Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models

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.Comment: 35 page

Free Random Levy Variables and Financial Probabilities

We suggest that Free Random Variables, represented here by large random matrices with spectral Levy disorder, may be relevant for several problems related to the modeling of financial systems. In particular, we consider a financial covariance matrix composed of asymmetric and free random Levy matrices. We derive an algebraic equation for the resolvent and solve it to extract the spectral density. The free eigenvalue spectrum is in remarkable agreement with the one obtained from the covariance matrix of the SP500 financial market.Comment: 8 pages with 2 EPS figures; talk given by M.A. Nowak at NATO Advanced Research Workshop ``Applications of Physics to Economic Modeling'', Prague, 8-10 February, 200