CHARDIRS - visualizing eigen-manifolds

- 1.
- If all the eigenvalues are real, different from each other, and different from zero, three pairs of stream lines are integrated into the direction of the corresponding eigenvectors. Thereby the (locally) most significant trajectories are depicted (see Fig. 6.1(a)).
- 2.
- If all the eigenvalues are real, different from zero, but two of them are equal, a 1-manifold and a 2-manifold corresponding to the double eigenvalue build up the geometry of behavior near the critical point. In addition to a pair of stream lines we use three stream lines within the 2-manifold plus an optional stream surface to encode this special flow topology (see Fig. 6.1(b)).
- 3.
- If two eigenvalues are complex and the real parts of all eigenvalues are different from zero, the same geometry of behavior is present as in the second case. Thus the same visualization technique is used (see Fig. 6.1(c)). However, the flow characteristics are different - spiraling vs. radial attraction/repulsion occurs.

In Fig. 6.2(a) the Lorenz
system [63] was
visualized by the use of this technique. Two saddle foci with
(each) a pair of conjugated complex eigenvalues and
a large negative real eigenvalue drive the rotating
characteristic of this chaotic dynamical system. A third saddle
coordinates the alternating dominance of these two foci.

Helwig Löffelmann, November 1998,