Stability and attraction domains of traffic equilibria in a day-to-day dynamical system formulation

We formulate the traffic assignment problem from a dynamical system approach. All exogenous factors are considered to be constant over time and user equilibrium is being pursued through a day-to-day adjustment process. The traffic dynamics is represented by a recurrence function, which governs the system evolution over time. Equilibrium stability and attraction domain are then analyzed by studying the topological properties of the system evolution. Stability is important because unstable equilibrium is transient. Even for stable equilibrium, only points within its attraction domain are attracted to the equilibrium. We show that the attraction domain of a stable equilibrium is always open. Furthermore, its boundary is formed by trajectories towards unstable equilibria. Through an understanding of these properties, computation schemes can be devised to determine the ranges of the attraction domains, as demonstrated in this study. Once this is accomplished, a partition chart can be drawn on the state space where each part represents the attraction domain of an equilibrium point. We trust that charting the attraction domains of user equilibria, as presented in this paper, will open up innovative ways for transportation network management.