Visualizing dynamical systems

In the left graph three critical points (`A', `B', and `C') are denoted. Characteristic trajectories coinciding with the eigenvectors of the Jacobian matrices associated with the critical points are added. Small arrows indicate the orientation of flow. In addition to the critical points also a cycle (`D') appears. Finally, a few additional trajectories are plotted to give an impression about the important features of the dynamical system being visualized. This type of sketch is quite usual for illustrating the most important structures of low-dimensional dynamical systems.

Another interesting book on dynamical systems is ``Dynamics - The
Geometry of Behavior'' [1] by Abraham, a
mathematician, and Shaw, an artist. Hand-drawn images are used to
visualize certain characteristics of special dynamical systems.
The mathematician provides knowledge concerning the important
structures of the dynamical systems. The artist has an ability to
clearly convey complex spatial arrangements through only a small set
of visual cues. The cooperation of both results in effective
depictions of dynamical systems.
Two examples out of this book can be seen in Fig. 2.2.

Fig. 2.2(a) gives a sketch of a
two-dimensional system. A
cycle (red trajectory) around a critical point in the center is
shown together with a few accompanying trajectories.
Fig. 2.2(b) visualizes a dynamical system with
three variables. A
saddle critical point and a saddle cycle are shown in red and
white. The surface structures in-between these two characteristic
sub-sets make up the main part of the image. The sketch
illustrates a rather
complex relation between the critical point and the saddle cycle.

Helwig Löffelmann, November 1998,