Browsing by Subject "Bifurcation theory"
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Item Fitzhugh-Nagumo model and signal processing in the visual cortex of fly(Texas Tech University, 2004-05) Chen, BailiIn this thesis, FitzHugh-Nagumo equations are used to model the dynamics of tangential cells in fly's visual cortex. Membrane current produced by the visual stimuli is modeled as the external input. An asympototic method is used to show that the amplitude of the oscillation depends critically on the magnitude of the external input. Furthermore, in a neighborhood of a bifurcation, the amplitude increases monotonically as a function of the external input, this result helps to explain how the visual cortex of fly encodes visual stimulus and detects motion.Item Modeling, control, and optimization of fixed bed reactors(Texas Tech University, 2002-12) Gudekar, Kishor G.In this work, modeling and optimization of an industrial vinyl acetate reactor, and modeling, optimization, control and bifurcation analysis of industrial ethylene oxide process is performed. For a vinyl acetate reactor, a steady state two-dimensional homogeneous model is developed. The catalyst activity is expressed as a nonlinear function of catalyst age, shell side coolant temperature and the moderator used in the reaction. Offline optimization is carried out for the vinyl acetate reactor using a steady state reactor model to find an optimal operating temperature profile, which maximizes the profit of the process. Updating the model parameters online does online optimization. The ethylene oxide process studied consists of a feed effluent heat exchanger, a multitubular fixed bed reactor, a steam generator, and a separation system. The exothermic heat of reaction from the reactor is removed by passing coolant on the shell side of the reactor. A portion of the heated coolant is passed through a steam generator to produce steam, and the total coolant stream is recycled back to the shell side of the reactor. A single-loop PID control system uses the flow rate of the coolant that is passed through the steam generator to maintain the inlet temperature of the coolant to the reactor. A two-dimensional heterogeneous dynamic model is developed for a catalytic multitubular ethylene oxide reactor. The catalyst deactivation is modeled as a nonlinear function of operating time and temperature of the reactor. Sequential quadratic programming (SQP) is used to solve this nonlinear programming problem. An optimal temperature profile is found which maximizes the profit over the existing operating conditions for the fixed run length of the reactor. The open-loop and closed-loop stability studies are conducted using the benchmarked model of an ethylene oxide reactor system. Steady-state nonlinear bifurcation analysis is performed to identify the multiplicity in the heat integrated ethylene oxide reactor system. The effect of manipulated (flow through steam generator) and disturbance (reactor inlet carbon dioxide composition) variables are addressed. An analysis of the stable control region of the system is developed as a function of operating temperature, catalyst activity, and disturbance direction and magnitude.Item Numerical studies of the standard nontwist map and a renormalization group framework for breakup of invariant tori(2004) Apte, Amit Shriram; Morrison, Philip J.Item Periodic orbit bifurcations and breakup of shearless invariant tori in nontwist systems(2006) Fuchss, Kathrin; Morrison, Philip J.This thesis explores two nontwist systems: the spherical pendulum as an example of a continuous one and the standard nontwist map (SNM) as an example of a discrete one. Whereas the spherical pendulum is a concrete example of a physical system exhibiting nontwist phenomena, the SNM is an abstract, numerically easily accessible model permitting systematic studies of nontwist effects characteristic of a wide range of applications. For the spherical pendulum, a system that has captured physicists' and mathematicians' interest for centuries, the gradual progress in understanding this seemingly simple, but still not fully explored problem is outlined. The known solutions for the unforced (integrable) spherical pendulum are reviewed and approximated by power series. The approximations are then used to analytically calculate, for the vertically forced case, xed points and low-period periodic orbits. These are found to undergo collision phenomena typical for nontwist systems. For the SNM, a detailed cartography of parameter space is developed, based on periodic orbit collision curves and their branching thresholds, hyperbolic manifold reconnection thresholds, and the boundary for the onset of global chaos. This is used to nd meanders, multiple shearless curves, and extended scenarios for periodic orbit reconnection/collision. Based on Greene's residue criterion, the breakup of new types of shearless orbits: meanders, outer shearless tori, and a nonnoble torus is studied in detail within the framework of renormalization theory.