Introduction

As already addressed in Chapter 1, there are many ways to investigate a dynamical system. An abstract approach to analysis is the investigation of the topological skeleton of a dynamical system [5,6]. The topology of behavior is built from characteristic elements, such as, for example, critical points and separatrices (see also Section 6.2). Once the topological structure of a dynamical system is derived, the long term evolution is known for all the points in phase space qualitatively. However, almost no quantitative information, such as, e.g., spatial extent of trajectories, is provided through this approach.

Spatially quantitative insights are gained from building the geometry of behavior. Location and shape of the topological entities are composed to a geometry within phase space [1,31]. The topological structure is extended by the spatial attributes of its elements. Thereby at least some quantitative information is provided. The geometry of behavior is a rather dense description of the flow without flow details apart from characteristic structures.

Detailed (local) information is gained by directly visualizing the dynamics using stream lines, stream surfaces, particle systems, or other integral objects [67]. Selected trajectories are visualized through these techniques. This kind of visualization technique always lacks completeness.

 

Table 6.1: density and completeness of information, and amount of quantities, provided through different approaches of dynamical system analysis.

	density	completeness	quantities
the topological structure	high	usually yes	almost non
the geometry of behavior	medium	potentially yes	just a few
selective direct visualization	regular	no	yes, details

 

As the previously described approaches have their advantages and disadvantages each (see Tab. 6.1 for a summary), a combination of them is appropriate. Most approaches found in literature, on the other hand, concentrate on either the one or the other.

In this chapter we first present two new techniques for the visualization of a continuous dynamical system in 3D space, which belong to two different approaches concerning the analysis of the flow data. Furthermore, we demonstrate that combining both methods yields better results.

mailto:helwig@cg.tuwien.ac.at.