Abstract
Recently a method has been developed by Jen to enumerate limit cycles in cellular automata (CA) with periodic boundary conditions. This involves operations on a connectivity matrix whose elements are related to the invariance of a site in a particular neighborhood to application of the CA rule. We extend this method to the case of fixed boundary conditions, of interest in simulations. In this case, translational invariance is lost, and the enumeration procedure is much more tedious than with periodic boundary conditions. We show examples for a fixed-point, a period-two, and a period-three enumeration in considerable detail, and give results-in agreement with simulations—for the number of fixed points and period-two cycles in selected two-state, nearest-neighbor CA rules.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. von Neumann,The theory of self-reproducing automata, A. Burks, ed. (University of Illinois Press, Urbana, 1966).
J. D. Farmer, T. Toffoli and S. Wolfram, eds.Cellular automata (North-Holland, Amsterdam, 1984).
S. Wolfram, ed.Theory and applications of cellular automata (World Scientific, Singapore, 1986).
T. Toffoli and N. H. Margolus,Cellular automata machines (MIT Press, Cambridge, 1987).
P. Manneville et al., eds.Cellular Automata and Modeling of Complex Physical Systems (Springer, Berlin, 1989).
H. Gutowitz, ed.Cellular automata: theory and applications (North-Holland, Amsterdam, 1990).
O. Martin, A. Odlyzko and S. Wolfram,Commun. Math. Phys. 93:219 (1984).
P. Guan and Y. He,J. Stat. Phys. 43:463 (1986);Compl. Systems 1:181 (1987).
E. Jen,J. Stat. Phys. 43:219, 243 (1986);Commun. Math. Phys. 118:569 (1988), and119:13 (1988).
D. Rothman,Geophysics 53:509 (1988).
S. Chen et al.,Physica D 47:72 (1991).
P. M. Binder,J. Phys. A: Math. Gen. 24:L31 (1991), and unpublished.
P. M. Binder,J. Phys. A: Math. Gen. 24:1677 (1991).
P. M. Binder, C. J. Twining, and D. Sherrington,Compl. Systems 5:127 (1991).
R. B. Stinchcombe, Private communication.
M. Kerszberg and D. Mukamel,J. Slat. Phys. 51:777 (1988).
N. Pitsianis, Ph. Tsalidas, G. L. Bleris, A. Thanailakis, and H. C. Card,J. Stat. Phys. 56:99 (1989).
E. Jen, Private communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Twining, C.J., Binder, P.M. Enumeration of limit cycles in noncylindrical cellular automata. J Stat Phys 66, 385–401 (1992). https://doi.org/10.1007/BF01060073
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01060073