 2 µm regime with femtosecond pulse durations enable a wide range of applications such as precise micro-machining, spectroscopy and metrology. Applications such as metrology and spectroscopy rely on the stability of mode-locked lasers (MLLs) which provide extreme precision, yet, the complex dynamics of such highly nonlinear systems result in unstable events which could hinder the normal operation of a MLL. MLL as a nonlinear system inherently exists a wide variety of complex attractors, which are sets of states that the system tends to evolve toward, exhibiting unique behaviors. Complex phenomena including pulsating solitons, chaotic solitons, period-doubling, soliton explosion, etc., have been predicted theoretically and observed experimentally in the past decade. However, most experimental observations rely on conventional characterization methods, which are limited to the scanning speed of the spectrometer and the electronic speed of photodetector and digitizer, so that the details of the non-repetitive events can be buried. In recent years, a technique called dispersive Fourier transform (DFT) has been developed and allows consecutive recordings of the pulse-to-pulse spectral evolution of a femtosecond pulse train, opening a whole new world of nonlinear dynamics in MLL. In this dissertation, we first demonstrate the ability of scaling the repetition rate of a Tm MLL to repetition rate as high as 1.25 GHz through miniaturizing the cavity. Our approach of maintaining comparable pulse energies while scaling the repetition rates allows a high-quality femtosecond mode-locking performance with low noise performance in Tm soliton lasers. Then we experimentally study the transition dynamics between consecutive multi-pulsing states through adjusting pump power with a constant rate in an erbium-doped fiber laser, specifically the build-up and annihilation of soliton pulses between a double pulsing and a three-pulse state utilizing DFT. To investigate real-time laser dynamics in Tm based laser systems, we propose and develop a DFT system that up-converts the signal to the 1 µm regime via second harmonics generation (SHG) and stretches the signal in a long spool of single-mode fiber to realize DFT. This approach overcomes the limitation of bandwidth of 2 µm photodetector and high intrinsic absorption of 2 µm light in fused silica fibers. The SHG-DFT system is used to study dynamics of both explosions in a chaotic state between stable single-pulsing and double-pulsing state, and explosions induced by soliton collision in a dual-wavelength vector soliton state. We also study dynamics of transient regimes in a Tm-doped fiber ring laser that can be switched between conventional soliton and dissipative soliton, revealing how spectral filtering plays a role in obtaining stable stationary states. / 2022-11-23T00:00:00Z