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.
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References
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Carrillo, A., Delgado, J., Saavedra, P., Velasco, R.M., Verduzco, F. (2013). A Bogdanov–Takens Bifurcation in Generic Continuous Second Order Traffic Flow Models. In: Kozlov, V., Buslaev, A., Bugaev, A., Yashina, M., Schadschneider, A., Schreckenberg, M. (eds) Traffic and Granular Flow '11. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39669-4_2
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DOI: https://doi.org/10.1007/978-3-642-39669-4_2
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