2.2. SphereTufts

This technique is an extension of the IntLenPins approach, where most of the technique is the same, but the main difference is the following: the integration is not only done for getting the spatial length, but every integration step is taken into account by visualizing the entire result of the integration by a streamline. So this visualization shows not only the differences in the length of the integration (and therefore in the speed around the fixed point), but also the directional behavior of the "flow" near the fixed point.

This approach is of course a much more direct visualization approach, because the abstraction of the integration (which can be rather strange bent or even tangled sometimes) by plotting only a straight line is replaced by visualizing the actual integration itself. Of course, this can sometimes (when integrating very accurately) lead to getting too much data for displaying. This can be overcome by simply decoupling the process of the calculation from the visualization by calculating very accurately , but displaying only coarser data (for example: connecting every 10.th step of the calculation with straight lines).
That is why the user can choose interactively for this visualization technique, how accurate he wants the data to be displayed (not to be calculated!).

Here are now some results obtained by using this technique with different dynamical systems. One can clearly see the advantage of not only drawing straight lines for each startpoint of the integration by comapring these results with the one shown in the last section (IntLenPins).

In the two pictures above, two fixed points are visualized with this technique, the two fixed points have clearly visible different behavior and characteristics. In all the pictures generated with this visualization technique, the green streamlets give the direction, where the fixed point is repelling, the red streamlets are used for the attracting streamlets. This can be interpreted as if green means "GO!", that is, the integration moves on (away from the fixed point), red means "STOP!", as the integration for these streamlets ends at the fixed point itself (after infinite time, of course), so the integration stops there.

In this picture the three fixed points of the Lorentz system are visualized using this fixed points visualization technique. As you can see, this technique not only visualizes the directions, but also the speed of the different directions and the speed at the different fixed points, as all the streamlets are integrated over the same length of time, and so the differnces in the spatial lengths shows the difference in the speeds.