Visualization of Strange Attractors

project duration: 1993-1994

Typically strange attractors are depicted by taking a one-dimensional trajectory as approximation. This can be done fast and easily but does have some disadvantages. With a one-dimensional trajectory it is rather difficult to get an accurate visual impression of the geometric shape of a strange attractor. This problem is even more severe if only a still image of an attractor is given. Furthermore a one-dimensional trajectory does not contain enough information to clearly illustrate the behavior of the underlying dynamical system. We implemented more complex visualization techniques for displaying strange attractors and their chaotic properties. Some of the techniques we investigated were: sweep representation of trajectories, placing 3D solids at discrete positions along a trajectory, visualization of the flow dynamics, volume visualization, Poincare sections, color coding chaotic properties, computer animation. Further information is given in the technical report Application of Visualization Techniques to Complex and Chaotic Dynamical Systems

The local stability of chaotic dynamical systems is especially interesting as it desribes the limited predictability of such systems. Lyapunov exponents (an n-dimensional system does have n Lyapunov exponents) which are due to a local linearization of the system correspond to converging and diverging behavior of nearby trajectories. We investigated some methods for visualizing Lyapunov exponents. The magnitude of Lyapunove exponents may be encoded in the size of spheres positioned along a trajectory. Positive Lyapunov exponents describe diverging behavior (image with red spheres), negative Lyapunov exponents describe converging behavior (image with green spheres). The evolvment of adjacent trajectories describes the local dynamics as well (see both images to the right). Further information is given in the technical report Visualization of Local Stability of Dynamical Systems

This page is maintained by Eduard Gröller. It was last updated on November 20, 1998.
If you have any comments, please send a message to