J. P. Crutchfield and J. E. Hanson
Center for Complex Systems Research
Beckman Institute for Advanced Science and Technology
University of Illinois
405 North Mathews Avenue
Urbana, Illinois 61801 USA
We study the temporal decay of an attractor's vicinity for a domain-wall dominated CA. Using selected initial pattern ensembles, state space structures in the high-dimensional nonlinear spatial system can be identified via the resulting decay processes. Vicinity decay breaks into two epochs. The first is governed by ideal diffusive annihilation of the walls and is described by the stochastic dynamics of a random cliff walker. The second decay epoch consists of deviations from the ideal due to accumulated space-time correlations coming from the boundary conditions, lattice size, and the deterministic CA rule. The decay behavior in this regime -- considered over a range of lattice sizes — falls into two main classes. The first is a decelerating decay to small nonattracted fractions. The second, more populous, class is a catastrophic decay to very small or vanishing nonattracted fractions. Small amounts of additive noise move all lattices into this second class. In the end we find it overwhelmingly likely that the recently-proposed attractor-basin portrait captures the qualitative dynamics of the CA.
J. P. Crutchfield and J. E. Hanson, Attractor Vicinity Decay for a Cellular Automaton, CHAOS 3:2 (1993) 215 - 224.University of Illinois, Beckman Institute, Center for Complex Systems Research, Technical Report UIUC-BI-CCSR-92-13.