Threshold-range scaling of excitable cellular automata |
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Authors: | Robert Fisch Janko Gravner David Griffeath |
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Institution: | (1) Department of Mathematics, Colby College, 04901 Waterville, ME, USA;(2) Department of Mathematics, University of Wisconsin, 53706 Madison, WI, USA |
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Abstract: | Each cell of a two-dimensional lattice is painted one of colors, arranged in a color wheel. The colors advance (k tok+1 mod ) either automatically or by contact with at least a threshold number of successor colors in a prescribed local neighborhood. Discrete-time parallel systems of this sort in which color 0 updates by contact and the rest update automatically are called Greenberg-Hastings (GH) rules. A system in which all colors update by contact is called a cyclic cellular automation (CCA). Started from appropriate initial conditions, these models generate periodic traveling waves. Started from random configurations the same rules exhibit complex self-organization, typically characterized by nucleation of locally periodic ram's horns or spirals. Corresponding random processes give rise to a variety of forest fire equilibria that display large-scale stochastic wave fronts. This paper describes a framework, theoretically based, but relying on extensive interactive computer graphics experimentation, for investigation of the complex dynamics shared by excitable media in a broad spectrum of scientific contexts. By focusing on simple mathematical prototypes we hope to obtain a better understanding of the basic organizational principles underlying spatially distributed oscillating systems. |
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Keywords: | Cellular automation excitable medium locally periodic spiral stable periodic object Greenberg-Hastings model cyclic CA self-organization phase transition turbulence bug macaroni |
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