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

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  • 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.

    Mathematics Subject Classification: Primary:34K20, 90B20;Secondary:34K17, 34K19, 34K60.

    Citation:

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  • Figure 1.  The schematic setting of the car-following model

    Figure 2.  Function $V_q$ and its derivative for $V^{\max}=1$ and $d_S=0.5$

    Figure 3.  The region $S$ from Proposition 5

    Figure 4.  Numerically calculated solution $z$ and its first derivative from Example 1 ($c\approx 0.0501$, $h=0.2$, and $V^{\max}=100$)

    Figure 5.  Numerical computation of the disturbed solution $z^*$ and its first derivative from Example 1 ($c^{*}_e\approx 0.0451$)

    Figure 6.  Numerical computation of solution $z$ and its first derivative from Example 2 ($c\approx 19.9499$, $h=0.2$, and $V^{\max}=100$)

    Figure 7.  Numerical computation of solution $z$ and its first derivative from Example 3 ($c\approx 0.2492$, $h=1.5$, and $V^{\max}=2.841$)

    Figure 8.  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 $V^{\max}=2.841$)

  •   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.
      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.
      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.
      I. Gasser and E. Stumpf, work in progress.
      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.
      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.
      MATLAB R2016a, The MathWorks Inc. , Natick, Massachusetts, 2016.
      E. Stumpf, work in progress.
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