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Figure 5.13:
Extreme phase relations as difficult cases for the
visualization of Poincaré maps. [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.
Next: Visualization of critical points
Up: Poincaré maps and visualization
Previous: Animation aspects
Helwig Löffelmann, November 1998, mailto:helwig@cg.tuwien.ac.at.