2013/01/31 by Christian Marschler, Jan Sieber, Rainer Berkemer +2 · 1 citation
Computer Science · Economics, Econometrics and Finance · Engineering · Mathematics · #Applied mathematics #Bifurcation #Computer science #Convergence (economics) #Mathematical analysis #Mathematical optimization #Mathematics #Measure (data warehouse) #Nonlinear Dynamics and Pattern Formation #Nonlinear system #Physics #Saddle #Slow manifold #Statistical physics #Stochastic processes and financial applications #Traffic control and management #Traffic flow (computer networking) #math.DS #msc:34E13 #msc:37M20 #msc:37Mxx #msc:65P30
paper · pdf · doi:10.1137/130913961
published as SIAM J. Appl. Dyn. Syst., 13(3), 1202-1238 (2014) · 35 pages
openalex publication_date 2014/01/01 · arxiv created 2014/07/09 · arxiv updated 2015/08/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold within an error that is exponentially small with respect to the small parameter measuring time scale separation. Second, we apply this result to the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. The traffic jams are waves that travel slowly against the direction of traffic. Equation-free analysis enables us to investigate the behavior of the microscopic traffic model on a macroscopic level. The standard deviation of cars' headways is chosen as the macroscopic measure of the underlying dynamics such that traveling wave solutions correspond to equilibria on the macroscopic level in the equation-free setup. The collapse of the traffic jam to the free flow then corresponds to a saddle-node bifurcation of this macroscopic equilibrium. We continue this bifurcation in two parameters using equation-free analysis.