# American Institute of Mathematical Sciences

November  2017, 22(9): 3317-3340. doi: 10.3934/dcdsb.2017139

## On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability

 University of Hamburg, Department of Mathematics, Bundesstrasse 55,20146 Hamburg, Germany

Received  September 2016 Revised  February 2017 Published  April 2017

This work is concerned with the study of the scalar delay differential equation
 $z^{\prime\prime}(t)=h^2\,V(z(t-1)-z(t))+h\,z^\prime(t)$
motivated by a simple car-following model on an unbounded straight line. Here, the positive real
 $h$
denotes some parameter, and
 $V$
is some so-called optimal velocity function of the traffic model involved. We analyze the existence and local stability properties of solutions
 $z(t)=c\,t+d$
,
 $t∈\mathbb{R}$
, with
 $c,d∈\mathbb{R}$
. In the case
 $c\not=0$
, such a solution of the differential equation forms a wavefront solution of the car-following model where all cars are uniformly spaced on the line and move with the same constant velocity. In particular, it is shown that all but one of these wavefront solutions are located on two branches parametrized by
 $h$
. Furthermore, we prove that along the one branch all solutions are unstable due to the principle of linearized instability, whereas along the other branch some of the solutions may be stable. The last point is done by carrying out a center manifold reduction as the linearization does always have a zero eigenvalue. Finally, we provide some numerical examples demonstrating the obtained analytical results.
Citation: Eugen Stumpf. On a delay differential equation arising from a car-following model: Wavefront solutions with constant-speed and their stability. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3317-3340. doi: 10.3934/dcdsb.2017139
##### References:
 [1] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Structure stability of congestion in traffic dynamics, Japan Journal of Industrial and Applied Mathematics, 11 (1994), 203-223. doi: 10.1007/BF03167222. Google Scholar [2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035. Google Scholar [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. -O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar [4] I. Gasser and E. Stumpf, work in progress.Google Scholar [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [6] T. Insperger and G. Stépán, Semi-discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4614-0335-7. Google Scholar [7] MATLAB R2016a, The MathWorks Inc. , Natick, Massachusetts, 2016.Google Scholar [8] E. Stumpf, work in progress.Google Scholar

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##### References:
 [1] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Structure stability of congestion in traffic dynamics, Japan Journal of Industrial and Applied Mathematics, 11 (1994), 203-223. doi: 10.1007/BF03167222. Google Scholar [2] M. Bando, K. Hasebe, A. Nakayama, A. Shibata and Y. Sugiyama, Dynamical model of traffic congestion and numerical simulation, Physical Review E, 51 (1995), 1035-1042. doi: 10.1103/PhysRevE.51.1035. Google Scholar [3] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H. -O. Walther, Delay Equations. Functional, Complex, and Nonlinear Analysis, Applied Mathematical Sciences, 110, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4206-2. Google Scholar [4] I. Gasser and E. Stumpf, work in progress.Google Scholar [5] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [6] T. Insperger and G. Stépán, Semi-discretization for Time-Delay Systems. Stability and Engineering Applications, Applied Mathematical Sciences, 178, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4614-0335-7. Google Scholar [7] MATLAB R2016a, The MathWorks Inc. , Natick, Massachusetts, 2016.Google Scholar [8] E. Stumpf, work in progress.Google Scholar
The schematic setting of the car-following model
Function $V_q$ and its derivative for $V^{\max}=1$ and $d_S=0.5$
The region $S$ from Proposition 5
Numerically calculated solution $z$ and its first derivative from Example 1 ($c\approx 0.0501$, $h=0.2$, and $V^{\max}=100$)
Numerical computation of the disturbed solution $z^*$ and its first derivative from Example 1 ($c^{*}_e\approx 0.0451$)
Numerical computation of solution $z$ and its first derivative from Example 2 ($c\approx 19.9499$, $h=0.2$, and $V^{\max}=100$)
Numerical computation of solution $z$ and its first derivative from Example 3 ($c\approx 0.2492$, $h=1.5$, and $V^{\max}=2.841$)
The final stage of the numerical computation of solution $z$ and its first derivative from Example 3 ($c\approx 0.2492$, $h=1.5$, and $V^{\max}=2.841$)
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