Equation-Free Detection and Continuation of a Hopf Bifurcation Point in a Particle Model of Pedestrian Flow

Using an equation-free analysis approach we identify a Hopf bifurcation point and perform a twoparameter continuation of the Hopf point for the macroscopic dynamical behavior of an interacting particle model. Due to the nature of systems with a moderate number of particles and noise, the quality of the available numerical information requires the use of very robust numerical algorithms for each of the building blocks of the equation-free methodology. As an example, we consider a particle model of a crowd of pedestrians where particles interact through pairwise “social forces.” The pedestrians move along a corridor where they are constrained by the walls of the corridor, and two crowds are aiming, from opposite directions, to pass through a narrowing doorway perpendicular to the corridor. We focus our investigation on the collective behavior of the model. As the width of the doorway is increased, we observe an onset of oscillations of the net pedestrian flux through the doorway, described by a Hopf bifurcation. An equation-free continuation of the Hopf point in the two parameters, door width and ratio of the pedestrian velocities of the two crowds, is performed.