Skip to main content
SpringerLink
Log in

Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The phenomenon of chaos has been exhibited in mathematical nonlinear models that describe traffic flows, see, for instance (Li and Gao in Modern Phys Lett B 18(26–27):1395–1402, 2004; Li in Phys. D Nonlinear Phenom 207(1–2):41–51, 2005). At microscopic level, Devaney chaos and distributional chaos have been exhibited for some car-following models, such as the quick-thinking-driver model and the forward and backward control model (Barrachina et al. in 2015; Conejero et al. in Semigroup Forum, 2015). We present here the existence of chaos for the macroscopic model given by the Lighthill–Whitham–Richards equation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Albanese, A.A., Barrachina, X., Mangino, E.M., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12(5), 2069–2082 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647–655 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris Sér. II 329, 439–444 (2001)

    Google Scholar 

  4. Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755–775 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criteria for chaos. Discrete Contin. Dyn. Syst. 12(5), 959–972 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation—stability and chaos. Discrete Contin. Dyn. Syst. 29(1), 67–79 (2011)

    MathSciNet  MATH  Google Scholar 

  7. Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. Art. ID 457019, 11 (2012)

  8. Barrachina, X., Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Distributional chaos for the forward and backward control traffic model (2015, preprint)

  9. Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)

    Book  MATH  Google Scholar 

  10. Bayart, F., Matheron, É.: Mixing operators and small subsets of the circle. J Reine Angew. Math. (2015, to appear)

  11. Bermúdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Stud. Math. 170(1), 57–75 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bernardes Jr, N.C., Bonilla, A., Müller, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265(9), 2143–2163 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  14. Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. Part F Traffic Psychol. Behav. 2(4), 181–196 (1999)

    Article  Google Scholar 

  15. Conejero, J.A., Lizama, C., Rodenas, F.: Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation. Appl. Math. Inf. Sci. 9(5), 1–6 (2015)

    MathSciNet  Google Scholar 

  16. Conejero, J.A., Mangino, E.M.: Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators. Mediterr. J. Math. 7(1), 101–109 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Conejero, J.A., Müller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic \(C_0\)-semigroup. J. Funct. Anal. 244, 342–348 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Linear chaos for the quick-thinking-driver model. Semigroup Forum (2015). doi:10.1007/s00233-015-9704-6

  19. Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions. Int. J. Bifur. Chaos Appl. Sci. Eng. 20(9), 2943–2947 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discrete Contin. Dyn. Syst. 35(2), 653–668 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793–819 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  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 S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt

  23. Grosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Universitext. Springer, London (2011)

  24. Herzog, G.: On a universality of the heat equation. Math. Nachr. 188, 169–171 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Li, K., Gao, Z.: Nonlinear dynamics analysis of traffic time series. Modern Phys. Lett. B 18(26–27), 1395–1402 (2004)

    Article  MATH  Google Scholar 

  26. Li, T.: Nonlinear dynamics of traffic jams. Phys. D Nonlinear Phenom. 207(1–2), 41–51 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lustri, C.: Continuum Modelling of Traffic Flow. Special Topic Report. Oxford University, Oxford (2010)

    Google Scholar 

  28. Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317–345 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  29. Maerivoet, S., De Moor, B.: Cellular automata models of road traffic. Phys. Rep. 419(1), 1–64 (2005)

    Article  MathSciNet  Google Scholar 

  30. Mangino, E.M., Peris, A.: Frequently hypercyclic semigroups. Stud. Math. 202(3), 227–242 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Murillo-Arcila, M., Peris, A.: Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398, 462–465 (2013)

  32. Murillo-Arcila, M., Peris, A.: Strong mixing measures for \(C_0\)-semigroups. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 109(1), 101–115 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44. Springer, New York (1983)

  34. Protopopescu, V., Azmy, Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  35. Richards, P.I.: Shock waves on the highway. Oper. Res. 4, 42–51 (1956)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The authors are supported by MEC Project MTM2013-47093-P. The second and third authors are supported by GVA, Project PROMETEOII/2013/013.

Author information

Authors and Affiliations

  1. Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, València, Spain

    J. Alberto Conejero, Félix Martínez-Giménez, Alfredo Peris & Francisco Ródenas

Authors
  1. J. Alberto Conejero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Félix Martínez-Giménez
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alfredo Peris
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Francisco Ródenas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alfredo Peris.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Conejero, J.A., Martínez-Giménez, F., Peris, A. et al. Chaotic asymptotic behaviour of the solutions of the Lighthill–Whitham–Richards equation. Nonlinear Dyn 84, 127–133 (2016). https://doi.org/10.1007/s11071-015-2245-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2245-4

Keywords

Search

Navigation