2.4. CharDirs

The charDirs approach is the oldest of the fixed points visualizations developed during this project.

This approach is completely different to the three sphere-based approaches described in the last three sections. In this technique, the eigenvectors and eigenvalues are analytically calculated, and used as the basis of the visualization. With the calculated eigenvectors and the corresponding eigenvalues the characteristics of the fixed point are determined, and then visualized. The directions of the eigenvectors together with the corresponding eigenvalues are called the characteristic directions of the fixed point, hence the name of this technique.

For every eigenvector a streamline is integrated (either in positive or negative time, corresponding to the behavior along this eigenvector). If there exists a plane, where all directions are of the same importance, as there don't exist any two special eigenvectors for this plane, but any vectors are possible, then this plane is shown as a streamsurface, starting at a circle in the plane around the fixed point. The fixed point itself is again shown as a small sphere, but this time the sphere has nothing to do with all the other visualization elements (streamlines, steramsurfaces, arrows, ...).

As mentioned above, the eigenvectors are taken as the startingpoints for streamlines, these streamlines are either shown green (repelling direction) or red (attracting direction). To enhance the visualization, and to get a little better feeling over the speed in the characteristic directions, arrows are added to the streamlines, which show a logarithmic scale, that is two arrows represent 10 times the speed of one arrow. Besides the arrows help also to better see if the charcteristic direction is attracting or repelling.