Deterministic and stochastic chaos

I have a question about chaos, but first a foreword of what I understood.

Noise refers to the random variation of values. Usually unwanted, noise causes a measurement to fluctuate over time.

Chaos happens when starting the system in a slightly different way will lead to drastically different outcomes.

The fundamental difference between noise and chaos is that noise is stochastic whilst chaos is deterministic.

Stochastic means the changes in a system depends on a probability. Suppose you were standing on a line and flipped a coin every second. If it were head you moved right, and if it were tails you moved left. You can’t predict the future because there is only a chance that you go one way or another. This property usually arises from Quantum Mechanics, where things are not for certain, but very likely. Yet most randomness we deal with in most applications is on a level much higher than the Quantum one, e.g., inhomogeneities of nature and materials, air fluctuations (chaotic), particle billiards…

Deterministic means that the system will change the same way from the same starting conditions every time. In this way we could predict the chaotic behaviour if one were to know all the decimal points on a measurement. However, we cannot have perfect information (also restricted by Quantum Mechanics), so the tiny immeasurable differences will be amplified until the system is effectively unpredictable.

Question: can chaos be both stochastic and deterministic? If yes: how/why?

• I think that the idea of noise isn't pertinent. What you want to say seems to me that in one case it is just matter of practical knowledge, and in other domains, namely at Q level, we must deal with probability nevertheless. What is exactly chaotic and stochastic honestly I don't know. Also the above aspect seems to lead to analogue scenario's in practice: throwing a coin isn't quantum, practically one has do a number of experiments to end up with 50:50. I am not sure noise per sé enter in this discussion, it seems more related to the act of measuring than the underlying phenomenon. May 25 '21 at 11:10
• Maybe reading the wiki article will clear up matters en.wikipedia.org/wiki/Chaos_theory May 25 '21 at 11:29
• Perhaps this could help: physics.stackexchange.com/a/629397/247642 May 25 '21 at 16:54

Question: can chaos be both stochastic and deterministic? If yes: how/why?

No, not at the same time, since these words are antonyms in dynamical systems theory. A chaotic model can be either stochastic or deterministic, but not both simultaneously, since the model either includes some source of randomness in it or not.

Now, if the question is not about a given model, but rather on how to model a given process/system, then it's worthy it checking this answer to the question Stochastic process vs high dimensional chaos in models, which discusses deterministic (chaotic) and random modelling.

For a more philosophical take, don't miss the answers to Is throwing dice a stochastic or a deterministic process? (or many of other questions with the tag determinism).

Most of the time one will focus on either the chaotic or the statistical aspect of systems that display both (investigating, say, the effect of noise on the chaotic dynamics), but sometimes the interplay of both is crucial: For instance, random/stochastic chaotic systems have been found to model fully developed turbulence using few degrees of freedom, and stochastic chaos has been linked to quantum chaos in restricted settings.

• That was an awesome answer. Thank you for all the related discussions! May 25 '21 at 16:37

There is no such thing as deterministic chaos in reality. It is only a theoretical concept. Even in a computer simulation, like Conway's Game of Life, the initial conditions would have to be randomized or deliberately designed.

In reality, all chaos is stochastic, there is quantum randomness is every single event.

I think the answer depends on context.

When a real world behaviour is described as “chaotic” then it is probably stochastic, since it will ultimately arise from the exponential growth of quantum-level events (this is the quantum chaos theory).

On the other hand, when a mathematical model or computer simulation is described as “chaotic” then it is probably ultimately deterministic. Even if it uses random parameters, these are usually only pseudo-random. The one exception to this is if the computer program uses a hardware random number generator that is linked to a source of real world noise.

I think the answer is no. "Chaos" usually refers to deterministic chaos: i.e. chaotic phenomena are governed by well known laws and equations, but they are not predictable because they are exponentially divergent as time passes.

Stochastic phenomena are also not predictable with certainty because they are probabilistic by definition, as you noted. They are, however, not deterministic like chaotic phenomena are.