Analysis of Macroscopic Behavior in Complex Systems - with Applications to Traffic, Pedestrian Flow and Mechanical Laboratory experiments

Physikalisches Kolloquium

Understanding the macroscopic behavior of given complex systems and their parameter dependence is a typical aim to achieve in science. The presentation will focus on two numerical methods which can be applied in cases where analytical approaches are impossible or too difficult to apply. First, an implicit numerical multiscale approach is introduced in the framework of slow-fast dynamical systems. The method allows to perform numerical investigations of the macroscopic behavior of microscopically defined complex systems including continuation and bifurcation analysis on the coarse or macroscopic level where no explicit equations are available. This approach fills a gap in the analysis of many complex real-world applications including particle models with intermediate number of particles where the microscopic system is too large for direct investigations of the full system and too small to justify large-particle limits. The results include an equation-free continuation of traveling wave solutions, identification of saddle-node and Hopf-bifurcations as well as two-parameter continuations of bifurcation points.
Second, a control-based continuation technique will be presented which allows to track also unstable branches of bifurcation diagrams. This method can be applied not only for microscopically defined complex systems but also to laboratory experiments. The methods are demonstrated with applications to particle models of traffic as well as pedestrian flow situations and periodically forced nonlinear pendulum experiment.




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