Chaos, attractors and the Lorenz conjecture: Noninvertible transitive maps of ivariant sets are sensitive.

Abstract

In 1989, Edward Lorenz published a paper entitled, “Computational chaos- a prelude to computational instability” [L]. His paper looked at Euler approximations to differential equations. If the time increment of the approximating function was increased, he found that computational chaos set in. Since the numerics suggested transitivity and noninvertibility, he conjectured that transitive, noninvertible maps of an attractor were chaotic. To set the stage for investigating this conjecture, this thesis looked to examine the relationships between some of the standard definitions of chaos and attractor used throughout the literature. In addition to offering a proof of the Lorenz conjecture, a review of a number of related results was conducted. A side product of the work done was a partial result that tried to address whether topological transitivity carries sensitivity at a point to sensitivity on a set.