Global bifurcation diagram for the Kerner-Konhauser traffic flow model

Abstract

We study traveling wave solutions of the Kerner--Konh\"auser PDE for traffic flow. By a standard change of variables, the problem is reduced to a dynamical system in the plane with three parameters. In a previous paper (Carrillo, F.A., J. Delgado, P. Saavedra, R.M. Velasco and F. Verduzco, (2013). Traveling waves, catastrophes and bifurcations in a generic second order traffic flow model --to appear in \textit{International Journal of Bifurcation and Chaos}--, it was shown that under general hypotheses on the fundamental diagram, the dynamical system has a surface of critical points showing either a fold or cusp catastrophe when projected under a two dimensional plane of parameters named $q_g$--$v_g$. In any case a one parameter family of Bogdanov--Takens (BT) bifurcation takes place, and therefore local families of Hopf and homoclinic bifurcation arising from each BT point exist. Here we prove the existence of a degenerate Bogdanov--Takens bifurcation (DBT) which in turn implies the existence of Generalized Hopf or Bautin bifurcations (GH). We describe numerically the global lines of bifurcations continued from the local ones, inside a cuspidal region of the parameter space. In particular, we compute the first Lyapunov exponent, and compare with the GH bifurcation curve. We present some families of stable limit cycles which are taken as initial conditions in the PDE leading to stable traveling waves.