  Reaction delays are important in determining the qualitative dynamical
properties of a platoon of vehicles traveling on a straight road. In this
paper, we investigate the impact of delayed feedback on the dynamics of the
Modified Optimal Velocity Model (MOVM). Specifically, we analyze the MOVM in
three regimes -- no delay, small delay and arbitrary delay. In the absence of
reaction delays, we show that the MOVM is locally stable. For small delays, we
then derive a sufficient condition for the MOVM to be locally stable. Next, for
an arbitrary delay, we derive the necessary and sufficient condition for the
local stability of the MOVM. We show that the traffic flow transits from the
locally stable to the locally unstable regime via a Hopf bifurcation. We also
derive the necessary and sufficient condition for non-oscillatory convergence
and characterize the rate of convergence of the MOVM. These conditions help
ensure smooth traffic flow, good ride quality and quick equilibration to the
uniform flow. Further, since a Hopf bifurcation results in the emergence of
limit cycles, we provide an analytical framework to characterize the type of
the Hopf bifurcation and the asymptotic orbital stability of the resulting
non-linear oscillations. Finally, we corroborate our analyses using stability
charts, bifurcation diagrams, numerical computations and simulations conducted
using MATLAB.
