Macroscopic Relations of Urban Traffic Variables: An Analysis of Instability

For networks consisting of a single route the Macroscopic Fundamental Diagram (MFD) can be predicted analytically; but when the networks consist of multiple overlapping routes the flows observed in congestion for a given density are less than those one would predict if the routes were homogeneously congested and did not overlap. These types of networks also tend to jam at densities that are only a fraction of their routes’ average jam density. This paper provides an explanation for this phenomena. It shows that, even for perfectly homogeneous networks with spatially uniform travel patterns, symmetric equilibrium patterns with equal flows and densities across all links are unstable if the average network density is sufficiently high. Instead, the stable equilibrium patterns are asymmetric. Analysis of small idealized networks that can be treated as simple dynamical systems shows that these networks undergo a bifurcation at a network-specific critical density such that for lower densities the MFDs have predictably high flows and are univalued, and for higher densities the order breaks down. Simulations show that this bifurcation also manifests itself in large symmetric networks. It is important in real-world applications that a network’s density never be allowed to approach this critical value.