Deterministic chaos and randomness

[Giulio Prisco]

The term "deterministic chaos" emphasizes that chaotic behavior is random in-practice but deterministic in-principle. But does this even make sense in a finite universe where computation is physical? There ain't no such thing as a Turing machine with an infinite tape. If computing the behavior of a chaotic system requires more input information and more computing than the physical universe can provide, does it make sense to insist that the computation is feasible in-principle? Shouldn't we just accept that chaotic behavior is random?

 

[Asymptotic]

The term "deterministic chaos" emphasizes that chaotic behavior is random in-practice but deterministic in-principle.

Is this so? Chaos ≠ Random, at least, not for the meaning imbued in the phrase 'deterministic chaos'.

 

[Giulio Prisco]

In the first sentence "unpredictable" would be more correct, but in the last I really mean random.

 

[Asymptotic]

Randomness is white noise. It is without form. One thing does not lead to another (data points are serially uncorrelated). Perform an autocorrelation on random data, and the resulting output is indistinguishable from the input.

Deterministic chaos forms patterns. Patterns emerge when chaotic data is autocorrelated.

 

[sophiecentaur]

But does this even make sense in a finite universe where computation is physical?

This is true of all computational Maths. We use Maths to map the Universe and the way it works but all our calculations involve discrete values, even when the Theory assumes a continuum and monotonicity. Chaotic behaviour is different. It is the result of Non linear systems under some conditions. All systems are subject to noise.
We can often detect when a relationship is not continuous by using autocorrelation (as @Asymptotic has pointed out). But what has all that got to do with your claim that Randomness and Chaos are the same thing? Noise, add noise to a continuous relationship and you get a continuous relationship. Add noise to a chaotic function and the result can have variations that are far from continuous.
You should read a bit more about Chaos and not just apply your intuition. It is a revolutionary subject and has revealed a side to our Universe that is not like other processes. Look at this entertaining link about Fox and Rabbit populations and then try to apply what you know about Noise and Randomness to the problem.

 

[Giulio Prisco]

Thanks @Asymptotic and @sophiecentaur. I guess I typed too fast. What I mean is:

Consider a chaotic system near a critical point. The system will fall into an attractor, but we can't compute which one because the computation is too complicated and sensitive to small variations. Yet, the evolution is assumed to be deterministic, with small fluctuation determining the outcome.

I guess the computing resources that would be needed to compute the outcome (which attractor) diverge to infinity as the system approaches criticality. Right?

If so, since we don't have infinite computing resources, does it even make sense to insist that the outcome is predetermined?

@sophiecentaur what should I read? I can follow maths but prefer conceptual explanations?

 

[sophiecentaur]

we can't compute which one because the computation is too complicated and sensitive to small variations.

If you cannot specify the initial conditions precisely then no amount of computational power can help you. Your input parameters are not (quite) the same as those of the system you are trying to predict.

CHAOS by James Gleich is what I read. But what's wrong with that link I posted? If the Maths in that is too hard for you then you can't expect the alternative of a "Conceptual Explanation". Would you expect a "Conceptual Explanation" about how to win at Poker? You can get a "conceptual" flavour of the business from a downloaded Mandelbrot pattern generator and work your way deeper and deeper into the pattern. That exercise should convince you that you are in a different world from the regular Maths we are all used to.

 

[sophiecentaur]

One occasion where vast computing power becomes handy for Chaotic weather situations. I suggest you do a google search on Weather and Chaos and find something you feel ok about reading. Things have progressed a lot since the 1950s and now we can now work 8 days ahead with pretty reasonable accuracy in the weather forecasts and, more than that, the likely errors can be forecast too.

 

[Giulio Prisco]

After some reading on fractal structures in chaos theory I can make my point more precisely.

A riddled basin is a “basin of attraction [with] the property that every point in the basin has pieces of another attractor’s basin arbitrarily nearby,” explains the review paper “Fractal structures in nonlinear dynamics” (2009), by J. Aguirre et al.

In other words, a riddled basin is a region of the phase space that can be thought of as a bulky “fat fractal” boundary between different attraction basins. Every neighborhood of a point in a riddled basin, no matter how small, contains points that will eventually reach different attractors. Therefore, no matter how accurate is the specification of the starting point, the attractor that the system will eventually reach is undetermined.

Riddled basins, which have been found in many dissipative systems, “show that totally deterministic systems might present in practice an absolute lack of predictability,” note Aguirre at al. See also the book “Transient Chaos: Complex Dynamics on Finite Time Scales” (2011), by Ying-Cheng Lai and Tamás Tél.

I suspect that the fractal depth of riddled basins might be widespread in real-world, dissipative dynamical systems, and perhaps be the rule rather than the exception. If so, chaotic evolution is really nondeterministic.

Nature “knows” the starting point of the system as an infinitely precise real number. But we can’t know the starting point with infinite precision, and any finitely precise starting point contains the possibility of different outcomes.

By the way, yes there is a conceptual explanation of how to win at poker. It is: If you know how to play reasonably well, and you can afford to lose much more money than the other players, you always win in the long run ;-)