Notes

Chapter 4: Systems Based on Numbers

Section 8: Continuous Cellular Automata


History [of continuous cellular automata]

Continuous cellular automata have been introduced independently several times, under several different names. In all cases the rules have been at least slightly more complicated than the ones I consider here, and behavior starting from simple initial conditions does not appear to have been studied before. Versions of continuous cellular automata arose in the mid-1970s as idealizations of coupled ordinary differential equations for arrays of nonlinear oscillators, and implicitly in finite difference approximations to partial differential equations. They began to be studied with extensive computer simulations in the early 1980s, probably following my work on ordinary cellular automata. Most often considered, notably by Kunihiko Kaneko and co-workers, were so-called "coupled map lattices" or "lattice dynamical systems" in which an iterated map (typically a logistic map) was applied at each step to a combination of neighboring cell value. A transition from regular class 2 to irregular class 3 behavior, with class 4 behavior involving localized structures in between, was observed, and was studied in detail by Hugues Chaté and Paul Manneville, starting in the late 1980s.

From Stephen Wolfram: A New Kind of Science [citation]