Visualizing the dynamical behavior of Wonderland

Abstract

The article concentrates on the visualization of a four dimensional continuous dynamical system, the Wonderland model. This system poses some interesting analysis challenges. Its temporal evolution, for example, is characterized by slow-fast dynamics. This means that some of the system variables vary much faster than others. The velocity varies by two magnitudes and thus makes visualization difficult. Due to the underlying slow-fast dynamics, some equilibrium surfaces play an essential role in the model's analysis. We consider equilibrium surfaces on which the derivative of the fast system variable equals zero. Due to their definition, these equilibrium surfaces do not describe stream surfaces, which means that trajectories intersect and cross them. For these surfaces we cannot use visualization techniques designed for stream surfaces, which are based on the assumption that a flow starting on the surface will stay within it. Econometrics research typically uses only elementary visualization techniques such as still images, simple line drawings, and 2D projections or cross-sections of higher dimensional phase spaces. Color use is scarce. We collaborated with econometricians to apply more sophisticated visualization techniques and adaptations to the Wonderland model using standard visualization software (AVS by Advanced Visual Systems). The visualizations provided new insights into the system's slow-fast dynamics. Using a commercial rendering package (Autodesk's 3D Studio), we also turned the visualizations into high quality images for presentations.