In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
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