Physical Sciences and Mathematics Commons^™

Dances And Escape Of The Vortex Quartet, Brandon Behring

Dissertations

This dissertation considers the linear stability of a one-parameter family of periodic solutions of the four-vortex problem known as 'leapfrogging' orbits. These solutions, which consist of two pairs of identical yet oppositely-signed vortices, were known to W. Gröbli (1877) and A. E. H. Love (1883) and can be parameterized by a dimensionless parameter related to the geometry of the initial configuration. Simulations by Acheson and numerical Floquet analysis by Tophøj and Aref both indicate, to many digits, that the bifurcation occurs at a value related to the inverse square of the golen ratio. Acheson observed that, after an initial period …

Qualitative Modeling Of Chaotic Logical Circuits And Walking Droplets: A Dynamical Systems Approach, Aminur Rahman

Dissertations

Logical circuits and wave-particle duality have been studied for most of the 20th century. During the current century scientists have been thinking differently about these well-studied systems. Specifically, there has been great interest in chaotic logical circuits and hydrodynamic quantum analogs.

Traditional logical circuits are designed with minimal uncertainty. While this is straightforward to achieve with electronic logic, other logic families such as fluidic, chemical, and biological, naturally exhibit uncertainties due to their inherent nonlinearity. In recent years, engineers have been designing electronic logical systems via chaotic circuits. While traditional boolean circuits have easily determined outputs, which renders dynamical models …

Using Feed-Forward Networks To Infer The Activity Of Feedback Neuronal Networks, Xinxian Huang

Dissertations

The nervous system is one of the most important organ systems in a multicellular body. Animals, including human beings perceive, learn, think and deliver motion instructions through their nervous system. The basic structural units of the nervous system are individual neurons which constitute different neuronal networks with distinct functions. In each network, constituent neurons are coupled with different connection patterns, for example, some neurons send feed-forward information to the coupling neurons while others are mutually coupled. Because it is often difficult to analyze large interconnected feedback neuronal networks, it is important to derive techniques to reduce the complexity of the …

Mathematical Problems Arising In Interfacial Electrohydrodynamics, Dmitri Tseluiko

Dissertations

In this work we consider the nonlinear stability of thin films in the presence of electric fields. We study a perfectly conducting thin film flow down an inclined plane in the presence of an electric field which is uniform in its undisturbed state, and normal to the plate at infinity. In addition, the effect of normal electric fields on films lying above, or hanging from, horizontal substrates is considered. Systematic asymptotic expansions are used to derive fully nonlinear long wave model equations for the scaled interface motion and corresponding flow fields. For the case of an inclined plane, higher order …

The Control Of Frequency Of A Conditional Oscillator Simultaneously Subjected To Multiple Oscillatory Inputs, Christina L. Ambrosio

Dissertations

A conditional oscillator is one that requires input to oscillate. An example of such is the gastric mill network of the stomatogastric ganglion of the crab Cancer borealis which requires modulatory input from outside the stomatogastric ganglion and fast input from the pyloric network of the animal in order to become active. This dissertation studies how the frequency of the gastric mill network is determined when it is simultaneously subjected to two different rhythmic inputs whose timing may be mismatched. We derive a mathematical model of the gastric mill network and deduce that the difference in timing between the pyloric …

Pitchfork Bifurcations Of Invariant Manifolds, Jyoti Champanerkar

Dissertations

In a parameter dependent, dynamical system, when the qualitative structure of the solutions changes due to a small change in the parameter, the system is said to have undergone a bifurcation. Bifurcations have been classified on the basis of the topological properties of fixed points and invariant manifolds of dynamical systems. A pitchfork bifurcation in R is said to have occurred when a stable fixed point becomes unstable and two new stable fixed points, separated by the unstable fixed point come into existence.

In this thesis, a pitchfork bifurcation of an (in- 1)-dimensional invariant submani-fold of a dynamical system in …

Temporal Synchronization Of Ca1 Pyramidal Cells By High-Frequency, Depressing Inhibition, In The Presence Of Intracellular Noise, Stephen A. Kunec

Dissertations

The Sharp Wave-associated Ripple is a high-frequency, extracellular recording observed in the rat hippocampus during periods of immobility. During the ripple, pyramidal cells synchronize over a short period of time despite the fact that these cells have sparse recurrent connections. Additionally, the timing of synchronized pyramidal cell spiking may be critical for encoding information that is passed on to post-hippocampal targets. Both the synchronization and precision of pyramidal cells is believed to be coordinated by inhibition provided by a vast array of interneurons. This dissertation proposes a minimal model consisting of a single interneuron which synapses onto a network of …