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		2.3.4— Chaos: Deterministic but Not Predictable

		1— Non-Chaotic: Deterministic and Predictable

		For a deterministic system that is not chaotic, the value of a variable X at a given time can be used to calculate accurately the value of X at all times in the future.

		2— Chaotic: Deterministic and Not Predictable

		For a deterministic system that is chaotic, the value of a variable X at a given time can be used to calculate the value of X at a brief time in the future.

		However, as we calculate the value of X at longer times into the future, the sensitivity to initial conditions means that the calculation becomes less accurate.

		For example, if the starting value of X was known to 5 significant digits, as we calculate the value of X at longer times into the future, the number of accurate digits in the result will become 4, then 3, then 2, then 1, then zero. That is, the values of X at long times into the future cannot be predicted.

		If we could know the values of the initial conditions to infinite accuracy, we could predict the future values of X. However, we can only measure the initial conditions to finite accuracy, for example, 5 significant digits. The sensitivity to initial conditions amplifies this initial inaccuracy so that it is not possible to predict the values of X far into the future.

		A chaotic system is deterministic but not predictable in the long run. It is deterministic because the values of the variables at the next instant in time are determined by their present values. However, it is not predictable in the long run, because the sensitivity to initial conditions amplifies the inaccuracy that is always present in the initial conditions so that we cannot accurately predict the values of the variables far into the future.

 

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