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State of the art

Lord grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference.
St. Francis of Assisi (1181-1226)

The visualization of dynamical systems must be viewed in the context of a few related fields. Flow visualization, for example, is tightly related to it, as flow data can be seen as a special class of dynamical systems - flow, for example, usually is considered to be compressible only up to a certain level. No attracting nodes can be found in such a system. Dynamical systems, on the other hand, principally do not have any restrictions and, thus, can be considered as a super-class of flows.

Nevertheless, it is useful to distinguish between flows and dynamical systems, since often different aspects of interest are investigated through visualization. Another difference between flows and dynamical systems, which significantly influences visualization, is that flows usually are given discretely on large-sized grids whereas dynamical systems usually are given analytically by a few equations. Next to computational methods, experimental flow visualization techniques are also of interest. They provide the possibility to evaluate computational methods. Furthermore they have been inspirational for quite a few computer-based visualization techniques.

Another field related to the visualization of dynamical systems is visualizing tensor fields. In this case the type of data (vectors vs. tensors) is not compatible. However, the methods used to visualize either vector or tensor fields have several concepts in common, e.g., the use of integral curves. The aim for extracting the field topology in order to condense the content of information transported via visualization is also common to both fields.

The mathematical theory about ordinary differential equations (ODEs) is another important related field. It provides a common language to effectively describe dynamical systems. Furthermore, the field of computational fluid dynamics (CFD) provides a number of useful terms to characterize and describe dynamical systems. Its main focus is the simulation of flows. Both fields are kind of a basis the visualization of dynamical systems is built on.


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Next: Visualizing dynamical systems Up: Helwig's PhD thesis Previous: Visualization of dynamical systems
Helwig Löffelmann, November 1998,