Limit of a collection of dynamical systems: An application to modeling the flow of traffic

The flow of traffic is usually described using a continuum approach as that of a compressible fluid, a statistical approach via the kinetic theory of gases or cellular automata models. These approaches are not suitable for modeling dynamical systems such as traffic. While such systems are large collections, they are not large enough to be treated as a continuum. We provide a rationale for why they cannot be appropriately described using a continuum model, the kinetic theory of gases, or by appealing to cellular automata models. As an alternative, we develop a discrete dynamical systems approach that is particularly well suited to describe the dynamics of large systems such as traffic.