Chaotic Behavior in a Forecast Model
This paper is the product of an Undergraduate Research Project held at the University of
Houston and will appear in the Pi Mu Epsilon Journal. Devaney defines a function
f : X --> X to be chaotic if all of the following three properties hold: (1) f
is topologically transitive, (2) the periodic points of f are dense in X, and
(3) f has sensitive dependence on initial conditions. In addition to proving
the main result of the paper, we also give self-contained proofs of the fact that
(1) + (2) implies (3) when X is a metric space with no isolated points, and that
(1) implies (2) when X is an interval of the real numbers. These results are well
known and we prove them by methods similar to the existing literature. However, we
write the proofs to be completely self-contained and to have greater detail than what
is in the literature; in particular, we write them at a level that is accessible to
the average undergraduate reader. Prior to publication, the referee requested that
these results have been removed because they are well known and have already appeared
in the literature. We are posting an unabridged version of the paper here for readers
who may wish to see them.
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