Document Type

Dissertation

Date of Award

Spring 2017

Degree Name

Doctor of Philosophy in Mathematical Sciences - (Ph.D.)

Department

Mathematical Sciences

First Advisor

Denis L. Blackmore

Second Advisor

Victor Victorovich Matveev

Third Advisor

Roy Goodman

Fourth Advisor

Casey Diekman

Fifth Advisor

Ivan C. Christov

Abstract

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 unnecessary, chaotic logical circuits employ components that behave erratically for certain inputs.

There has been an equally dramatic paradigm shift for studying wave-particle systems. In recent years, experiments with bouncing droplets (called walkers) on a vibrating fluid bath have shown that quantum analogs can be studied at the macro scale. These analogs help us ask questions about quantum mechanics that otherwise would have been inaccessible. They may eventually reveal some unforeseen properties of quantum mechanics that would close the gap between philosophical interpretations and scientific results.

Both chaotic logical circuits and walking droplets have been modeled as differential equations. While many of these models are very good in reproducing the behavior observed in experiments, the equations are often too complex to analyze in detail and sometimes even too complex for tractable numerical solution. These problems can be simplified if the models are reduced to discrete dynamical systems. Fortunately, both systems are very naturally time-discrete. For the circuits, the states change very rapidly and therefore the information during the process of change is not of importance. And for the walkers, the position when a wave is produced is important, but the dynamics of the droplets in the air are not.

This dissertation is an amalgam of results on chaotic logical circuits and walking droplets in the form of experimental investigations, mathematical modeling, and dynamical systems analysis. Furthermore, this thesis makes connections between the two topics and the various scientific disciplines involved in their studies.

Included in

Mathematics Commons

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