The dynamic behaviour of road traffic flow: stability or chaos?

The objective of this thesis is to investigate the dynamic behaviour of road traffic flow based on theoretical traffic models. Three traffic models are examined: the classical car-following model which describes the variations of speeds of cars and distances between the cars on a road link, the logit-based trip assignment model which describes the variations of traffic flows on road links in a road network, and the dynamic gravity trip distribution model which describes the variations of
flows between O-D pairs in an O-D network.
Some dynamic analyses have been made of the car-following model in the literature (Chandler et al., 1958, Herman et al., 1959, Disbro & Frame, 1990, and Kirby and Smith, 1991). The dynamic gravity model and the logit-based trip assignment model are both suggested by Dendrinos and Sonis (1990) without detailed analysis. There is virtually no previous dynamic analysis of trip distribution, although there are some dynamic considerations of trip assignment based on other assignment models (Smith, 1984 and Horowitz, 1984).
In this thesis, the three traffic models are considered as dynamical systems. The variations of traffic characteristics are investigated in the context of nonlinear dynamics. Equilibria and oscillatory behaviour are found in all three traffic models; complicated behaviour including period doubling and chaos is found in the gravity model. Values of parameters for different types of behaviour in the models are given. Conditions for the stability of equilibria in the models are established. The stability analysis of the equilibrium in the car-following model
is more general here than that in the literature (Chandler et al., 1958, Herman et al., 1959). Chaotic attractors found in the gravity model are characterized by Liapunov exponents and fractal dimension.
The research in this thesis aims at understanding and predicting traffic behaviour under various conditions. Traffic systems may be monitored, based on these results, to achieve a stable equilibrium and to avoid instabilities and chaos.

A thesis submitted to Middlesex University in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Two papers were written based on the research of the thesis during the term of this study. The first paper was published (Jarrett and Zhang, 1993) and the second paper was presented in the 26th Annual Conference of Universities Transport Study Group (Zhang, 1994). Copies of the two papers can be found at the end of this thesis.