1993/03/22 by Richard Durrett, Durrett, Richard, David Griffeath +1 · 1 citation
Computer Science · Physics and Astronomy · #Adaptation and Self-Organizing Systems (nlin.AO) #Cellular Automata and Applications #FOS: Physical sciences #Nonlinear Dynamics and Pattern Formation #Pattern Formation and Solitons (nlin.PS) #Theoretical and Computational Physics #adap-org #nlin.AO #nlin.PS #patt-sol
paper · pdf · doi:10.48550/arxiv.patt-sol/9303002
38 pages. arXiv admin note, 17Jun2003: original PCL file and PDF version now available, see HTML page
arxiv created 1993/03/22 · openalex publication_date 1993/03/22 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Z2, 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.