Stability of Micro- and Macroscopic Traffic Flow Models on the Transition from Circular Road to Infinite Lane

Abstract

Mathematical models of traffic flow have been successfully used to describe, understand and predict congestion, behaviour at bottlenecks, and other phenomena. Models for single-lane vehicular traffic are often formulated either “microscopically” as systems of ordinary differential equations, trying to capture the dynamics at the level of a single vehicle, or “macroscopically” as (systems of) partial differential equations, describing e.g. car density and flow velocity. For microscopic models of a finite number of cars on a ring road, detailed stability and bifurcation analysis can be done. In this way it can be explained why and how a slight variation of circumstances like mean density, reaction time, or driving behaviour can lead to an abrupt change from smoothly flowing traffic to congestion. It may be suspected that the fact that the information about a driver’s decisions influences his actions at a later point of time by travelling from vehicle to vehicle upstream around the circle might lead to unrealistic effects. If, however, an open road of infinite length is considered instead, the situation becomes more involving mathematically. The aim of this dissertation is to study how stability properties of traffic flow models change on the transition from circular road to infinite lane and from microscopic to macroscopic description. Prior applications of the concepts of convective and absolute instability to microscopic models are reviewed. These results are compared to those obtained for related macroscopic models. The notions of transient and remnant instability, well-known from partial differential equations, are introduced for microscopic models by considering their behaviour under certain exponentially weighted norms. Analysis of car-following models on the circular road has shown that periodic solutions correspond- ing to stop-and-go-waves may emerge from Hopf bifurcations and can be numerically continued through parameter space, sometimes even into regions for which the quasi-stationary solutions are locally stable. We examine how these solutions behave and how they move with respect to different reference frames when the ring is opened and the number of cars is infinite.