Visualizing dynamical systems

The investigation of dynamical systems spans a wide research area. Models of systems with a state that varies over time, often are formalized using dynamical systems. Examples are food chains, econometric models, chemical reaction systems, meteorologic models, and stock market models. Usually researchers start with an in-depth analysis of the dynamical system. Afterwards visualization is used to communicate the results of the analysis. Critical sub-sets, e.g., critical points, separatrices, or bifurcation lines, are combined with additional integral cues as, for example, stream lines, stream surfaces, etc. In many text books about dynamical systems one can find images accompanying the theoretical analysis. Mapping phase space to image space is an intuitive way to visualize specific sub-sets of a two-dimensional phase space. Critical points and characteristic trajectories usually make up an important part of such illustrations. See Fig. 2.1 for two examples out of ``Nonlinear Dynamics and Chaos'' by Strogatz [80].

  

Figure 2.1: Two examples of visualization used for the communication of results gained from in-depth analysis of two-dimensional dynamical systems (images by Strogatz [80]).

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.

  

Figure 2.2: Two examples of hand-drawn flow visualization (images by Abraham and Shaw [1]).

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.

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