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The first visualization technique presented here, i.e.,
CHARDIRS, is similar
to the iconic representation of flow
near critical points [31,74].
Instead of using a glyph to encode the flow topology, we directly
represent the geometry of behavior by the
use of stream lines and stream surfaces. We first inspect the
eigenvalues of the Jacobian matrix and distinguish between
topological different cases:
Figure 6.1:
Visualizing the geometry of behavior near the critical
point of a linear dynamical system - three different saddle
configurations. [left image] [center image] [right image]
![\framebox[\textwidth]{
\begin{tabular*}{.93\linewidth}{@{}@{\extracolsep{\fill}...
...arDirs.ps}
\\ {\small{}(a)}
& {\small{}(b)}
& {\small{}(c)}
\end{tabular*} }](img246.gif) |
- 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.
To add more quantitative information we encode the order of
magnitude of the eigenvalues by a certain number of arrows along
the characteristic trajectories. Thereby the geometry of behavior
near critical points is visualized for the most important cases.
Degeneracies of flow geometry, e.g.,
non-hyperbolic critical points, are not considered through this
approach.
Figure 6.2:
Visualizing the flow characteristics near the critical
points of the Lorenz system:
(a) CHARDIRS and
(b) SPHERETUFTS.
![\framebox[\textwidth]{
\begin{tabular*}{.93\linewidth}{@{}@{\extracolsep{\fill}...
...pics/Lorenz.sphereTufts.ps}
\\ {\small{}(a)}
& {\small{}(b)}
\end{tabular*} }](img247.gif) |
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.
Next: SPHERETUFTS - using many
Up: Visualization of critical points
Previous: Vector field topology and
Helwig Löffelmann, November 1998, mailto:helwig@cg.tuwien.ac.at.