Strange Attractors, Chaotic Behavior, and Information Flow

TLDR

Turbulent systems are analyzed through information theory, linking the average information production rate (λ̄) to Lyapunov exponents, explaining strange attractor geometry and the prevalence of information‑producing dynamics such as 1/f noise in real‑world systems. The study argues that physical realizations of turbulent equations can serve as information sources, and it examines the phenomenology and classification of three‑dimensional strange attractors. The authors analyze three‑dimensional strange attractor phenomenology and propose a classification framework based on their dynamical properties. They find that the shift from laminar to turbulent behavior corresponds to λ̄ changing from negative to positive, turning the system from an information sink into a source, and that accumulating new information limits predictability beyond a finite horizon.

Abstract

Simple system equations displaying turbulent behavior are reviewed in the light of information theory. It is argued that a physical implementation of such equations is capable of acting as an information source, bringing into the macroscopic variables information not implicit in initial conditions. The average rate of information production λ̄ is a system state function, and is given for simple cases by a "Liapunov characteristic exponent", developed by Oseledec. The transition of a system from laminar to turbulent behavior is understandable in terms of the change of λ̄ from negative to positive, corresponding to the change of the system from an information sink to a source. The new information of turbulent systems precludes predictability past a certain time; when new information accumulates to displace the initial data, the system is undetermined. The observed geometry of strange attractors is seen to arise naturally from a rule allowing joining but not splitting of trajectories in phase space. The phenomenology of strange attractors in three dimensions is discussed, and a basis for their classification suggested. A comment is made on the commonplace occurrence of information producing systems in the real world, and on their possible relation to 1/f noise