On the stability of stationary states in general road networks

• Define the criticality of junctions in stationary states.
• Discuss information propagation in stationary links and junctions.
• Examine information propagation in the discrete map in critical demand levels.
• Analytically study the stability of stationary states with circular information propagation.
• Stationary states in general networks have the same stability as fixed points of the map.

In [Jin, W.-L., 2015. On the existence of stationary states in general road networks. Transportation Research Part B 81, 917–929.], with a discrete map in critical demand levels, it was proved that there exist stationary states for the kinematic wave model of general road networks with constant origin demands, route choice proportions, and destination supplies. In this study we further examine the stability property of stationary states with the same map, and the results will help us to understand the long-term trend of a network traffic system. We first review a network kinematic wave model and properties of stationary states on a link, define the criticality of junctions in stationary states, and discuss information propagation in stationary states on links and junctions. We then present the map and examine information propagation in the map. We apply the map to analytically study the stability of stationary states on ring roads and diverge-merge networks with circular information propagation and compare them with results obtained from the Poincaré map [Jin, W.-L., 2013. Stability and bifurcation in network traffic flow: A Poincaré map approach. Transportation Research Part B 57, 191–208]. We further study the stability property of general stationary states in a grid network. We find that the stability of fixed points of the map is the same as that of stationary states in a network, and the new approach is more general than the Poincaré map approach. We conclude the study with future directions and implications.