2.6. Combinations of the different techniques

In the system developed during this project, all these five different visualization techniques presented in the last few subsections, can be combined and therefore even further enhance the understanding of the behavior of the fixed points. These combinations can be applied in different ways. If a dynamical system has several fixed points (at least more than one), than the different fixed points can be visualized with different techniques, so that for each behavior the optimal visualization technique can be applied. But it is also possible, to combine different visualization techniques for the visualization of one fixed point at the same time. This sometimes improves the understanding of the behavior, too. Examples for these two methods of visualization techniques combinations will be shown now:

In the two pictures above, always one fixed point has been visualized by using 2 methods at the same time, and therefore the combination of the two visualization techniques enhances the understaning of the behavior of this fixed point.



In the above picture the three fixed points of the Lorentz system have been visualized each by using a different technique. By combinig the visualization methods in this way, the most appropriate visualization technique for each fixed point can be chosen independently. And therefore the whole system and its behavior is even easier to understand.

In the next three pictures the "CharDirs" technique is again combined with the "SphereTufts" technique for the visualization of the fixed points, but also colored streamlines are used this time, to give an even better understanding of the behavior around the fixed point.