Discrete vs. continuous time implementation of a ring car following model in which overtaking is allowed

Car following models seek to describe the interactions between individual vehicles as they move along a stretch of road where the behaviour of each vehicle is dependent on the motion of the vehicle directly in front and overtaking is typically not permitted. In this work we study a modified version of the traditional car following model in which the vehicles are travelling on a closed loop and the ‘no overtaking’ restriction has been removed. The resulting model is described firstly in terms of a set of coupled continuous time delay differential equations and then in terms of their discrete time equivalents and both forms of the model are then solved numerically to analyse their post transient behaviour under a periodic perturbation. For certain parameter choices both the continuous and discrete forms of the model can exhibit chaotic behaviour but a comparison of the behaviour of the two models over a wide range of parameter values shows that the discretization can dramatically affect the type of post transient behaviour exhibited. This becomes increasingly evident as the time step used in the discrete time model is increased.