| Автор | Durrett, Richard |
| Автор | Griffeath, David |
| Дата выпуска | 1993 |
| dc.description | We study two families of excitable cellular automata known as the Greenberg-Hastings model and the cyclic cellular automaton. Each family consists of focal deterministic oscillating lattice dynamics, with parallel discrete-time updating, parametrized by the range ρ of interaction, l <sup> p </sup> shape of its neighbor set, threshold θ for contact updating, and number K. of possible states per site. These models are mathematically tractable prototypes for the spatially distributed periodic wave activity of so-called excitable media observed in diverse disciplines of experimental science.Fisch, Gravner and Griffeath [Fisch et al. 1991] studied experimentally the ergodic behavior of these models on Z <sup>2</sup>, started from random initial states. Among other phenomena, they noted the emergence of asymptotic phase diagrams (and dynamics on R <sup>2</sup>) in the threshold-range scaling limit as ρ, θ → ∊ with θ/ρ<sup>2</sup> constant.Mere we present several rigorous results and some experimental findings concerning various phase transitions in the asymptotic diagrams. Our efforts focus on evaluating bend(p), the limiting threshold cutoff for existence of the spirals that characterize many excitable media. Our main results are formulated in terms of spo(p), the cutoff for existence of stable periodic objects that arise asspiral cores. Somesubtle consequences of anisotropic neighbor sets (p ≠ 2) are also discussed; the case of box neighborhoods (p = ∊) is examined in detail. |
| Формат | application.pdf |
| Издатель | Taylor & Francis Group |
| Копирайт | Copyright Taylor and Francis Group, LLC |
| Название | Asymptotic Behavior of Excitable Cellular Automata |
| Тип | research-article |
| DOI | 10.1080/10586458.1993.10504277 |
| Electronic ISSN | 1944-950X |
| Print ISSN | 1058-6458 |
| Журнал | Experimental Mathematics |
| Том | 2 |
| Первая страница | 183 |
| Последняя страница | 208 |
| Аффилиация | Durrett, Richard; Department of Mathematics, Cornell University |
| Аффилиация | Griffeath, David; Department of Mathematics, University of Wisconsin |
| Выпуск | 3 |
| Библиографическая ссылка | Dewdney, A. K. 1988. “Computer recreations: The hodgepodge machine makes waves”. Scientific American, August: 104–107. [Dewdney 1988] |
| Библиографическая ссылка | Dewdney, A. K. 1989. “Computer recreations: A cellular universe of debris, droplets, defects and demons”. Scientific American, August: 102–105. [Dewdney 1989] |
| Библиографическая ссылка | Durrett, R. 1991. Probability: Theory and Examples Pacific Grove, CA: Wadsworth & Brooks/Cole.. [Durrett 1991] |
| Библиографическая ссылка | Durrett, R. 1992. “Multicolor particle systems with large threshold and range”. J. Theoretical Prob., 4: 127–154. [Durrett 1992] |
| Библиографическая ссылка | Durrett, R. “Ten Lectures on Particle Systems”.”. In 1993 Saint Flour Probability Summer School Berlin: Lecture Notes in Math., Springer-Verlag.. [Durrett 1993], (to appear) |
| Библиографическая ссылка | Durrett, R. and Neuhauser, C. 1991. “Epidemics with regrowth in d = 2”. Ann. Appl. Probability, 1: 189–206. [Durrett and Neuhauser 1991] |
| Библиографическая ссылка | Durrett, R. and Steif, J. 1991. “Some rigorous results for the Greenberg–Hastings model”. J. Theoretical Prob. bf 3, : 669–690. [Durrett and Steif 1991] |
| Библиографическая ссылка | Durrett, R. and Steif, J. 1993. “Fixation results for threshold voter systems”. Ann. Probability, 21: 232–247. [Durrett and Steif 1993] |
