dorsal/arxiv
View SchemaAsymptotic Behavior of Excitable Cellular Automata
| Authors | Richard Durrett, David Griffeath |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9303002 |
| URL | https://arxiv.org/abs/patt-sol/9303002 |
Abstract
We study two families of excitable cellular automata known as the Greenberg-Hastings Model (GHM) and the Cyclic Cellular Automaton (CCA). Each family consists of local deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range of interaction, the "shape" of its neighbor set, threshold value for contact updating, and number of possible states per site. GHM and CCA are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science. Earlier work by Fisch, Gravner, and Griffeath studied the ergodic behavior of these excitable cellular automata on Z^2, and identified two distinct (but closely-related) elaborate phase portraits as the parameters vary. In particular, they noted the emergence of asymptotic phase diagrams (and Euclidean dynamics) in a well-defined threshold-range scaling limit. In this study we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams, focusing on evaluating the limiting threshold cutoff for existence of the spirals that characterize many excitable media. For mathematical expediency our main results are formulated in terms of spo(p), the cutoff for existence of stable periodic objects that arise as spiral cores.
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"abstract": "We study two families of excitable cellular automata known as the\nGreenberg-Hastings Model (GHM) and the Cyclic Cellular Automaton (CCA). Each\nfamily consists of local deterministic oscillating lattice dynamics, with\nparallel discrete-time updating, parametrized by the range of interaction, the\n\"shape\" of its neighbor set, threshold value for contact updating, and number\nof possible states per site. GHM and CCA are mathematically tractable\nprototypes for the spatially distributed periodic wave activity of so-called\nexcitable media observed in diverse disciplines of experimental science.\nEarlier work by Fisch, Gravner, and Griffeath studied the ergodic behavior of\nthese excitable cellular automata on Z^2, and identified two distinct (but\nclosely-related) elaborate phase portraits as the parameters vary. In\nparticular, they noted the emergence of asymptotic phase diagrams (and\nEuclidean dynamics) in a well-defined threshold-range scaling limit. In this\nstudy we present several rigorous results and some experimental findings\nconcerning various phase transitions in the asymptotic diagrams, focusing on\nevaluating the limiting threshold cutoff for existence of the spirals that\ncharacterize many excitable media. For mathematical expediency our main results\nare formulated in terms of spo(p), the cutoff for existence of stable periodic\nobjects that arise as spiral cores.",
"arxiv_id": "patt-sol/9303002",
"authors": [
"Richard Durrett",
"David Griffeath"
],
"categories": [
"patt-sol",
"adap-org",
"nlin.AO",
"nlin.PS"
],
"title": "Asymptotic Behavior of Excitable Cellular Automata",
"url": "https://arxiv.org/abs/patt-sol/9303002"
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