This dissertation creates practical methods for Poincaré return mappings of individual and networked neuron models. Elliptic bursting models are found in numerous biological systems, including the external Globus Pallidus (GPe) section of the brain; the focus for studies of epileptic seizures and Parkinson's disease. However, the bifurcation structure for changes in dynamics remains incomplete. This dissertation develops computer-assisted Poincaré ́maps for mathematical and biologically relevant elliptic bursting neuron models and central pattern generators (CPGs). The first method, used for individual neurons, offers the advantage of an entire family of computationally smooth and complete mappings, which can explain all of the systems dynamical transitions. A complete bifurcation analysis was performed detailing the mechanisms for the transitions from tonic spiking to quiescence in elliptic bursters. A previously unknown, unstable torus bifurcation was found to give rise to small amplitude oscillations. The focus of the dissertation shifts from individual neuron models to small networks of neuron models, particularly 3-cell CPGs. A CPG is a small network which is able to produce specific phasic relationships between the cells. The output rhythms represent a number of biologically observable actions, i.e. walking or running gates. A 2-dimensional map is derived from the CPGs phase-lags. The cells are endogenously bursting neuron models mutually coupled with reciprocal inhibitory connections using the fast threshold synaptic paradigm. The mappings generate clear explanations for rhythmic outcomes, as well as basins of attraction for specific rhythms and possible mechanisms for switching between rhythms.

Le but principal de cette thèse est de développer un modèle numérique pour modéliser les phénomènesde bifurcation et du branchement des fissures. Pour réaliser cet objectif, il était indispensablede posséder un modèle permettant un couplage fort entre le modèle de Plasticité cristalline etcelui de l'Endommagement régularisé. Dans un premier temps, quelques outils de post-traitement ont été développés pour analyser les systèmes de glissement actifs. Ces outils ont été utilisés surdes simulations d'éprouvettes réelles, et comparés à des résultats expérimentaux. Par ces comparaisons, l'application du modèle de Plasticité cristalline aux superalliages monocristallins a été validée. Ce modèle a ensuite été couplé avec le modèle d'endommagement régularisé. Le couplage a été réalisé dans les deux sens, c'est-à-dire que l'évolution de la plasticité a une influence sur l'endommagement et vice-versa. Le nouveau modèle peut être implémenté simplement, avec la méthode traditionnelle des Éléments Finis. Des expériences étudiant la propagation de fissure sous des chargements de types différents ont été simulées à l'aide de ce nouveau modèle :éprouvettes CT,fissuration en Mode II et rupture en fluage. Une méthode pour l'identification des paramètres matériaux a également été proposée.

A simple asset pricing model with two types of adaptively learning traders, fundamentalists and technical analysts, is studied. Fractions of these trader types, which are both boundedly rational, change over time according to evolutionary learning, with technical analysts conditioning their forecasting rule upon deviations from a benchmark fundamental. Volatility clustering arises endogenously in this model. Two mechanisms are proposed as an explanation. The first is coexistence of a stable steady state and a stable limit cycle, which arise as a consequence of a so-called Chenciner bifurcation of the system. The second is intermittency and associated bifurcation routes to strange attractors. Both phenomena are persistent and occur generically in nonlinear multi-agent evolutionary systems. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

I present a simple model of an evolutionary financial market with heterogeneous agents, based on the concept of adaptive belief systems introduced by Brock and Hommes (1997a). Agents choose between different forecast rules based on past performance, resulting in an evolutionary dynamics across predictor choice coupled to the equilibrium dynamics. The model generates endogenous price fluctuations with similar statistical properties as those observed in real return data, such as fat tails and volatility clustering. These similarities are demonstrated for data from the British, German, and Austrian stock market. (author's abstract) / Series: Working Papers SFB "Adaptive Information Systems and Modelling in Economics and Management Science"

An analog controller has been analyzed and built for an electrostatic micro-cantilever beam. The closed loop MEMS device can be used as both actuator and sensor. As an actuator it will have the advantage of large stable travel range up to 90% of the gap. As a sensor the beam is to be driven into chaotic motion which is very sensitive changes in the system parameters. Two versions of the controller have been analyzed and implemented, one for the actuator and one for the sensor. For the actuator, preliminary experiments show good matching with the model. As for the sensor, the dynamic behavior have been studied and the best operating regions have been determined.