| Библиографическая ссылка | Fisch, R. and Griffeath, D. Excite!: a periodic wave modeling environment. [Fisch and Griffeath 1991], See section on software availability above |
| Библиографическая ссылка | Fisch, R., Gravner, J. and Griffeath, D. 1991. “Threshold-range scaling of excitable cellular automata”. Statistics and Computing, 1: 23–39. [Fisch et al. 1991] = [FGG] |
| Библиографическая ссылка | Fisch, R., Gravner, J. and Griffeath, D. 1992. “Cyclic cellular automata in two dimensions”.”. In Spatial Stochastic Processes Edited by: Alexander, K. and Watkins, J. Boston: Birkhäuser.. [Fisch et al. 1992] |
| Библиографическая ссылка | Fisch, R., Gravner, J. and Griffeath, D. 1993. “Metastability in the Greenberg–Hastings model”. Ann. Appl. Probability, 3 [Fisch et al. 1993] |
| Библиографическая ссылка | Gerhardt, M., Schuster, H. and Tyson, J. 1990. “A cellular automaton model of excitable media, II: Curvature, dispersion, rotating waves and meandering waves”. Physica, D46: 392–415. [Gerhardt et al. 1990] |
| Библиографическая ссылка | Gravner, J. “Ring dynamics in the Greenberg–Hastings model” [Gravner], in preparation |
| Библиографическая ссылка | Gravner, J. and Griffeath, D. 1994. “Threshold growth dynamics”. Trans. Amer. Math. Soc., [Gravner and Griffeath 1994] |
| Библиографическая ссылка | Greenberg, J., Hassard, B. and Hastings, S. 1978. “Pattern formation and periodic structures in systems modeled by reaction-diffusion equations”. Bull. Amer. Math. Soc., 84: 1296–1327. [Greenberg et al. 1978] |
| Библиографическая ссылка | Greenberg, J. and Hastings, S. 1978. “Spatial patterns for discrete models of diffusion in excitable media”. SIAM J. Appl. Math., 34: 515–523. [Greenberg and Hastings 1978] |
| Библиографическая ссылка | Griffeath, D. 1988. “Cyclic random competition: a case history in experimental mathematics”, in “Computers and Mathematics”. Notices Amer. Math. Soc., : 1472–1480. [Griffeath 1988] |
| Библиографическая ссылка | Kapral, R. 1991. “Discrete models for chemically reacting systems”. J. Math. Chem., 6: 113–163. [Kapral 1991] |
| Библиографическая ссылка | Markus, M., Krafczyk, M. and Hess, B. 1991. “Randomized automata for isotropic modelling of two-and three-dimensional waves and spatiotem-poral chaos in excitable media”.”. In Nonlinear Wave Processes in Excitable Media Edited by: Holden, A., Markus, M. and Othmer, H. New York: Plenum Press.. [Marcus et al. 1991] |
| Библиографическая ссылка | Mikhailov, A. 1991. “Heart waves and the flutter that follows when they break down”. Quantum, November/December: 12–17. [Mikhailov 1991] |
| Библиографическая ссылка | Muller, S. 1986. “Two-dimensional spectrophotometry of spiral waves”. Physica, D24: 71–86. [Muller et al. 1986] |
| Библиографическая ссылка | Newell, P. C. 1983. “Attraction and adhesion in the slime mold Dictyostelium”.”. In Fungal Differentiation: A Contemporary Synthesis Edited by: Smith, J. E. 43–71. New York: Marcel Dekker.. [Newell 1983] |
| Библиографическая ссылка | Toffoli, T. and Margolus, N. 1987. Cellular Automata Machines Cambridge, MA: MIT Press.. [Toffoli and Margolus 1987] |
| Библиографическая ссылка | Weiner, N. and Rosenblueth, A. 1946. “The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle”. Arch. Inst. Cardiol. Mexico, 16: 205–265. [Weiner and Rosenblueth 1946] |
| Библиографическая ссылка | Winfree, A. 1974. “Rotating chemical reactions”. Scientific American, June: 82–95. [Winfree 1974] |
| Библиографическая ссылка | Winfree, A. 1987. When Time Breaks Down: The Three Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias. Princeton, NJ: Princeton Univ. Press.. [Winfree 1987] |
