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		2.7.1— Summary of Chaos

		1— Deterministic System with Complex Behavior

		A chaotic system is deterministic. It can be described by a small number of independent variables. However, the output of the system is so complex that it mimics the behavior of a random system based on chance. The word chaos was chosen to describe this complex behavior. It does not mean that a chaotic system is driven by disorder, randomness, or chance.

		2— Dynamical System

		Deterministic means that the subsequent values of the variables can be computed from their previous values. This is called a dynamical system.

		3— Sensitivity to Initial Conditions

		The subsequent values of the variables computed depend sensitively on the exact starting values of the variables. If the computation is rerun with slightly different starting values, then the final values of the variables will be significantly different. The values of the variables at each small time step into the future can be computed from their present values. However, the values of the variables far into the distant future cannot be computed from their present values. Thus, the system is deterministic over each small time step, but it is not predictable in the long run.

		4— Phase Space

		The values of the variables in time can be transformed into an object in space. The space is called the phase space. The object is called the phase space set. It is easier to determine some properties of the system by analyzing the spatial properties of the phase space set rather than the temporal properties of the values of the variables. For example, if the fractal dimension of the phase set is high, then the values of the variables in time were generated by a random mechanism. If the fractal dimension of the phase space set is low, then the values of the variables in time were generated by a deterministic mechanism.

 

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