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This paper analyzes the global dynamics of 1-dimensional agent arrays withnearest neighbor linear couplings. The equations of motion are second orderlinear ODEs with constant coeffcients. The novel part of this research is thatthe couplings are different for each distinct agent. We allow the forces todepend on the positions and velocity (damping terms) but the magnitudes of boththe position and velocity couplings are different for each agent. We, also, donot assume that the forces are "Newtonian" (i.e. the force due to A on B equalsthe minus the force of B on A) as this assumption does not apply to certainsituations, such as traffic modeling. For example, driver A reacting to driverB does not imply the opposite reaction in driver B. There are no knownanalytical means to solve these systems, even though they are linear, and sorelatively little is known about them. This paper is a generalization ofprevious work that computed the global dynamics of 1-dimensional sequences ofidentical agents [3] assuming periodic boundary conditions. In this paper, wepush that method further, similar to [2], and use an extended periodic boundarycondition to to gain quantitative insights to the systems under consideration.We find that we can approximate the global dynamics of such a system bycarefully analyzing the low-frequency behavior of the system with (generalized)periodic boundary conditions.

Linear Nearest Neighbor Flocks with All Distinct Agents