Abstract :
In the past, iterated function systems have been used [M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. R. Soc. Lond. A 399, 243–275 (1985); M. F. Barnsley, V. Ervin, D. Hardin and J. Lancaster, Solution of an inverse problem for fractals and other sets, Proc. Natl. Acad. Sci. USA 83, 1975–1977 (1986); M. F. Barnsley, Fractals Everywhere. Academic Press Inc., New York (1988)] for the generation of fractal images. It has also been shown by Berger [M. A. Berger, Random affine iterated function systems: curve generation and wavelets, SIAM Rev. 34, 361–385 (1992)] and Massopust [P. R. Massopust, Smooth interpolating curves and surfaces generated by iterated function systems, J. Analysis and Applications (Z. Anal. Anwendungen) 12, 201–210 (1993)] that curves and wavelets may be generated by iterated function systems. In the following paper we introduce an iterated function system (IFS) which exhibits bifurcation from a fixed point, and an IFS which generates a closed curve which undergoes period doubling, exhibits a period-3 orbit, and generates a chaotic attractor as parameters are varied.
