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The ``circle'' attractor

Now consider a time series that consists of samples of a sine wave. Such a time series could be obtained, for instance, by measuring the position of a forced pendulum as a function of time when transients have died out. If we reconstruct a two-dimensional phase space with a delay $\tau = T/4$, where $T$ is the period of the sine, we obtain a circle. That is, if we choose the sample frequency in such a way that it is incommensurable with the frequency of the sine, and if we have an infinitely long time series. The dimension of this circle is one. Now, define a homogeneous probability measure on the circle so that the distance, measured along the circle, between two random chosen points is uniformly distributed. By applying a coordinate transformation, we obtain for the probability density function of the Euclidean distance (divided by the diameter of the circle):
\begin{displaymath}
p(r) = \frac{2}{ \pi \sqrt(1-r^2) }
\end{displaymath} (4.10)

Then, an analytic expression for the correlation integral is:
\begin{displaymath}
C(r) = P(r) = \frac{2}{\pi} \arcsin (r) \quad (\approx \frac{2}{\pi} r
\mbox{ for small r})
\end{displaymath} (4.11)

and
\begin{displaymath}
\nu(r) = \frac{d \ln (C(r))} {d \ln (r)} =
\frac{r}{\arcsin(r) \sqrt(1-r^2)} \quad (\approx 1 \mbox{ for small r})
\end{displaymath} (4.12)

In figure 4.1 the correlation integral and the correlation dimension are plotted as a function of $r$. We observe that we will overestimate the dimension. In figure 5.9 the correlation integrals were plotted which were calculated using a time series consisting of 10000 points. Compare this figure with figure 4.1! It would be interesting to know what happens if we would choose a higher embedding dimension, another lag, a sum of sines, introduce noise, or if we would use methods of dimension estimation from Chapter 3 or 5. Fortunately, in these cases the problem gets very complicated.


next up previous contents
Next: Implementation Up: Two analytical examples Previous: The ``hypercube''   Contents
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