Bifurcation in car-following models with time delays and driver and mechanic sensitivities

In this work, we study a model of traffic flow along a one-way, one lane, road or street, the so-called car-following problem. We first present a historical evolution of models of this type corresponding to a successive improvement of requirements, to explain some real traffic phenomena. For both mathematical reasons and a better explanation of some of those phenomena, we consider more convenient and accurate requirements which lead to a better non-linear model with reaction delays, from several sources. The model can be written as an ordinary nonlinear delay differential equation. It has equilibrium solutions, which correspond to steady traffic. The mentioned reaction delays introduce perturbation terms in the equation, leading to of instabilities of equilibria and changes of the structure of the solutions. For some of the values of the delays, they may become oscillatory. We make a number of simulations to show these changes for different values of delays. We also show that, for certain values of the delays the above mentioned change of structure (representing regimes of real traffic) corresponds to a Hopf bifurcation.