Duration: 1996 -
This page assembles some examples for visualization of three-dimensional dynamical systems. This work is done in the scope of our research topic ``Visualization of Complex Dynamical Systems''. The images are provided in JPEG format.
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Plate 1 is a visualization of the Lorenz attractor. Three different techniques have been used for the generation of this image. A streamline approximates the attractor, whereas the fixed points of this dynamical system are visualized separately. Two of them, which exhibit one attracting 1D component and a rotating 2D component each, are visualized by presenting locally the principal directions. The third fixed point, which is a saddle as well, but does not contain any rotational component, is visualized by integrating short streamlines from seed points evenly distributed over a sphere around the fixed point.
Plate 2 shows a visualization of a dynamical system, which was modelled on the basis of a biochemical process, namely the break-down of sugar by cells to generate energy. The cyclic behavior of the system can be easily obtained by this visualization, which consists of streamlines and streamsurfaces.
Plates 3, 4, and 5 present the only fixed point of three different linear dynamical systems, each visualized using the same techniques as for the fixed points of Lorenz in Plate 1.
Plate 6 is another visualization of the Lorenz attractor. Just the fixed points are visualized using the same technique: Starting a bunch of streamlines from a sphere around the fixed points.
Plate 7 is again a visualization of a fixed point of a linear dynamical system using streamlines starting on a sphere around the fixed point. Red streamlines are integrated backwards and the green ones forwards, therefore it is easy to see where the flow comes from and where it goes in the vicinity of the fixed point.
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helwig löffelmann, last update on march 4, 1997.