Figures 7.3 show plots of (part of) the time series and the power spectrum. The spike at the beginning of the signal is caused by a sigh. From the largest peak in the power spectrum we see that the duration of one breath is about 3 seconds. The time delay for the phase space reconstruction was determined to be (see figure 7.4c and 7.5c). In MLDK2, the time series is multiplied by a factor such that the maximum distance on the object (attractor) will be 1. Due to the sighs in the signal, the object appears much smaller in the phase portrait. From that portrait, the estimated maximum distance is about 0.4. The system seems to have settled on a limit cycle, though being disturbed by noise: the width of the line is about 0.04. The dimension of the object will obviously depend on the length scale. For distances the object is approximately one-dimensional. For distances the estimated dimension should equal the embedding dimension if the disturbances from the limit cycle are due to noise, or lower and saturating if the disturbances are caused by a sufficiently low-dimensional deterministic chaotic system. So this region is probably the most interesting because at these length scales information may be extracted about the cause of small disturbances on an apparently periodic process.
To estimate the dimension at these length scales, was set at 0.04 and at 0.02 which is the minimum possible. So the boundaries of the ``scaling'' region differ a factor 2 which is barely enough. Figure 7.9 (figure 7.9b) and table 7.4 show the results from the MLDK2 program. In the plot of the correlation integrals we observe the following: 1) The edge effects: the peak in the numerical derivatives at distance , the size of the object. Compare such a peak to the example of the sine wave - figure 4.1 and 5.9. 2) In the range the dimension is low but increasing as a function of the embedding dimension. This is caused by the non-neglegible thickness of the line. 3) For distances the estimated dimension does not saturate and if we would be able to measure at length scales , we would probably obtain the embedding dimension. Here we obtain something like an average of dimensions between the embedding dimension and 1, the dimension of a line.