P. Bak, C. Tang, and K. Wiesenfeld, Self-organized criticality: An explanation of 1/f
noise,Phys. Rev. Lett.
:381–384 (1987).Google Scholar
E. Berlekamp, J. Conway, and R. Guy,Winning Ways for Your Mathematical Plays, Vol. 2 (Academic Press, 1984).
T. Bohr, G. Grinstein, Y. He, and C. Jayaprakash, Coherence, chaos, and broken symmetry in classical, many-body dynamical systems,Phys. Rev. Lett.
(21):2155–2158 (1987).Google Scholar
T. Bohr and O. B. Christensen, Size dependence, coherence, and scaling in turbulent coupled-map lattices,Phys. Rev. Lett.
(20):2161–2164 (1989).Google Scholar
J. Carlson, J. Chayes, E. Grannan, and G. Swindle, Self-organized criticality in sandpiles: Nature of the critical phenomenon,Phys. Rev. A
(4):2467–2470 (1990).Google Scholar
H. Chaté and P. Manneville, Collective behaviors in spatially extended systems with local interactions and synchromous updating,Prog. in Theor. Phys.
:1–60 (1992).Google Scholar
A. Chhabra, M. Feigenbaum, L. Kadanoff, A. Kolan, and I. Procaccia, Sandpiles, avalanches, and the statistical mechanics of non-equilibrium stationary states, preprint (1992).
E. F. Codd,Cellular Automata
(Academic Press, New York, 1968).Google Scholar
J. P. Crutchfield and K. Kaneko, Phenomenology of spatio-temporal chaos, inDirections in Chaos
, Bailin Hao, ed. (World Scientific, Singapore, 1987).Google Scholar
J. P. Crutchfield and K. Kaneko, Are attractors relevant to turbulence?Phys. Rev. Lett.
(26):2715–2718 (1988).Google Scholar
J. P. Crutchfield, Hunting for transients and cycles, unpublished notes (March 1988).
J. P. Crutchfield, Subbasins, portals, and mazes: Transients in high dimensions,Nucl. Phys. B (Proc. Suppl.)
:287–292 (1988).Google Scholar
F. Fogelman-Soulié, Parallel and sequential computation on Boolean networks,Theor. Computer Sci.
:275–300 (1985).Google Scholar
E. Fredkin and T. Toffoli, Conservative logic,Int. J. Theor. Phys.
(3/4):219–253 (1982).Google Scholar
P. Gacs, G. L. Kurdyumov, and L. A. Levin, One-dimensional uniform arrays that wash out finite islands,Prob. Peredachi. Inf.
:92–98 (1978).Google Scholar
J. Guckenheimer and P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
H. Gutowitz, J. Victor, and B. Knight, Local structure theory for cellular automata,Physica D
:18–48 (1987).Google Scholar
H. Gutowitz, Hierarchical classification of cellular automata,Physica D
(1–3):136–156 (1990).Google Scholar
G. Grinstein, Stability of nonstationary states of classical, many-body dynamical systems,J. Stat. Phys.
(5/6):803–815 (1988).Google Scholar
E. Jen, Global properties of cellular automata,J. Stat. Phys.
(1/2):219–242 (1986); Invariant strings and pattern-recognizing properties of one-dimensional cellular automata,J. Stat. Phys.
(1/2):243–265 (1986).Google Scholar
K. Kaneko, Lyapunov analysis and information flow in coupled map lattices,Physica D
:436–447 (1986).Google Scholar
K. Kaneko, Chaotic but regular posi-nega switch among coded attractors by cluster-size variation,Phys. Rev. Lett.
(3):219–223 (1989).Google Scholar
K. Kaneko, Clustering, coding, switching, hierarchical ordering, and control in a network of chaotic elements,Physica D
:137–172 (1990).Google Scholar
K. Kaneko, Super-transients, spatio-temporal intermittency and stability of fully developed spatio-temporal chaos,Phys. Lett. A
(2, 3):105–112 (1990).Google Scholar
F. Kaspar and H. G. Schuster, Scaling at the onset of spatial disorder in coupled piecewise linear map,Phys. Lett. A
:451–453 (1986).Google Scholar
S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets,J. Theor. Biol.
:437–167 (1969).Google Scholar
S. A. Kauffman, Emergent properties in randomly complex automata,Physica D
:145–156 (1984).Google Scholar
D. E. Knuth,The Art of Computer Programming, Vol. 2:Seminumerical Algorithms (Addison-Wesley, 1981).
J. Krug, J. Socolar, and G. Grinstein, Surface fluctuations and criticality in a class of 1d sandpile models, preprint (1992).
C. Langton, Studying artificial life with cellular automata,Physica D
(1–3):120–149 (1986).Google Scholar
C. Langton, Computation at the edge of chaos: Phase transitions and emergent computation,Physica D
:12–37 (1990).Google Scholar
C. Langton, Computation at the Edge of Chaos, Ph.D. Thesis, University of Michigan (1990).
W. Li, Power spectra of regular languages and cellular automata,Complex Systems
(1):107–130 (1987).Google Scholar
W. Li, Complex patterns generated by next nearest neighbors cellular automata,Computer Graphics
(4):531–537 (1989).Google Scholar
W. Li, Problems in Complex Systems, Ph.D. Thesis, Columbia University, New York (1989) (Available from University Microfilm International, Ann Arbor, Michigan).Google Scholar
