Subcritical Hopf bifurcations in a car-following model with reaction-time delay

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences

Volumn 462, Issue 2073, 2006, Pages 2643-2670

  Orosz, Gábor ^a,c   Stépán, Gábor ^b,d  

^a UNIVERSITY OF BRISTOL   (United Kingdom)

^b BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

^c UNIVERSITY OF EXETER   (United Kingdom)

^d INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

Author keywords

Bistability; Reaction time delay; Stop and go waves; Subcritical Hopf bifurcation; Translational symmetry; Vehicular traffic

Indexed keywords

EID: 33748103156     PISSN: 13645021     EISSN: 14712946     Source Type: Journal    
DOI: 10.1098/rspa.2006.1660     Document Type: Article

References (27)

1. Bando, M., Hasebe, K., Nakayama, A., Shibata, A. & Sugiyama, Y. 1995 Dynamical model of traffic congestion and numerical simulation. Phys. Rev. E 51, 1035-1042. (doi:10.1103/PhysRevE.51.1035)
2. Bando, M., Hasebe, K., Nakanishi, K. & Nakayama, A. 1998 Analysis of optimal velocity model with explicit delay. Phys. Rev. E 58, 5429-5435. (doi:10.1103/PhysRevE.58.5429)
3. Berg, P. & Wilson, R. E. 2005 Bifurcation analysis of meta-stability and waves of the OV model. In Traffic and granular flow '03 (ed. S. P. Hoogendoorn, S. Luding, P. H. L. Bovy, M. Schreckenberg & D. E. Wolf). Berlin: Springer.
4. Campbell, S. A. & Bélair, J. 1995 Analytical and symbolically-assisted investigations of Hopf bifurcations in delay-differential equations. Can. Appl. Math. Q. 3, 137-154.
5. Davis, L. C. 2003 Modification of the optimal velocity traffic model to include delay due to driver reaction time. Physica A 319, 557-567. (doi:10.1016/S0378-4371(02)01457-7)
6. Diekmann, O., van Gils, S. A., Verduyn Lunel, S. M. & Walther, H. O. 1995 Delay equations: functional-, complex-, and nonlinear analysis. Applied mathematical sciences, vol. 110. New York: Springer.
7. Doedel E. J., Champneys A. R., Fairgrieve T. F., Kuznetsov Yu. A., Sandstede B. & Wang X. 1997 Auto97: continuation and bifurcation software for ordinary differential equations. Technical report, Department of Computer Science, Concordia University, http://indy.cs.concordia.ca/auto/.
8. Engelborghs K., Luzyanina T. & Samaey G. 2001 Dde-Biftool v. 2.00: a Matlab package for bifurcation analysis of delay differential equations. Technical Report TW-330, Department of Computer Science, Katholieke Universiteit Leuven, Belgium, http://www.cs.kuleuven.ac.be/koen/delay/ddebiftool.shtml.
9. Gasser, I., Sirito, G. & Werner, B. 2004 Bifurcation analysis of a class of 'car-following' traffic models. Physica D 197, 222-241. (doi:10.1016/j.physd.2004.07.008)
10. Guckenheimer, J. & Holmes, P. 1997 Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences, vol. 42, 3rd edn. New York: Springer.
11. Hale, J. K. & Verduyn Lunel, S. M. 1993 Introduction to functional differential equations. Applied mathematical sciences, vol. 99. New York: Springer.
12. Hale, J. K., Magelhães, L. T. & Oliva, W. M. 2002 Dynamics in infinite dimensions. Applied mathematical sciences, vol. 47. 2nd edn. New York: Springer.
13. Hassard, B. D., Kazarinoff, N. D. & Wan, Y.-H. 1981 Theory and applications of Hopf bifurcation. London Mathematical Society Lecture Note Series, vol. 41. Cambridge: Cambridge University Press.
14. Kerner, B. S. 1999 The physics of traffic. Phys. World 8, 25-30.
15. Kolmanovskii, V. B. & Myshkis, A. D. 1999 Introduction to the theory and applications of functional differential equations. Mathematics and its applications, vol. 463. London: Kluwer Academic Publishers.
16. Kuznetsov, Yu. A. 1998 Elements of applied bifurcation theory. Applied mathematical sciences. vol. 112, 2nd edn. New York: Springer.
17. Lunt G. & Wilson R. E. 2003. New data sets and improved models of highway traffic. In Proc. 35th UTSG Conf. University of Loughborough, England.
18. Orosz, G. 2004 Hopf bifurcation calculations in delayed systems. Periodica Polytechnica 48, 189-200.
19. Orosz, G. & Stépán, G. 2004 Hopf bifurcation calculations in delayed systems with translational symmetry. J. Nonlin. Sci. 14, 505-528. (doi:10.1007/s00332-004-0625-4)
20. Orosz, G., Wilson, R. E. & Krauskopf, B. 2004 Global bifurcation investigation of an optimal velocity traffic model with driver reaction time. Phys. Rev. E 70, 026 207. (doi:10.1103/PhysRevE.70.026207)
21. Orosz, G., Krauskopf, B. & Wilson, R. E. 2005 Bifurcations and multiple traffic jams in a car-following model with reaction-time delay. Physica D 211, 277-293. (doi:10.1016/j.physd.2005. 09.004)
22. Rottschäfer, V. & Krauskopf, B. 2004 A three-parameter study of external cavity modes in semiconductor lasers with optical feedback. In Fifth IFAC workshop on time-delay systems (ed. W. Michiels & D. Roose). Leuven, Belgium: International Federation of Automatic Control (IFAC).
23. Stépán, G. 1986 Great delay in a predator-prey model. Nonlin. Anal. TMA 10, 913-929. (doi:10.1016/0362-546X(86)90078-7)
24. Stépán, G. 1989 Retarded dynamical systems: stability and characteristic functions. Pitman research notes in mathematics, vol. 210. Essex, England: Longman.
25. Stone, E. & Campbell, S. A. 2004 Stability and bifurcation analysis of a nonlinear DDE model for drilling. J. Nonlin. Sci. 14, 27-57. (doi:10.1007/s00332-003-0553-1)
26. Verduyn Lunel, S. M. & Krauskopf, B. 2000 The mathematics of delay equations with an application to the Lang-Kobayashi equations. In Fundamental issues of nonlinear laser dynamics (ed. B. Krauskopf & D. Lenstra), vol. 548, pp. 66-86. Melville, New York: American Institute of Physics.
27. Whitham, G. B. 1999 Linear and nonlinear waves. New York: Wiley.