Randomness of deterministic dynamical system

Question

I was told by someone that there are results which show that an observation of certain deterministic dynamical systems can not be statistically distinguished from stochastic noise. Can someone point some such results to me?

Answer 1

The most blatant example for this are pseudo-random number generators. Their output is the observable of a deterministic dynamical system, but if they have a high quality, you need a lot of data to actually find a pattern.

Many chaotic systems (which are deterministic) appear stochastic in at least some respect, for example having a decaying autocorrelation function and a broad frequency spectrum. One of the goals of chaos theory is to find patterns despite of this. To give a specific example, consider the output of the logistic map:

$$x_{t+1} = f(x_t) = 4 x_t (1-x_t).$$

At first glance it will look random. At second glance, you will see some patterns. These are owed to the simplicity of the map. For more complex maps, these will become very difficult to discern. For example, suppose you look only at every $n$-th time point, i.e., at the map $f^n$. (This is already not so different from what pseudorandom number generators do.)

From another point of view, one of the features is that the dynamics becomes unpredictable on longer time scales. As soon as you are looking at statistical properties on these time scales, they will be indistinguishable from randomness – even though the dynamics may be look not random at all on shorter time scales.

In an example from my own research (preprint), we investigate a deterministic system that exhibits oscillations, some of which have an exceptionally high amplitude (extreme events). On short time scales, you just see the oscillations, which are clearly not random. However, the times of the events are statistically barely distinguishable from a Poisson process.

Sidenote: This is actually my scientific motto: Any sufficiently complex deterministicity is indistinguishable from stochasticity.