Semigroup Forum

, Volume 92, Issue 2, pp 486–493 | Cite as

Linear chaos for the Quick-Thinking-Driver model

  • J. Alberto ConejeroEmail author
  • Marina Murillo-Arcila
  • Juan B. Seoane-Sepúlveda


In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars (Google driverless cars, Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behaviour using the same approach as for studying chaos of death models of cell growth.


Death model Birth-and-death problem Car-following Quick-Thinking-Driver Devaney chaos Distributional chaos \(C_0\)-semigroups 



The authors were supported by a grant from the FPU program of MEC and MEC Project MTM2013-47093-P.


  1. 1.
    Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647–655 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris Série II 329, 439–444 (2001)Google Scholar
  3. 3.
    Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755–775 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Banasiak, J., Lachowicz, M., Moszyński, M.: Topological chaos: when topology meets medicine. Appl. Math. Lett. 16(3), 303–308 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criteria for chaos. Discret. Contin. Dyn. Syst. 12(5), 959–972 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation–stability and chaos. Discret. Contin. Dyn. Syst. 29(1), 67–79 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Banks, J., Brooks, J., Cairns, G., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Am. Math. Mon. 99(4), 332–334 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. 457,019, 11 (2012)Google Scholar
  9. 9.
    Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. Part F 2(4), 181–196 (1999)CrossRefGoogle Scholar
  11. 11.
    Brzeźniak, Z., Dawidowicz, A.L.: On periodic solutions to the von Foerster–Lasota equation. Semigroup Forum 78, 118–137 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Chandler, R.E., Herman, R., Montroll, E.W.: Traffic dynamics: studies in car following. Op. Res. 6, 165–184 (1958)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Chung, C.C., Gartner, N.: Acceleration noise as a measure of effectiveness in the operation of traffic control systems. Operations Research Center. Massachusetts Institute of Technology. Cambridge (1973)Google Scholar
  14. 14.
    CNN (2014) Driverless car tech gets serious at CES. Accessed 7 Apr 2014
  15. 15.
    Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discret. Contin. Dyn. Syst. 35(2), 653–668 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
  17. 17.
    de Laubenfels, R., Emamirad, H., Protopopescu, V.: Linear chaos and approximation. J. Approx. Theory 105(1), 176–187 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793–819 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    El Mourchid, S.: The imaginary point spectrum and hypercyclicity. Semigroup Forum 73(2), 313–316 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    El Mourchid, S., Metafune, G., Rhandi, A., Voigt, J.: On the chaotic behaviour of size structured cell populations. J. Math. Anal. Appl. 339(2), 918–924 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    El Mourchid, S., Rhandi, A., Vogt, H., Voigt, J.: A sharp condition for the chaotic behaviour of a size structured cell population. Differ. Integral Equ. 22(7–8), 797–800 (2009)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics, vol. 194. Springer, New York, 2000. With contributions by Brendle S., Campiti M., Hahn T., Metafune G., Nickel G., Pallara D., Perazzoli C., Rhandi A., Romanelli S., and Schnaubelt RGoogle Scholar
  23. 23.
    Godefroy, G., Shapiro, J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Greenshields, B.D.: The photographic method of studying traffic behavior. In: Proceedings of the 13th Annual Meeting of the Highway Research Board, pp. 382–399 (1934)Google Scholar
  25. 25.
    Greenshields, B.D.: A study of traffic capacity. In: Proceedings of the 14th Annual Meeting of the Highway Research Board, pp. 448–477 (1935)Google Scholar
  26. 26.
    Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Universitext. Springer, London (2011)CrossRefzbMATHGoogle Scholar
  27. 27.
    Herman, R., Montroll, E.W., Potts, R.B., Rothery, R.W.: Traffic dynamics: analysis of stability in car following. Op. Res. 7, 86–106 (1959)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Helly, W.: Simulation of Bottleneckes in Single-Lane Traffic Flow. Research Laboratories, General Motors. Elsevier, New York (1953)Google Scholar
  29. 29.
    Li, T.: Nonlinear dynamics of traffic jams. Phys. D 207(1–2), 41–51 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Lo, S.C., Cho, H.J.: Chaos and control of discrete dynamic traffic model. J. Franklin Inst. 342(7), 839–851 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Martínez-Giménez, F., Oprocha, P., Peris, A.: Distributional chaos for backward shifts. J. Math. Anal. Appl. 351(2), 607–615 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Pipes, L.A.: An operational analysis of traffic dynamics. J. Appl. Phys. 24, 274–281 (1953)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • J. Alberto Conejero
    • 1
    Email author
  • Marina Murillo-Arcila
    • 1
  • Juan B. Seoane-Sepúlveda
    • 2
  1. 1.Instituto Universitario de Matemática Pura y Aplicada (IUMPA)Universitat Politècnica de ValènciaValenciaSpain
  2. 2.Departamento de Análisis Matemático, Facultad de Ciencias MatemáticasUniversidad Complutense de MadridMadridSpain

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