  This paper analyzes the global dynamics of 1-dimensional agent arrays with
nearest neighbor linear couplings. The equations of motion are second order
linear ODEs with constant coeffcients. The novel part of this research is that
the couplings are different for each distinct agent. We allow the forces to
depend on the positions and velocity (damping terms) but the magnitudes of both
the position and velocity couplings are different for each agent. We, also, do
not assume that the forces are "Newtonian" (i.e. the force due to A on B equals
the 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, and so
relatively little is known about them. This paper is a generalization of
previous work that computed the global dynamics of 1-dimensional sequences of
identical agents [3] assuming periodic boundary conditions. In this paper, we
push that method further, similar to [2], and use an extended periodic boundary
condition to to gain quantitative insights to the systems under consideration.
We find that we can approximate the global dynamics of such a system by
carefully analyzing the low-frequency behavior of the system with (generalized)
periodic boundary conditions.
