Introduction

The stream arrows technique was developed during a cooperation with mathematicians who investigated dynamical systems that exhibit mixed-mode oscillations [56,57]. Specifically, a simplified model, called the three-dimensional autocatalator, which describes the interactions of three chemical entities, is investigated. See Eq. 4.1 for the mathematical description of this three-dimensional dynamical system. See Fig. 4.1(a) for a stream line typical for this dynamical system.

$$\begin{array}{rcl} \dot{a} & = & \mu \cdot ( \kappa + c ) - ... ...appa\!=\!2.5$ }, \mbox{$\varepsilon\!=\!0.013$ }} \end{array}$$ (4.1)

Stream surfaces were determined to be a proper representation of the principal dynamics of this dynamical system. However, stream surfaces usually are spatially extensive, and thus occlusion often becomes a problem when this visualization technique is used. See Fig. 4.1(b) for two stream surfaces calculated for the same system. By introducing semi-transparent stream arrows and selectively removing parts of the stream surface two improvements are provided (see Sects. 4.2, 4.3, and 4.5).

  

Figure 4.1: A typical stream line (a)  and stream surface (b)  calculated for a dynamical system exhibiting mixed-mode oscillations.

A major inspiration concerning a solution of how to deal with the problem of occlusion caused by complex shaped (for example, curly) stream surfaces comes from a book on the ``Geometry of Behavior'' by Abraham and Shaw [1]. The authors discuss various types of three-dimensional dynamical systems by using hand-drawn illustrations which represent the topological structure of their systems. Stream surfaces make up an important part of most of their images. To reduce the negative effects caused by occlusion they use only arrow-shaped parts of a stream surface instead of the whole surface. Additionally they use arrow-shaped holes within stream surfaces to diminish occlusion. Using this approach directional information is also added to the stream surfaces. This enables the viewer to obtain a better feeling for the flow within a stream surface. They also use simple textures in their hand-drawn illustrations to convey a better understanding of the shape of objects in phase space. Refer to Fig. 4.2 for a typical image out of this book.

  

Figure 4.2: Visualization of a dynamical system by using stream arrows [1].

Another issue of this work is to increase the information provided by a stream surface. Plain stream surfaces, for example, do not represent the direction of flow within the stream surface. Also, neither temporal cues about the integration (time lines) nor information about the flow in the vicinity of a plain stream surface is available.

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