Chaos and Information

Chaos is a word that gets thrown around a lot, and I know nothing about Chaos Theory, which is at this point probably a serious branch of mathematics, but again, I know nothing about it. However, it dawned on me, that you can think about chaos in terms of information, using Cellular Automata as the intuition. Specifically, consider the initial conditions of a binary automaton, which together form a single vector of N binary digits. Then you have the rule, R, which takes those initial conditions, and mechanistically generates the rows that follow. Let X be a particular set of initial conditions (i.e., a particular binary vector), so that M = R(X,K) is the result of applying R to the initial conditions X, for some fixed number of K iterations. Now change exactly one bit of X, producing \bar{X}, which in turn produces \bar{M} = R(\bar{X},K). Now count the number of unequal bits between \bar{M} and M, which must be at least one, since we changed exactly one bit of X to produce \bar{X}. Let a be the number of unequal bits between \bar{M} and M, and let b be the number of bits changed in X, to produce \bar{X}. We can measure how chaotic R is in bits, as the ratio,

\Theta = \frac{a}{b}.

Said in words, \Theta is the ratio of the total number of bits that change divided by the number of bits changed in the initial conditions.

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