W. Li and N. Packard, Structure of the elementary cellular automata rule space,Complex Systems
(3):281–297 (1990).Google Scholar
W. Li, N. Packard, and C. Langton, Transition phenomena in cellular automata rule space,Physica D
(1–3):77–94 (1990).Google Scholar
W. Li and M. Nordahl, Transient behavior of cellular automaton rule 110,Phys. Lett. A, to appear (1992).
W. Li, Parametrizations of cellular automata rule space, in preparation.
W. Li, Group meeting problems, in preparation.
W. Li, Dynamical behavior of coupled selectors, in preparation.
K. Lindgren and M. Nordahl, Universal computation in simple one-dimensional cellular automata,Complex Systems
(3):299–318 (1990).Google Scholar
B. D. Lubachesky, Efficient parallel simulations of asynchronous cellular arrays,Complex Systems
(6):1099–1123 (1987).Google Scholar
J. L. Marroquín and A. Ramírez, Stochastic cellular automata with Gibbsian invariant measures,IEEE Trans. Information Theory
(3):541–551 (1991).Google Scholar
P. C. Matthews and S. H. Strogatz, Phase diagram for the collective behavior of limitingcycle oscillators,Phys. Rev. Lett.
:1701–1704 (1990).Google Scholar
R. May, Simple mathematical models with very complicated dynamics,Nature
:459–467 (1976).Google Scholar
N. Packard, Complexity of growing patterns in cellular automata, inDynamical Systems and Cellular Automata, J. Demongeot, E. Goles, and M. Techuente, eds. (Academic Press, 1985).
N. Packard, Adaptation toward the edge of chaos, inComplexity in Biological Modeling
, S. Kelso and M. Shlesinger, eds. (World Scientific, Singapore, 1988).Google Scholar
P. Peretto and J.-J. Niez, Stochastic dynamics of neural networks,IEEE Trans. Systems, Man, Cybernet.
(1):73–83 (1986).Google Scholar
C. Prado and Z. Olam, Inertia and break of self-organized criticality in sandpile cellularautomata model,Phys. Rev. A
(2):665–669 (1992).Google Scholar
W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling,Numerical Recipes in C
(Cambridge University Press, Cambridge, 1988).Google Scholar
D. R. Rasmussen and T. Bohr, Temporal chaos and spatial disorder,Phys. Lett. A
(2, 3):107–110 (1987).Google Scholar
L. S. Schulman and P. E. Seiden, Statistical mechanics of a dynamical system based on Conway's Game of Life,J. Stat. Phys.
:293–314 (1978).Google Scholar
R. Shaw,The Dripping Faucet as a Model Chaotic System (Aerial Press, 1984).
R. J. Smith,Circuits, Devices, and Systems — A First Course in Electrical Engineering (Wiley, 1966, 1984).
J. Theiler, Mean field analysis of systems that exhibit self-organized criticality, Center for Nonlinear Studies preprint, Los Alamos National Lab (1991).
J. Theiler, private communication.
T. Toffoli and N. Margolus,Cellular Automata Machine—A New Environment for Modeling (MIT Press, 1987).
K. Tsang, R. Mirollo, S. Strogatz, and K. Wiesenfeld, Dynamics of a globally coupled oscillator array,Physica D
:102–112 (1991).Google Scholar
D. K. Umberger, C. Grebogi, E. Ott, and B. Afeyan, Spatiotemporal dynamics in a dispersively coupled chain of nonlinear oscillators,Phys. Rev. A
:4835–4842 (1989).Google Scholar
J. von Neumann,Theory of Self-Reproducing Automata, A. W. Burks, ed. (University of Illinois Press, 1966).
G. Y. Vichniac, P. Tamayo, and H. Hartman, Annealed and quenched inhomogeneous cellular automata (ICA),J. Stat. Phys.
(5/6):875–883 (1986).Google Scholar
C. C. Walker and W. R. Ashby, On temporal characteristics of behavior in a class of complex systems,Kybernetik
:100–108 (1966).Google Scholar
C. C. Walker, Behavior of a class of complex systems: The effect of system size on properties of terminal cycles,J. Cybernet.
(4):57–67 (1971).Google Scholar
C. C. Walker, Predictability of transient and steady-state behavior in a class of complex binary sets,IEEE Trans. Systems, Man, Cybernet.
(4):433–436 (1973).Google Scholar
C. C. Walker, Stability of equilibrial states and limit cycles in sparsely connected, structurally complex Boolean nets,Complex Systems
(6):1063–1086 (1987).Google Scholar
C. C. Walker, Attractor dominance patterns is sparsely connected Boolean nets,Physica D
(1–3):441–451 (1990).Google Scholar
G. Weisbuch,Complex Systems Dynamics (Addison-Wesley, 1991).
K. Wiesenfeld and P. Hadley, Attractor crowding in oscillator arrays,Phys. Rev. Lett.
:1335–1338 (1989).Google Scholar
W. Wilbur, D. Lipman, and S. Shamma, On the prediction of local patterns in cellular automata,Physica D
:397–410 (1986).Google Scholar
S. Wolfram, Statistical mechanics of cellular automata,Rev. Mod. Phys.
:601–644 (1983).Google Scholar
S. Wolfram, Universality and complexity in cellular automata,Physica D
:1–35 (1984).Google Scholar
S. Wolfram, Computation theory of cellular automata,Commun. Math. Phys.
:15–57 (1984).Google Scholar
S. Wolfram, Twenty problems in the theory of cellular automata,Physica Scripta
:170–183 (1985).Google Scholar
Appendix: Properties of thek=2r=1 cellular automata, inTheory and Applications of Cellular Automata, S. Wolfram, ed. (World Scientific, Singapore, 1986).
W. Wootters and C. G. Langton, Is there a sharp phase transition for deterministic cellular automata?Physica D
(1–3):95–104 (1990).Google Scholar