Automatic Generation of Iterated Function Systems
J. C. Sprott
Department of Physics, University
of Wisconsin, Madison, WI 53706, USA
ABSTRACT
A set of affine mappings with randomly chosen coefficients is
repeatedly
iterated numerically using the random iteration algorithm to produce an
attractor with fractal characteristics. The attractor is tested for
boundedness,
sensitivity to initial conditions, and correlation dimension. In this
way,
a computer can generate a large collection of fractal patterns 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
18, 417-425 (1994)
The complete paper is available
in PDF format.
Return to Sprott's Books and Publications.
Fig. 1. Examples of iterated function systems produced by pairs of
two-dimensional
affine maps.
![[Figure 1]](210fig1.gif)
Fig. 2. Results of evaluating 7500 iterated function systems,
showing
that the most visually appealing cases are those with large negative
Lyapunov
exponents (L) and with correlation dimensions (F) greater than one.
![[Figure 2]](210fig2.gif)
Fig. 3. Examples of iterated function systems produced by pairs of
three-dimensional
affine maps in which the color is determined by one of the variables.
![[Figure 3a]](210fig3a.gif)
![[Figure 3b]](210fig3b.gif)
The computer source code ifs.bas
from
the article is available along with an executable version ifs.exe.