American Institute of Mathematical Sciences

Advanced
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

Eugen Stumpf

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.
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.  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

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.  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

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

Figure 3.  The region $S$ from Proposition 5
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 $V^{\max}=2.841$)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157
[2]

Ábel Garab. Unique periodic orbits of a delay differential equation with piecewise linear feedback function. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2369-2387. doi: 10.3934/dcds.2013.33.2369
[3]

Samuel Bernard, Fabien Crauste. Optimal linear stability condition for scalar differential equations with distributed delay. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 1855-1876. doi: 10.3934/dcdsb.2015.20.1855
[4]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[5]

Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
[6]

Leonid Shaikhet. Stability of equilibriums of stochastically perturbed delay differential neoclassical growth model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1565-1573. doi: 10.3934/dcdsb.2017075
[7]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[8]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[9]

Azmy S. Ackleh, Keng Deng. Stability of a delay equation arising from a juvenile-adult model. Mathematical Biosciences & Engineering, 2010, 7 (4) : 729-737. doi: 10.3934/mbe.2010.7.729
[10]

Pham Huu Anh Ngoc. Stability of nonlinear differential systems with delay. Evolution Equations & Control Theory, 2015, 4 (4) : 493-505. doi: 10.3934/eect.2015.4.493
[11]

Dimitri Breda, Sara Della Schiava. Pseudospectral reduction to compute Lyapunov exponents of delay differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2727-2741. doi: 10.3934/dcdsb.2018092
[12]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[13]

Huaying Guo, Jin Liang. An optimal control model of carbon reduction and trading. Mathematical Control & Related Fields, 2016, 6 (4) : 535-550. doi: 10.3934/mcrf.2016015
[14]

Teresa Faria, José J. Oliveira. On stability for impulsive delay differential equations and application to a periodic Lasota-Wazewska model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2451-2472. doi: 10.3934/dcdsb.2016055
[15]

César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067
[16]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353
[17]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873
[18]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161
[19]

Leonid Berezansky, Elena Braverman. Stability of linear differential equations with a distributed delay. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1361-1375. doi: 10.3934/cpaa.2011.10.1361
[20]

Wouter Huberts, E. Marielle H. Bosboom, Frans N. van de Vosse. A lumped model for blood flow and pressure in the systemic arteries based on an approximate velocity profile function. Mathematical Biosciences & Engineering, 2009, 6 (1) : 27-40. doi: 10.3934/mbe.2009.6.27

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (13)
  • HTML views (23)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences