Discussion

  

Figure 5.13: Extreme phase relations as difficult cases for the visualization of Poincaré maps. [left image] [right image]

  

Figure 5.14: Combined visualization techniques to disambiguate results. [left image] [right image]

Poincaré maps are a useful mathematical concept for the analysis of periodic or quasi-periodic flows. Thus, it is a good idea to use this concept for the visualization of dynamical systems that exhibit such a periodic behavior. Similarly to the investigation of continuous dynamical system, also the examination of Poincaré maps depends on the investigation goal of the user. The technique proposed in this chapter allows to investigate the long-term behavior of specific points of interest, or one iteration of the map for many points of the Poincaré section (or even the entire set). Furthermore it is useful to integrate visualization on the basis of Poincaré maps and direct flow visualization.

There are, however, some difficult situations, where the application of the Poincaré technique itself is not useful. For instance, if extreme phase relations occur comparing main rotation and secondary rotation, then resulting images usually are difficult to interpret - see Fig. 5.13 for two examples. In cases of frequency coupling, for example, visualization composed of at least two of the presented techniques can help to disambiguate results. See Fig. 5.14 for two examples.

mailto:helwig@cg.tuwien.ac.at.