2. Fixed Points Visualization

The main concern of this project was to find new and interesting, but also very well suited visualization techniques for the visualization of fixed points in three-dimensional dynamical systems. That is, why the main part of this documentation is also on these new and advanced visualization techniques.

In the following sections five methods for the visualization of fixed points in dynamical systems will be described, each of them has been developed as an AVS-Module, and can be compiled with any of the dynamical systems, that are available in the DynSys3D system, and that have implemented knowledge about their fixed points.
The five methods are:

• IntLenPins: a first, more abstract test.
• SphereTufts: the first "real" tool, that showed our ideas.
• Advanced SphereTufts: the further development of the SphereTufts approach, by applying an adaptive algorithmn.
• CharDirs: a tool for visualizing the characteristic directions of the given fixed point.
• PlotXYZ: here you can plot the x, y, or z value over time as a functiongraph.

As already mentioned in the Introduction, one of the differences in visualizing dynamical systems is in wheter one chooses abstract visualization techniques or more direct ones. As far as the above mentioned techniques are concerned, we can now distinguish between these different approaches as to how abstract or direct they are:

• PlotXYZ: this technique is the most abstract technique, here functiongraphs in 2D are plotted, so you even don't have any 3D Information shown here.
• CharDirs: here the information from the Jacobimatrix is extracted, the eigenvalues and eigenvectors are calculated, and then this information is taken as starting point for the visualization, which has some direct aspects.
• IntLenPins: this was the first test for the next to described techniques, which are more direct than this one. Here a sphere is choosen, the center of which is the fixed point. On this sphere points are stochastically chosen, from there on integration is calculated over some fixed time, and the length of the total integration is shown as a straight line perpendicular to the surface of the sphere.
• SphereTufts: this technique is already a more direct visualization technique, as the straight lines from the IntLenPins approach are replaced by really integrated streamlines, so one can not only see the different lenght of the integration in different points, but also the directional behavior.
• Advanced SphereTufts: this is the most direct and also most advanced approach, and the one we were always working for. Here the points on the sphere were the streamlets are started are chosen evenly distributed as astarting configuration, and then adaptively the mesh of the startingpoints is refined, by classifying which regions are the most interesting ones, and there more streamlets are rendered.

In the system, all these five different visualization techniques 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 later in the Combinations Section.

These five methods will now be investigated. After that, there will be a short description of all the other visualization techniques and tools needed for those, that are not suitable to visualize fixed points, but nevertheless, are very useful for general visualizations of three-dimensional dynamical systems.