In this thesis we describe two instances of the nonlinear interaction between a fluid and an elastic solid to mimic or explain some phenomenon seen in nature. We focus on the phenomena of singing of small song birds and swimming of small fish. Song birds show a versatile range of tones and compositions that have highly complex spectral structure, while fish cut through water with amazing ease. The complexity of the Navier Stokes Equations that describe the fluid coupled with large deformation elasticity equations, makes analytic attempts intractable. Hence experiments were carried out with simple physical models to explain these observations. In the first case a cylindrical elastic tube (2.5cm long, 2.5mm diameter) was used to model the vocal organ, the syrinx, of a song bird. Muscle action was mimicked using a linear motor that implemented a squeezing action and the action of the lungs was modeled by a constant source of air flow. The combined fluid-elastic system behaved like a nonlinear dynamical system and produced sound under certain conditions of external parameters. Moreover the structure of the sound created depended sensitively on the control parameters, which in this case was dominated by the position of the linear motor. The motor was dynamically controlled to produce a range of songs from simple tonal ones of the Vireo to the complex chaotic songs of the Zebra Finch. In the second instance, a cantilever (5cm long, 1cm wide)suspended between and driven by magnetic coils arranged in an anti-Helmholtz configuration, was used to mimic the dynamics of the caudal fin of a small fish. Three different gaits were observed as a function of the control parameter, the drive frequency. One of these gaits maximized thrust and hydrodynamic efficiency, and velocities unto 6 body lengths per second were measured. / Engineering and Applied Sciences

Balance control is important for biped standing. Due to the time-varying control bounds induced by the foot constraints, and the lack of tools for analyzing stability of highly nonlinear systems, it is extremely difficult to design balance control strategies for a standing biped with a rigorous stability analysis in spite of large efforts. In this thesis, three important issues are fully considered for a standing biped: maintaining the postural stability, minimizing the energy consumption and satisfying the constraints between the biped feet and the ground. Both the theoretical and the experimental studies on the constrained and energy-efficient control are carried out systematically using the genetic algorithm (GA). The stability for the proposed balancing system is thoroughly investigated using the concept of Lyapunov exponents. On the other hand, the controlled standing biped is characterized by high nonlinearity and great complexity. For systems with such features, in general the Lyapunov exponents are hard to be estimated using the model-based method. Meanwhile the biped is supposed to be stabilized at the upright posture, indicating that the system should possess negative Lyapunov exponents only. However the accuracy of negative exponents is usually poor if following the traditional time-series-based methods. As it is nontrivial to examine the system stability for bipedal robots, the numerical accuracy of the estimated Lyapunov exponents is extremely demanding. In this research, two novel approaches are proposed based upon system approximation using different types of Radial-Basis-Function (RBF) networks. Both the proposed methods can estimate the exponents reliably with straightforward algorithms, yet no mathematical model is required in any newly developed method. The efficacies of both methods are demonstrated through a linear quadratic regulator (LQR) balancing system for a standing biped, as well as several other dynamical systems. The thesis as a whole, has set up a framework for developing more sophisticated controllers in more complex movement for robot models with less conservative assumptions. The systematic stability analysis shown in this thesis has a great potential for many other engineering systems.

The Atomic Force Microscope (AFM) is a tool for imaging surfaces at the microand nano meter scale. The microscope senses the force acting between a surfaceand a tip positioned at the end of a micro-cantilever, forming an image of the surface topography. Image contrast however, arises not only from surface topography, but also from variation in material composition. Improved material contrast, and improved interpretation of that contrast are two issues central to the further development of AFM. This thesis studies dynamic AFM where the cantilever is driven at multiple frequencies simultaneously. Due to the nonlinear dependence of the tip-surface force on the tip’s position, the cantilever will oscillate not only at the driven frequencies, but also at harmonics and at mixing frequencies of the drives, so-called intermodulation products. A mode of AFM called Intermodulation AFM (ImAFM) is primarily studied, which aims to make use of intermodulation products centered around the resonance frequency of the cantilever. With proper excitation many intermodulation products are generated near resonance where they can be measured with large signal-to-noise ratio. ImAFM is performed on samples containing two distinct domains of different material composition and a contrast metric is introduced to quantitatively evaluate images obtained at each response frequency. Although force sensitivity is highest on resonance, we found that weak intermodulation response off resonance can show larger material contrast. This result shows that the intermodulation images can be used to improve discrimination of materials. We develop a method to obtain material parameters from multifrequency AFM spectra by fitting a tip-surface force model. Together with ImAFM, this method allows high resolution imaging of material parameters. The method is very generalas it is not limited to a specific force model or particular mode of multifrequency AFM. Several models are discussed and applied to different samples. The parameter images have a direct physical interpretation and, if the model is appropriate, they can be used to relate the measurement to material properties such as the Young’s modulus. Force reconstruction is tested with simulations and on measured data. We use the reconstructed force to define the location of the surface so that we can address the issue of separating topographic contrast and material contrast. / Svepkraftmikroskop (eller atomkraftmikroskop från engelskans atomic forcemicroscope, AFM) är ett instrument för att avbilda ytor på mikro- och nanometer skalan. Mikroskopet känner av kraften som verkar mellan en yta och en spetsplacerad längst ut på ett mikrometerstort fjäderblad och kan därigenom skapa en topografisk bild av ytans form. Bildkontrast uppstår dock inte bara från ytans form utan även från variation i material. Förbättrad materialkontrast och förbättrad tolkning av denna kontrast är två centrala mål i vidareutvecklingen av AFM. Denna avhandling berör dynamisk AFM där fjädern drivs med flera frekvensersamtidigt. På grund av det ickelinjära förhållandet i yt-spets-kraften som funktion av spetsens position så kommer fjädern inte bara att svänga på de drivna frekvenserna utan också på övertoner och blandfrekvenser, så kallade intermodulationsprodukter. Vi undersöker primärt Intermodulation AFM (ImAFM) som ämnar att utnyttja intermodulationsprodukter nära fjäderns resonansfrekvens. Med en lämplig drivsignal genereras många intermodulationsprodukter nära resonansen, där de kan mätas med bra signal till brus förhållande. ImAFM utförs på ytor bestående av två distinkta domäner av olika material ochen kontrastmetrik introduceras för att kvantitativt utvärdera bilderna som skapas vid varje frekvens. Trots att känsligheten för kraftmätningen är högst på resonans-frekvensen, så fann vi att svaga intermodulationsprodukter bortanför resonansen kan visa hög materialkontrast. Detta resultat visar att intermodulationsbilderna kan användas för att bättre särskilja olika material. Vi har utvecklat en metod för att rekonstruera yt-spets-kraften från multifrekventa AFM spektra genom modellanpassning i frekvensrymden. Tillsammans med ImAFM leder detta till högupplösta bilder av materialparametrar. Metoden är generell och är applicerbar för olika kraftmodeller och AFM-varianter. Parametrarna har en direkt fysikalisk tolkning och, om lämpliga modeller används, kan egenskaper så som materialets elasticitetsmodul mätas. Metoden har testats på simulerat såvälsom experimentellt data, och den har också används för att särskilja topografisk kontrast från materialkontrast. / <p>QC 20150209</p>

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.