Fractal Scaling in Cellular Automata Simulations of Dissipative Dynamical Systems.

A thorough understanding of simple processes often provides insight into the nature of complex systems. In this investigation, cellular automata are used to study the natural evolution of dissipative dynamical systems and a new technique for extracting fractal scaling parameters is described. The cellular automata model the evolution of a complex structure with the properties of a self-organized critical SOC system as suggested by Bak, Tang, and Wiesenfeld. We have demonstrated that the evolving structures exhibit fractal scaling and that the fractal measures vary in response to changes in rules governing the dynamics of the system. We also demonstrate that the range over which fractal scaling occurs increases smoothly as the model evolves and that subcritical, as well as critical, avalanche distributions have the form predicted for a size-effect limited SOC system. The subcritical scaling is fractal over limited ranges of scales and the scaling range correlates with the size of subdomain structures.