Chaos in Nonlinear Difference Equations. I: Qualitative Study of (Formal) Chaos

Abstract

The interrelations among physicists' intuitive pictures of chaos are considered. Fortunately, in the case of C0-endomorphisms of an interval (1-dimensional nonlinear difference equations with continuous functions), almost all the intuitively plausible pictures of chaos —the mixing property (of some induced map), the positivity of the Kolmogorov (and topological) entropy, the existence of a homoclinic point, the existence of some natural correspondence with the “coin-tossing” process (formal chaos), etc.—are equivalent. The chaos in the sense of Li and Yorke follows from one of these properties.