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A Bogdanov–Takens Bifurcation in Generic Continuous Second Order Traffic Flow Models

In: Traffic and Granular Flow '11

Author

Listed:
  • Armando Carrillo

    (Universidad de Sonora, Mathematics Department)

  • Joaquín Delgado

    (UAM–Iztapalapa, Mathematics Department)

  • Patricia Saavedra

    (UAM–Iztapalapa, Mathematics Department)

  • Rosa Maria Velasco

    (UAM–Iztapalapa, Physics Department)

  • Fernando Verduzco

    (Universidad de Sonora, Mathematics Department)

Abstract

We consider the continuous model of Kerner–Konhäuser for traffic flow given by a second order PDE for the velocity and density. Assuming conservation of cars, traveling waves solution of the PDE are reduced to a dynamical system in the plane. We describe the bifurcations set of critical points and show that there is a curve in the set of parameters consisting of Bogdanov–Takens bifurcation points. In particular there exists Hopf, homoclinic and saddle node bifurcation curves. For each Hopf point a one parameter family of limit cyles exists. Thus we prove the existence of solitons solutions in the form of one bump traveling waves.

Suggested Citation

  • Armando Carrillo & Joaquín Delgado & Patricia Saavedra & Rosa Maria Velasco & Fernando Verduzco, 2013. "A Bogdanov–Takens Bifurcation in Generic Continuous Second Order Traffic Flow Models," Springer Books, in: Valery V. Kozlov & Alexander P. Buslaev & Alexander S. Bugaev & Marina V. Yashina & Andreas Schadsch (ed.), Traffic and Granular Flow '11, edition 127, pages 15-25, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-39669-4_2
    DOI: 10.1007/978-3-642-39669-4_2
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