Attractor vicinity decay for a cellular automaton |
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Authors: | Crutchfield James P Hanson James E |
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Institution: | Center for Complex Systems Research, Beckman Institute for Advanced Science and Technology, University of Illinois, Urbana, Illinois 61801(a)). |
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Abstract: | The temporal decay of an attractor's vicinity for a domain-wall dominated cellular automaton (CA) is studied. Using selected initial pattern ensembles, state space structures in this high-dimensional nonlinear spatial system can be identified via the resulting decay to its attractors. Considered over a range of lattice sizes, the decay behavior falls into three main classes, each of which shows a characteristic profile. The first consists of even-size lattices showing a decelerating decay to small nonattracted ensemble fractions. The second class, also for even lattices, is a catastrophic decay to very small or vanishing nonattracted fractions. The third class also shows catastrophic decay and contains all odd-size lattices. Stochastic models are constructed that mimic the behavior of typical lattices throughout their evolution until finite-size effects appear. Weak additive noise causes all states on all lattices to fall into the attractor. In the end we find it overwhelmingly likely that the recently proposed attractor-basin portrait captures the CA's qualitative dynamics. |
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