Figure 1.
The schematic setting of the car-following model
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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 |
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
| $h$ |
| $V$ |
| $z(t)=c\,t+d$ |
| $t∈\mathbb{R}$ |
| $c,d∈\mathbb{R}$ |
| $c\not=0$ |
| $h$ |
Keywords:
Delay differential equation,
stability,
center manifold reduction,
carfollowing model,
optimal velocity function.
Mathematics Subject Classification:
Primary:34K20, 90B20;Secondary:34K17, 34K19, 34K60.
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. |
| [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. |
| [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. |
| [4] |
I. Gasser and E. Stumpf, work in progress. |
| [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. |
| [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. |
| [7] |
MATLAB R2016a, The MathWorks Inc. , Natick, Massachusetts, 2016. |
| [8] |
E. Stumpf, work in progress. |
show all references
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. |
| [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. |
| [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. |
| [4] |
I. Gasser and E. Stumpf, work in progress. |
| [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. |
| [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. |
| [7] |
MATLAB R2016a, The MathWorks Inc. , Natick, Massachusetts, 2016. |
| [8] |
E. Stumpf, work in progress. |
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 $V^{\max}=2.841$ )
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