Automotive turbochargers (TCs) increase internal combustion engine power and efficiency in passenger and commercial vehicles. TC rotors are usually supported on floating ring bearings (FRBs) or semi-floating ring bearings (SFRBs), both of which are inexpensive to manufacture. However, fluid film bearings are highly nonlinear components of TC units and contribute to the complex behavior (i.e. bifurcations and frequency jumps between a first whirl frequency and a second whirl frequency) of the entire rotor-bearing system (RBS). The frequency jump phenomenon concerns the TC manufacturing industry due to increased levels of noise generation. This thesis presents progress on assessing the effects of some bearing parameters and operating conditions on the RBS dynamic forced performance and the frequency jump phenomenon. A fluid film bearing model is integrated into a finite element rotordynamics computational model for numerical prediction of the TC linear and nonlinear (time transient) forced response. Since automotive TCs operate with variable rotational speed, predictions are conducted with shaft acceleration/deceleration. Over most of its operating speed range, TC rotor nonlinear response predictions display two subsynchronous whirl frequencies w1 and w 2 representing a conical mode and a cylindrical bending mode, respectively. At low shaft speeds w1 is present up to a shaft speed (Omega bifurcation), where there is a frequency jump from w1 to w 2. The second whirl frequency may persist up to the highest shaft speeds (depending on operating conditions). Results show during rotor deceleration the Omega bifurcation may be different from the one during rotor acceleration (hysteresis). Predictions show the following factors delay the Omega bifurcation: increasing oil supply pressure, decreasing oil supply temperature, and increasing shaft acceleration. Also, rotor imbalance distribution greatly affects Omega bifurcation and the shaft amplitude of total motion. Overall, this study shows the sensitivity of bifurcations and frequency jump phenomenon in TC nonlinear response due to various bearing parameters and operating conditions. Further analysis is required to generalize these findings and to assess the effect of other bearing parameters (i.e. clearances, outer film length, ring rotation, etc.) on this phenomenon. In addition further validation of the predictions against test data is required for refinement of the predictive tool.

In this thesis, we introduce two new classes of differential equations, which essentially extend, in several directions, impulsive differential equations and equations on time scales. Basics of the theory for quasilinear systems are discussed, and particular results are obtained so that further investigations of the theory are guaranteed. Applications of the newly-introduced systems are shown through a center manifold theorem, and further, Hopf bifurcation Theorem is proved for a three-dimensional discontinuous dynamical system.

When excitable cardiac tissue is electrically paced at a sufficiently high rate, the duration of excitation can alternate from beat to beat despite a constant stimulation period. This rhythm, known as alternans, has been identified as an early stage in a sequence of increasingly complex instabilities leading to the lethal arrhythmia ventricular fibrillation (VF). This connection served as as a motivation for research into the control of alternans as a strategy to prevent VF. Control methods that do not use a model of the dynamics have been used for the suppression of alternans. However, these methods possess limitations. In this thesis we study theoretically model-based control techniques with the goal of developing protocols that would overcome the shortcomings of non model-based approaches. We consider one dimensional tissue in two different geometrical configurations: a ring and a fiber with free ends (open fiber). We apply standard control methods for linear time invariant systems to a stroboscopic map of the linearized dynamics around the normal rhythm. We found that, in the ring geometry, model-based control is able to suppress alternans faster and with lower current, thereby reducing the risk of tissue damage, compared with non-model-based control. In the open fiber, model-based control is able to suppress alternans for longer fibers and higher pacing frequencies in comparison with non-model-based control. The methodology presented here can be extended to two- and three-dimensional tissue, and could eventually lead to the suppression of alternans on the entire ventricles.

Photovoltaic (PV) DGs can be optimized to provide reactive power support to the grid, although this feature is currently rarely utilized as most DG systems are designed to operate with unity power factor and supply real power only to the grid. In this work, the voltage stability of a power system embedded with PV DG is examined in the context of the high reactive power requirement after a voltage sag or fault. A real-time dynamic multi-function power controller that enables renewable source PV DGs to provide the reactive power support necessary to maintain the voltage stability of the microgrid, and consequently, the wider power system is proposed. The loadability limit necessary to maintain the voltage stability of an interconnected microgrid is determined by using bifurcation analysis to test for the singularity of the network Jacobian and load differential equations with and without the contribution of the DG. The maximum and minimum real and reactive power support permissible from the DG is obtained from the loadability limit and used as the limiting factors in controlling the real and reactive power contribution from the PV source. The designed controller regulates the voltage output based on instantaneous power theory at the point-of-common coupling (PCC) while the reactive power supply is controlled by means of the power factor and reactive current droop method. The control method is implemented in a modified IEEE 13-bus test feeder system using PSCAD® power system analysis software and is applied to the model of a Tampa Electric® PV installation at Lowry Park Zoo in Tampa, FL. This dissertation accomplishes the systematic analysis of the voltage impact of a PV DGembedded power distribution system. The method employed in this work bases the contribution of the PV resource on the voltage stability margins of the microgrid rather than the commonly used loss-of-load probability (LOLP) and effective load-carrying capability (ELCC) measures. The results of the proposed method show good improvement in the before-, during-, and post-start voltage levels at the motor terminals. The voltage stability margin approach provides the utility a more useful measure in sizing and locating PV resources to support the overall power system stability in an emerging smart grid.

This thesis is based on the use of mathematical theories, modelling, and simulations to study the transmission dynamics of HIV/AIDS in the presence of PrEP (pre-exposure prophylaxis) in the MSM (men who have sex with men) population in the United States. A new deterministic model for HIV/AIDS that incorporates PrEP is designed and used to assess the population-level impact of the use of PrEP on the transmission dynamics within an MSM population. Conditions for the effective control (or elimination) and persistence of HIV/AIDS in the MSM population are determined by rigorously analyzing this model. Uncertainty and sensitivity analysis is carried out to determine the effect of the uncertainties in the parameter values on the response variable (the associated reproduction number) and to identify the top-five parameters that have the most effect on the disease transmission dynamics. Numerical simulations show that HIV burden decreases with increasing PrEP coverage. / October 2015