dorsal/arxiv
View SchemaThreshold-Range Scaling of Excitable Cellular Automata
| Authors | Robert Fisch, Janko Gravner, David Griffeath |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9304001 |
| URL | https://arxiv.org/abs/patt-sol/9304001 |
| Journal | Stat. and Comp. 1, 1991, Chapman and Hall, London |
Abstract
Each cell of a two-dimensional lattice is painted one of k colors, arranged in a "color wheel." The colors advance (0 to k-1 mod k) 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 automaton (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 article 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 obtain a better understanding of the basic organizational principles underlying spatially-distributed oscillating systems.
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"abstract": "Each cell of a two-dimensional lattice is painted one of k colors, arranged\nin a \"color wheel.\" The colors advance (0 to k-1 mod k) either automatically or\nby contact with at least a threshold number of successor colors in a prescribed\nlocal neighborhood. Discrete-time parallel systems of this sort in which color\n0 updates by contact and the rest update automatically are called\nGreenberg-Hastings (GH) rules. A system in which all colors update by contact\nis called a cyclic cellular automaton (CCA). Started from appropriate initial\nconditions these models generate periodic traveling waves. Started from random\nconfigurations the same rules exhibit complex self-organization, typically\ncharacterized by nucleation of locally periodic \"ram\u0027s horns\" or spirals.\nCorresponding random processes give rise to a variety of \"forest fire\"\nequilibria that display large-scale stochastic wave fronts. This article\ndescribes a framework, theoretically based, but relying on extensive\ninteractive computer graphics experimentation, for investigation of the complex\ndynamics shared by excitable media in a broad spectrum of scientific contexts.\nBy focusing on simple mathematical prototypes we obtain a better understanding\nof the basic organizational principles underlying spatially-distributed\noscillating systems.",
"arxiv_id": "patt-sol/9304001",
"authors": [
"Robert Fisch",
"Janko Gravner",
"David Griffeath"
],
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"journal_ref": "Stat. and Comp. 1, 1991, Chapman and Hall, London",
"title": "Threshold-Range Scaling of Excitable Cellular Automata",
"url": "https://arxiv.org/abs/patt-sol/9304001"
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