Abstract
In order to describe the dynamics of a group of road vehicles travelling in a single lane, car-following models attempt to mimic the interactions between individual vehicles where the behaviour of each vehicle is dependent upon the motion of the vehicle immediately ahead. In this paper we investigate a modified car-following model which features a new nonlinear term which attempts to adjust the inter-vehicle spacing to a certain desired value. In contrast to our earlier work, a desired time separation between vehicles is used rather than simply being a constant desired distance. In addition, we extend our previous work to include a non-zero driver vehicle reaction time, thus producing a more realistic mathematical model of congested road traffic. Numerical solution of the resulting coupled system of nonlinear delay differential equations is used to analyse the stability of the equilibrium solution to a periodic perturbation. For certain parameter values the post-transient response is a chaotic (non-periodic) oscillations consisting of a broad spectrum of frequency components. Such chaotic motion leads to highly complex dynamical behaviour which is inherently unpredictable. The model is analysed over a range of parameter values and, in each case, the nature of the response is indicated. In the case of a chaotic solution, the degree of chaos is estimated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lighthill, M. J. and Whitham, G. B., ‘On kinematic waves II. A theory of traffic flow on long crowded roads’, Proceedings of the Royal Society London, Series A 229, 1955, 317‐345.
Newell, G. F., ‘Mathematical models for freely-flowing highway traffic’, Operations Research 3, 1955, 176‐186.
Gazis, D. C., Herman, R., and Rothery, R. W., ‘Nonlinear follow-the-leader models of traffic flow’, Operations Research 9, 1961, 545‐567.
Leutzbach, W., Introduction to the Theory of Traffic Flow, Springer-Verlag, Berlin, 1988.
Reuschel, A., ‘Fahrzeugbewegungen in der Kolonne’, Oesterreichisches Ingenieur-Archiv 4, 1950, 193‐215.
Pipes, L. A., ‘An operational analysis of traffic dynamics’, Journal of Applied Physics 24(3), 1953, 274‐281.
Chandler, R. E., Herman, R., and Montroll, E. W., ‘Traffic dynamics: studies in car-following’, Operations Research 6, 1958, 165‐184.
Kometani, E. and Sasaki, T., ‘On the stability of traffic flow (Report-I)’, Journal of Operations Research Japan 2(1), 1958, 11‐26.
Gazis, D. C., Herman, R., and Potts, R. B., ‘Car-following theory of steady-state traffic flow’, Operations Research 7, 1959, 499‐505.
Edie, L. C., ‘Car-following and steady-state theory for noncongested traffic’, Operations Research 9, 1961, 66‐76.
May, A. D. and Keller, H. E. M., ‘Non-integer car-following models’, Washington Highway Research Board 199, 1967, 19‐32.
Addison, P. S. and Low, D. J., ‘The existence of chaotic behaviour in a separation-distance centred nonlinear car following model’, in Road Vehicle Automation II, C. Nwagboso (ed.), Wiley, Chichester, 1997, pp. 171‐180.
Low, D. J. and Addison, P. S., ‘Chaos in a car-following model including a desired inter-vehicle separation’, in Proceedings of the 28th ISATA Conference, Stuttgart, Germany, September 18‐22, 1995, pp. 539‐546.
Addison, P. S. and Low, D. J., ‘Order and chaos in the dynamics of vehicle platoons’, Traffic Engineering and Control 37(7/8), 1996, 456‐459.
Low, D. J. and Addison, P. S., ‘Chaos in a car-following model with a desired headway time’, in Proceedings of the 30th ISATA Conference, Florence, Italy, June 16‐19, 1997, pp. 175‐182.
Gerald, C. F. and Wheatly, P. O., Applied Numerical Analysis, Addison-Wesley, Reading, MA, 1985.
Cooley, J. W., Lewis, P. A. W., and Welsch, P. D., ‘The fast Fourier transform and its applications’, IEEE Transactions 12, 1969, 27‐34.
Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos, Wiley, Chichester, 1986.
Li, T.-Y. and Yorke, J. A., ‘Period three implies chaos’, American Mathematical Monthly 82(10), 1975, 985‐992.
Addison, P. S., Fractals and Chaos: An Illustrated Course, Institute of Physics Publishing, Bristol, 1997.
Addison, P. S., ‘On the characterisation of non-linear systems in chaotic mode’, Journal of Sound and Vibration 179(3), 1995, 385‐398.
Grassberger, P. and Procaccia, I., ‘Characterization of strange attractors’, Physical Review Letters 50, 1983, 346‐349.
Grassberger, P. and Procaccia, I., ‘Measuring the strangeness of strange attractors’, Physica D 9, 1983, 189‐208.
Fraser, A. M. and Swinney, H. L., ‘Independent coordinates for strange attractors from mutual information’, Physical Review A 33, 1986, 1134‐1140.
Fraser, A. M., ‘Reconstructing attractors from scalar time series: a comparison of singular system and redundancy criteria’, Physica D 34, 1989, 391‐404.
Guckenheimer, J., ‘Strange attractors in fluids: another view’, Annual Review of Fluid Mechanics 18, 1986, 15‐31.
Duncan, G., ‘Paramics-MP Final Report (project report)’, Edinburgh Parallel Computing Centre, University of Edinburgh, 1994.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Low, D.J., Addison, P.S. A Nonlinear Temporal Headway Model of Traffic Dynamics. Nonlinear Dynamics 16, 127–151 (1998). https://doi.org/10.1023/A:1008279031113
Issue Date:
DOI: https://doi.org/10.1023/A:1008279031113