On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability

Abstract

This work is concerned with the study of a scalar delay differential equation \begin{equation*} z^{\prime\prime}(t)=h^2\,V(z(t-1)-z(t))+h\,z^\prime(t) \end{equation*} motivated by a simple car-following model on an unbounded straight line. Here, the positive real $h$ denotes some parameter, and $V$ is a so-called \textit{optimal velocity function} of the traffic model involved. We analyze the existence and local stability properties of solutions $z(t)=c\,t+d$, $t\in\mathbb{R}$, with $c,d\in\mathbb{R}$. In the case $c\not=0$, such a solution of the differential equation forms a wavefront solution of the car-following model where all cars are uniformly spaced on the line and move with the same constant velocity. In particular, it is shown that all but one of these wavefront solutions are located on two branches parametrized by $h$. Furthermore, we prove that along the one branch all solutions are unstable due to the principle of linearized instability, whereas along the other branch some of the solutions may be stable. The last point is done by carrying out a center manifold reduction as the linearization does always have a zero eigenvalue. Finally, we provide some numerical examples demonstrating the obtained analytical results.