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		2.1.6— The Properties of Chaos

		1— Phase Space

		A sequence of values of data measured in time can be transformed into an object in space. This object is called the phase space set. Some properties are easier to determine by analyzing the phase space set rather than by analyzing the original data directly. For example, when the fractal dimension of the phase space set is high, then the data were generated by a random mechanism. When the fractal dimension of the phase space set is low, then the data were generated by a deterministic mechanism.

		2— Sensitivity to Initial Conditions

		If we rerun a non-chaotic system with almost the same starting values, we get almost the same values of the variables at the end. However, if we rerun a chaotic system with almost the same starting values, we get very different values of the variables at the end. This is called sensitivity to initial conditions. Chaotic systems amplify small differences in initial conditions into large differences. This is called the Butterfly Effect, because whether or not a butterfly beats its wings in Gliwice, Poland determines if there will be a thunderstorm a week later in Boca Raton, Florida.

		3— Bifurcations

		The behavior of a system can change abruptly with a small change in the value of a parameter. This is called a bifurcation.

		4— Good News and Bad News about Analyzing Data

		New methods now make it possible to analyze random-looking data to determine if they were produced by a random or a deterministic mechanism. However, there are practical difficulties in using these methods.

		5— Control of Chaos

		If the analysis shows that the data were produced by a deterministic mechanism, then we may be able understand and perhaps even control it.

 

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