First Advisor

J. J. P. Veerman

Term of Graduation

Winter 2022

Date of Publication

2-2-2022

Document Type

Dissertation

Degree Name

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

Department

Mathematics and Statistics

Language

English

DOI

10.15760/etd.7774

Physical Description

1 online resource (vi, 134 pages)

Abstract

This dissertation analyzes the global dynamics of 1-dimensional agent arrays with nearest neighbor linear couplings. The equations of motion are second order linear ODE's with constant coefficients. The novel part of this research is that the couplings are different for each agent. We allow the forces to depend on the relative position and relative velocity (damping terms) of the agents, and the coupling magnitudes differ for each agent. Further, we do not assume that the forces are "Newtonian'" (i.e., the force due to A on B equals minus the force of B on A) as this assumption does not apply to certain situations, such as traffic modeling. For example, driver A reacting to driver B does not imply the opposite reaction in driver B.

There are no known analytical means to solve these systems, even though they are linear. Relatively little is known about them. To estimate system behavior for large times we find an approximation for eigenvalues that are near the origin. The derivation of the estimate uses (generalized) periodic boundary conditions. We also present some stability conditions. Finally, we compare our estimate to simulated flocks.

Rights

© 2021 Robert G. Lyons

In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/ This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).

Persistent Identifier

https://archives.pdx.edu/ds/psu/37254

Available for download on Thursday, February 02, 2023

Included in

Mathematics Commons

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