First Advisor

J. J. P. Veerman

Term of Graduation

Winter 2022

Date of Publication


Document Type


Degree Name

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


Mathematics and Statistics







Physical Description

1 online resource (vi, 134 pages)


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.


© 2021 Robert G. Lyons

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