Rich dynamics in some discrete-time car-following models

There are a great number of works to study delay differential equations modeling road traffic, but fewer related to discrete-time car-following models. In this paper, we propose two classes of discrete-time car-following models, which can be viewed as leader–follower models or discretization version of classic continuous-time car-following models. Local stability analysis is established in details. Rich dynamical behavior is to be explored, including local stability analysis, chaotic behavior etc. Fractal properties are discovered by the computation of Lyapunov exponents and Lyapunov dimensions. High codimensional bifurcations can be expected. We find that one of the proposed models can admit infinite nontrivial fixed points in its equivalent form but the other cannot do. Moreover, if the leading vehicle presents a regular (steady states or periodic) or irregular (chaotic) oscillation pattern, the following is to do the same likely. In a sense, a synchronous/heredity property can be exhibited in the underlying model.