The dynamical system approach to traffic assignment: The attainability of equilibrium and its application to traffic system management

In this thesis we formulate the traffic assignment problem through a dynamical system approach. All exogenous factors are presumed to be constant over time and equilibrium is being pursued through a day-to-day learning process. Travellers' knowledge of the network is represented by their perceived costs on individual routes. Route choice on a day is determined by the perceived route costs on that day. If the actual travel costs are identical to the perceived costs, then equilibrium is achieved; otherwise, travellers update their perceived costs for the route choice next day.;The traffic dynamics can be formulated by a recurrence function of the vector of perceived costs. Fixed point in the dynamical system is equivalent to the stochastic user equilibrium in the static model. Equilibrium stability is analysed by a linearization of the dynamical system around equilibrium. Stability requires that the corresponding Jacobian matrix, when evaluated at the equilibrium point, has its eigenvalues (real or complex) all within the unit circle. Lyapunov function can also be utilized in stability analysis. Stability is important because unstable equilibrium is transient. In cases of instability, it is usually possible to shift the unstable equilibrium to a stable equilibrium by appropriately modifying the network.;Even for stable equilibrium, only points within its attraction basin are attracted to the equilibrium. Some topological analysis shows that the attraction basin of a stable equilibrium is always open. Furthermore, if all points in the state space are attracted to equilibria, one or another, then the boundary of the attraction basin for a stable equilibrium is formed by trajectories to unstable equilibria. Therefore, we can identify the exact range of attraction basins for stable equilibria by tracing back the dynamical evolution to unstable equilibria. Once this is done, the state space can be partitioned into a number of subsets, each representing the attraction basin of an equilibrium point.;The most important implication of network change for traffic system management is that temporary network change may have long term effects on the state of the system, a property which we call irreversibility. Particular attention should then be paid to irreversible temporary changes, whether planned or incidental. These changes, even though imposed on the system only temporarily, can permanently relocate the state of the system. Therefore it is essential that the planned changes should be well studied in advance and then carefully implemented, while remedies should be carried out for irreversible incidental changes.;Besides equilibrium, there are also cyclic and chaotic attractors. These nonequilibrium attractors share similar characteristics as equilibrium in terms of being the limit set of a trajectory. A cyclic attractor with a period of n days is an equilibrium point of the n-days-to- n-days mapping. Studying these attractors may help us understand the non-stationary flow in observed day-to-day traffic data. We also provide criteria for the case where equilibrium is the only type of attractor. When such criteria are satisfied we can eliminate the possibility of non-equilibrium attractors and focus instead on equilibrium attractors.;In summary, the concept of dynamic equilibrium is formulated to replace the traditional static equilibrium, which only concerns the state of equilibrium. The dynamical approach in this thesis addresses the process of pursuing and obtaining equilibrium, i.e. how disequilibrium states evolve towards equilibrium. It enables the analysis on equilibrium stability and attainability. In particular, this thesis shows how the equilibrium's attraction basin can be determined or estimated. Evolution starting from outside the attraction basin does not converge to the equilibrium. A temporary network alteration can then be made to divert the evolution to equilibrium. This implies that changes on the network, even temporary, can have long term effects on the system state. To avoid undesirable consequences, traffic management agency should assess the impact of planned network modification before implementing it. (Abstract shortened by UMI.)...