Link-based day-to-day network traffic dynamics and equilibria

• Propose a general dynamical system model with link-based variables.
• Establish several desirable properties of the model.
• Show three special cases of the model.
• Consider both the cases with fixed demand and elastic demand.

A general dynamical system model with link-based variables is formulated to characterize the processes of achieving equilibria from a non-equilibrium state in traffic networks. Several desirable properties of the dynamical system model are established, including the equivalence between its stationary state and user equilibrium, the invariance of its evolutionary trajectories, and the uniqueness and stability of its stationary points. Moreover, it is shown that not only a link-based version of two existing day-to-day traffic dynamics models but also two existing link-based dynamical system models of traffic flow are the special cases of the proposed model. The stabilities of stationary states of these special cases are also analyzed and discussed. In addition, an extension is made to the case with elastic demand. The study is helpful for better understanding the day-to-day adjustment mechanism of traffic flows in networks.