Noise versus Chaos

Look again at Fig.(3.12b). If you were not told that the time-series was from a deterministic system, would you have been able to decide if the figure was due to noise (random events) or to deterministic chaos ? This is an interesting issue as one would like to know whether many apparently noisy time-series in our daily lives (such as sunspots or brainwaves) are actually the result of deterministic chaos and hence whether there are relatively simple laws underlying those phenomena. If there is an underlying strange attractor to the time series then one can attempt to show this using the idea of attractor reconstruction. Suppose one is given a time series $x(n)$ obtained by sampling points at regular time intervals. For a deterministic dynamics, The simplest assumption then is that the value of $x(n+1)$ depends on its value at the previous time, $x(n)$ Thus one plots $x(n+1)$ against $x(n)$ to see if a pattern emerges.

Fig(3.15a) Fig(3.15b) Fig.(3.15c)

Fig(3.16a) Fig.(3.16b) Fig.(3.16c)

For the attractor reconstruction to be succesful there are two issues that one has to address. Firstly, the embedding dimension must be large enough for the attractor to become visible. In the above examples we plotted the vector $(x(n), x(n+1))$, meaning that we investigated a two-dimensional embedding space. Since the dimension of the Lorenz or Rossler strange attractors is three, a similar procedure for the latter cases would require an embedding space of at least three dimensions, so we would need to plot for example $(x(n),x(n+1),x(n+2))$. The second issue is the optimal time delay to be used for the above reconstruction. In the mentioned examples, the time delay was one. More generally one can consider other values, so that for the Rossler system one could plot $(x(n),x(n+N),x(n+N+1))$, corresponding to a time delay of $N$. The value of $N$ might be determined by trial. There have been several attempts to discover determinstic chaos in stock-market data and to so profit from that knowledge. You can read about the incredible and entertaining adventures of some physicists in this endeavour in Ref.[6].