Geodesic
Car-following model with explicit reaction-time delay~--- linear stability analysis of the uniform solution on a ring
Matematičeskoe modelirovanie, Tome 29 (2017) no. 4, pp. 88-100

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Car-following model with explicit reaction-time delay is considered. Vehicle's acceleration depends on the actual speed, the leader's speed, and the gap. The acceleration function includes the reaction-time delay of drivers explicitly. On the one hand, it is the advantage of the model, on the other hand, the mathematical analysis of the model becomes more complicated. We investigate the stability of the uniform flow on a ring. Via Hopf bifurcations linear stability conditions of the steady-state solution on a ring are obtained. Our investigations proof that model parameter values and reaction-time delay exist, which fulfill the stability conditions obtained and guarantee the realistic vehicles' dynamics simultaneously. We also show that the car-following model considered is able to reproduce such phenomenon as propagation of so-called stop-and-go waves, which present in real observations of traffic flow. This is another advantage of the model, because realistic models should have unstable uniform solutions.
Keywords: traffic flow theory, delay differential equations, linear stability analysis, uniform solutions on a ring, Hopf bifurcation. 
@article{MM_2017_29_4_a6,
     author = {V. V. Kurtc and I. E. Anufriev},
     title = {Car-following model with explicit reaction-time delay~--- linear stability analysis of the uniform solution on a ring},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {88--100},
     publisher = {mathdoc},
     volume = {29},
     number = {4},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2017_29_4_a6/}
}
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V. V. Kurtc; I. E. Anufriev. Car-following model with explicit reaction-time delay~--- linear stability analysis of the uniform solution on a ring. Matematičeskoe modelirovanie, Tome 29 (2017) no. 4, pp. 88-100. http://geodesic.mathdoc.fr/item/MM_2017_29_4_a6/