Chaos Automata: Iterated Function Systems with Memory
with James B. Golden
Submitted to Physica D

Abstract PDF eprint

Traditional iterated function system fractals, in which contraction maps are called in an iterated random or data-driven sequence, are forgetful. Random numbers or data items more than a few steps in the past have little influence on the position of the moving point used to generate the fractal. This is because the contraction associated with each of the iterated functions rapidly shrink the influence of past inputs below the pixel size of any displayed version of the fractal. To permit iterated function system fractals whose appearance can depend on relatively remote data items, e.g. changes of distribution in a data source, we introduce the chaos automata. A chaos automata is a finite state machine with a contraction map associated with each state. The random numbers or data previously used to drive the selection of contraction maps now drive the finite state machine with the contraction map on each state being applied as it would in a standard iterated function system. Chaos automata are used to create iterated function systems that yield fractals with different appearances when driven by different types of data. The parameter selection, including finite state transition diagram and selection of contraction maps, for the chaos automata is performed with an evolutionary algorithm in a data driven manner. Random sources over the DNA alphabet with distinct distributions and intron and exon sequences derived from Zea mays are used to test the chaos automata/evolutionary algorithm system.