Nonlinear day-to-day traffic dynamics with driver experience delay: Modeling, stability and bifurcation analysis

In day-to-day traffic assignment problems travelers’ past experiences have important impact on their cost prediction which influences their route choice and consequently the arising flow patterns in the network. Many travelers execute the same trip in every few days, not daily, which leads to time delays in the system. In this paper, we propose a nonlinear, discrete-time model with driver experience delay. The linear stability of the stochastic user equilibrium is analyzed by studying the eigenvalues of the Jacobian matrix of the system while the nonlinear oscillations arising at the bifurcations are investigated by normal form calculations, numerical continuation and simulation. The methods are demonstrated on a two-route example. By applying rigorous analysis we show that the linearly unstable parameter domain as well as the period of arising oscillations increase with the delay. Moreover, delays and nonlinearities result in an extended domain of bistability where the stochastic user equilibrium coexists with stable and unstable oscillations. This study explains the influence of initial conditions on the dynamics of transportation networks and may provide guidance for network design and management.