Abstract
Stop and go waves in granular flow can often be described mathematically by a dynamical system with a Hopf bifurcation. We show that a certain class of microscopic, ordinary differential equation-based models in crowd dynamics fulfil certain conditions of Hopf bifurcations. The class is based on the Gradient Navigation Model. An interesting phenomenon arises: the number of pedestrians in the system must be greater than nine for a bifurcation—and hence for stop and go waves to be possible at all, independent of the density. Below this number, no parameter setting will cause the system to exhibit stable stop and go behaviour. The result is also interesting for car traffic, where similar models exist. Numerical experiments of several parameter settings are used to illustrate the mathematical results.
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References
Dietrich, F., Köster, G.: Gradient navigation model for pedestrian dynamics. Phys. Rev. E 89(6), 062801 (2014)
Disselnkötter, S.: A bifurcation analysis for scenarios in the gradient navigation model. Master’s thesis, Technische Universität München (2013)
Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer (1986)
Kuznetsov, Y.A.: Andronov-hopf bifurcation. Scholarpedia 1(10), 1858 (2006)
Nishinari, K., Sugawara, K., Kazama, T., Schadschneider, A., Chowdhury, D.: Modelling of self-driven particles: Foraging ants and pedestrians. Physica A 372(1), 132–141 (2006)
Orosz, G., Wilson, R.E., Stepan, G.: Traffic jams: dynamics and control. Philos. Trans. A: Math. Phys. Eng. Sci. 368(1928), 4455–4479 (2010)
Portz, A., Seyfried, A.: Analyzing stop-and-go waves by experiment and modeling. In: Pedestrian and Evacuation Dynamics, vol. 1, pp. 577–586. Springer (2011)
Tadaki, S., Kikuchi, M., Fukui, M., Nakayama, A., Nishinari, K., Shibata, A., Sugiyama, Y., Yosida, T., Yukawa, S.: Phase transition in traffic jam experiment on a circuit. J. Phys. 15(10), 103034 (2013)
Tordeux, A., Seyfried, A.: Collision-free nonuniform dynamics within continuous optimal velocity models. Phys. Rev. E 90, 042812 (2014)
Acknowledgements
This work was partially funded by the German Federal Ministry of Education and Research through the project MultikOSi on assistance systems for urban events—multi criteria integration for openness and safety (Grant No. 13N12824). Support from the TopMath Graduate Center of TUM Graduate School at Technische Universität München, Germany is gratefully acknowledged.
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Dietrich, F., Disselnkötter, S., Köster, G. (2016). How to Get a Model in Pedestrian Dynamics to Produce Stop and Go Waves. In: Knoop, V., Daamen, W. (eds) Traffic and Granular Flow '15. Springer, Cham. https://doi.org/10.1007/978-3-319-33482-0_21
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DOI: https://doi.org/10.1007/978-3-319-33482-0_21
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