114.

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		2.5.9— Some of the Problems of the Phase Space Analysis

		1— How Much Data?

		It is not clear how many measurements are needed to construct the phase space set and determine its fractal dimension.

		It looks as if a lot of data are needed. It may be difficult to measure this much data from an experiment or to maintain a biological system under constant conditions long enough to record such a large amount of data.

		-->2— The Lag Dt

		The phase space set can be constructed from the measurements of one variable X(t). Each point in the N-dimensional phase space then has the coordinates X(t), X(t+Dt), X(t+2Dt), . . . . X(t+(N-l) Dt). It is often not clear what value of the lag Dt should be used to construct the phase space set accurately.

		3— The Fractal Dimension of the Phase Space Set

		There are different mathematical definitions of the fractal dimension. Any one dimension can be evaluated by different computational methods. The value found for the fractal dimension depends both on the definition and on the method used to evaluate it.

		4— Mathematical Limitations of the Embedding Theorems

		The mathematics of chaos is still being developed. Some of the existing theorems have assumptions that are not satisfied by the experimental data being analyzed. For example, the embedding theorems require that the sequence of values are "smooth," which is not true if the data are fractal.

 

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