Automatic Generation of Strange Attractors

J. C. Sprott
Department of Physics, University of Wisconsin, Madison, WI 53706


A pair of coupled quadratic difference equations with randomly chosen coefficients is repeatedly iterated by computer to produce a two-dimensional map. The map is tested for stability and sensitivity to initial conditions. The process is repeated until a chaotic solution is found. In this way a computer can generate a large collection of strange attractors that are all different, and most of which have considerable aesthetic appeal. A simple computer program and examples of its output are provided. Many of the attractors have been systematically evaluated for visual appeal, and a correlation is found with the Lyapunov exponent and correlation dimension.

Ref: J. C. Sprott, Comput. & Graphics 17, 325-332 (1993)

The complete paper is available in PDF format.

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Fig. 1. Examples of strange attractors produced by two-dimensional iterated quadratic maps.
[Figure 1]

Fig. 2. Results of evaluating 7500 strange attractors, showing that the most visually appealing cases are those with small Lyapunov exponents (L) and with correlation dimensions (F) somewhat greater than one.
[Figure 2]

Fig. 3. Examples of strange attractors produced by three-dimensional iterated quadratic maps in which the color is determined by one of the variables.
[Figure 3a]

[Figure 3b]

A variant of the computer source code prog06.bas from the article is available along with an executable version prog06.exe. A version of the program is also available as a Java applet.