next up previous contents
Next: Implementation Up: NEAREST NEIGHBOUR METHODS Previous: Methods to estimate the   Contents

An example

In this section we apply the nearest neighbour dimension estimator to the well-known Rössler system. It is governed by the following differential equations [Rössler, 1976]:
$\displaystyle \frac{ dx } { dt }$ $\textstyle =$ $\displaystyle -(y+z)$  
$\displaystyle \frac{ dy } { dt }$ $\textstyle =$ $\displaystyle x + ay$ (3.15)
$\displaystyle \frac{ dz } { dt }$ $\textstyle =$ $\displaystyle b + z(x-c)$  

This system has been shown to be chaotic for parameter values $a=0.15$, $b=0.2$ and $c=10$ [Wolf et al., 1985]; the Lyapunov dimension $D_L \leq D_2$ and entropy $K \geq K_2$ are about $2.01$ and $0.09$ [nats/s] respectively [Wolf et al., 1985,Procaccia, 1985]. The equations were solved with a Gear method (NAG routine D02EAF [Numerical Algorithms Group, 1983]) with tolerance parameter $10^{-12}$. The initial conditions were $[1,0,0]$ and the time step (or sample time) was 0.5 [s]. The first 1000 solutions were discarded to avoid transient effects. From the $x$ variable $10000$ solutions were used to obtain a time series. The time delay used for the reconstruction of phase space was determined with the mutual information criterion (see section 6.6); this yielded 1.5 [s]. See Ref. [Olofsen, 1987] for plots of the attractor, the time series, the reconstructed attractor, mutual information functions and power spectra. We used the $k(N)$ method with $\alpha=1$ and $\beta=0.6$

The results are presented in figure 3.1. We see that the estimated dimensions do not saturate, while the fits for the highest embedding dimension (20) do not look alarming, see figure 3.2.

Why do the dimensions increase this rapidly? To answer this question, we will use some concepts from Chapters 4 and 5. In figure 3.3 (figure 3.3b, table 3.1) we plotted correlation integrals and the averaged distance to the first and last nearest neighbour which we used for the fit (0.2 and 0.25 respectively for $\gamma=0$). We see that these distances are not in the scaling region. So $\alpha$ or $\beta$ was chosen too large. In figures 3.4 and 3.5 we repeated the experiment with $\beta=0.2$ (nearest neighbour distances between 0.05 and 0.1). The saturation has improved but there seem to be large statistical errors for negative values of $\gamma$. For reference, we also show the results of the correlation integral method for better scaling regions - figure 3.6 (figure 3.6b) and table 3.2.

To conclude, it seems to be unclear3.2which values to use for $\alpha$ and $\beta$. First we need to specify in what range the nearest neighbour distances are allowed to be. To do this, the method of correlation integrals may be of use; we will describe this method in the following chapter.


next up previous contents
Next: Implementation Up: NEAREST NEIGHBOUR METHODS Previous: Methods to estimate the   Contents
webmaster@rullf2.xs4all.nl