Strange Attractors, Chaotic Behavior, and Information Flow

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