Visualization of critical points

Besides the direct visualization of dynamical systems the visualization after analysis is an important field of research. Often the topology of a dynamical system is extracted in advance, and visualization is used afterwards to communicate the results. Usually the following procedure is used: First, the critical points are identified. Next, the Jacobian matrix is investigated for all critical points to classify their type (attractor, repellor, etc.), and to extract the characteristic trajectories emanating from these points. Further investigation is used to clarify the interrelation between the critical points. Then, critical elements of higher order, e.g., cycles or tori, are searched. Again local derivatives are investigated to identify the type of these higher order critical elements.

The identification of critical points as the most important step of flow topology analysis raises a demand on sophisticated visualization techniques to convey the results of the analysis. Location, type, as well as the local properties derived from the Jacobian matrix can be visualized. Depending on the type of the critical point different methods are useful:

Three characteristic trajectories -

if all eigenvalues of the Jacobian matrix are real, different from each other, and different from zero, three characteristic trajectories are connected to the critical point. If all of the eigenvalues are negative (positive), the critical point is an attracting (repelling) node. Mixed signs indicate real saddles.

An intuitive visualization is to (numerically) integrate the characteristic trajectories and indicate the order of attraction/repulsion by the number of arrow glyphs positioned onto them. As an example Fig. 6.1(a) shows the critical point of a linear dynamical system.

Two characteristic sets (1D & 2D) -

if two of three real eigenvalues are equal or a pair of complex numbers, one local characteristic structure is a surface. Again, the type of the critical point can be an attractor, repellor, or saddle.

Visualizing such a critical point, a stream surface is integrated to represent the 2D characteristic element, and two stream lines are integrated for the 1D characteristic direction associated with the critical point.

Other types -

Of course there are other, usually non-hyperbolic types of critical points. Arbitrary complex local structures can appear, especially if the Jacobian matrix degenerates and higher-order derivatives are to be investigated for analysis.

Specifying a general visualization technique for all these types is cumbersome and usually not necessary, since usually critical points are of one of the simple types described above - non-hyperbolic systems often are considered to be just a transitional element between two hyperbolic systems with differing behavior.

For the visualization of system abstractions it is useful to also include a certain amount of direct visualization to give a few intuitive visual hints for understanding the abstract structure. Streamlets, for example, usually are easily understood and, thus, well suited for being combined with the visualization of flow topology. See Fig. 6.5 for three examples of such a combination. Characteristic elements as well as few direct visual hints, i.e., streamlets originating near the critical points, are displayed.

mailto:helwig@cg.tuwien.ac.at.