Chapter 7: Mechanisms in Programs and Nature

Section 4: Chaos Theory and Randomness from Initial Conditions

Recognizing chaos

Any system that depends sensitively on digits in its initial conditions must necessarily be able to show behavior that is not purely repetitive (compare page 955). And when it is said that chaos has been found in a particular system in nature what this most often actually means is just that behavior with no specific repetition frequency has been seen (compare page 586). To give evidence that this is not merely a reflection of continual injection of randomness from the environment what is normally done is to show that at least some aspect of the behavior of the system can be fit by a definite simple iterated map or differential equation. But inevitably the fit will only be approximate, so there will always be room for effects from randomness in the environment. And in general this kind of approach can never establish that sensitive dependence on initial conditions is actually the dominant source of randomness in a given system—say as opposed to intrinsic randomness generation. (Attempts are sometimes made to detect sensitive dependence directly by watching whether a system can do different things after it appears to return to almost exactly the same state. But the problem is that it is hard to be sure that the system really is in the same state—and that there are not all sorts of large differences that do not happen to have been observed.)

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From Stephen Wolfram: A New Kind of Science [citation]