This is a good question many possible answer from both a philosophical and mathematical perspective.

One of the best overviews of chaos and chaotic dynamical systems is given in

http://mathworld.wolfram.com/Chaos.html

Briefly, a chaotic dynamical system S has the following properties:

density:  S is a dense collection of points with periodic orbits.

sensitivity:  S is sensitive to initial conditions, leading to the butterfly effect such initially nearby points can evolve quickly into very different states.

transitivity:  S is topologically transitive, i.e., neighbourhoods of points eventually get flung out to big sets so that they do not necessarily stick together in one localized clump.    For a more precise definition of topological transitivity, see

http://mathworld.wolfram.com/TopologicallyTransitive.html

The term stocastic is synonymous with random.   From its Greek origin, stochastic means pertaining to chance.   In effect, stochastic contrasts with deterministic.

Putting these things together, we can observe that chaotic phenomena are deterministic (from the sensitivity and transitivity properties).   By contrast, stochastic phenomena are random, not deterministic.   Philosophically, our observations about these two types of phenomena would be quite different.

Peter, you are right. However, if we have too few observations (and in social sciences we typically have few points and each may have substantial error of measurement), then we often cannot reject Gaussian hypothesis even for chaotic system.

It has a lot of implications for example in finance (and not only). There are theories of option valuations based on Gaussian assumption of stochasticity. At the same time, some econo-physicists (see for example the book of Mantegna and Stanley) have discovered the so called fat tails in probability distribution of changes in stock returns, that violate Gaussian hypothesis. Still, financial players continue to use Black-Scholes formula because there is nothing better. And this may lead to increasing instability of financial markets.

It is astonishing that this important question remained unnoticed by many researchers. 

Yuri mentioned econo-physics and the Stanley and Mantegna's work. They have detected that stock prices fluctuates according to a truncated Levy stable distribution. This distribution was discovered by taking time average and hence tie invariant. However, if we observe stock prices time-locally, their volatility also fluctuates in a considerable scale. This is why heteroskedastic processes were invented and studied (e.g. ARCH and GARCH model). We do not yet know how these process are generated. There should be some unknown (surely non-linear) mechanism that underlie them.

It is true that we have not yet a good theory (or a good athematics that treats it). This is a grave problem for us.

As Yuri put it,

• Still, financial players continue to use Black-Scholes formula because there is nothing better. And this may lead to increasing instability of financial markets.

However, we should also know the difficulty of our task. In my opinion, clarification of skedastic processes requires as new a type of mathematics as probability theory was to deterministic mathematics.

I have written on a related topic in the question page below. My answer starts from post number 123 which is an Introduction and 3 post follows there after:

How do the economists define self-interest and rationality?

https://www.researchgate.net/post/How_do_the_economists_define_self-interest_and_rationality/

I think that the difference relies in our capability to know the data that is also related to the initial conditions and sensitivity of the system with its abrupt changes. In other words, if we know to infinite precision past, we may know present and future into a Newtonian universe. However, we do not have enough information and we do not know initial conditions either. Last but not least, the sensitivity of the system is really important since it is the one that makes what we may know right know, something obsolete at the next second. Sensitivity makes its abrupt and random changes on the blink of an eye.

Thus, if we do identify a phenomenon as random, we can do some probability calculus and right after we may talk about its stochastic components, doing some statistical inference. In the other hand if there is sensitivity and the calculations that we do are obsolete to infer, predict, or forecast, since the errors are changing our present answer at every second, we are dealing with chaotic phenomena and we have to accept our limitations over such behaviors once trying to forecast, predict, or even repeat.  

With stochastic phenomena, we may at least give some accurate answers that we can trust. With chaotic phenomena, we are capable to know its estrange attractor and how its behaves.  Here I share with you an article that is related with topics about chaos, stochastic behaviors, with evolution sequences and complex Life. Perhaps you may want to read it.

Article Evolution through the stochastic dyadic Cantor Set: the uniq...

I have to disagree with some of what has been said here. In my experience, what is stochastic is a process that follows well-known laws of probability. What is termed chaotic MAY possibly be stochastic, but it's too darned complicated to figure it out.

One example of chaotic might be the unconstrained cases of ad-hoc, mobile networks. There's nothing deterministic about such networks, at all. However it's also not a problem that lends itself to simple, explicit equations. So, we essentially "give up" and model the problem instead. Or perhaps, do many real-world experiments.

Dear Albert, your opinion is interesting. People from different sciences can define basics in a different way. Yes, from a mathematical perspective a stochastic process can be described via the flow of probability distributions; see for example Kolmogorov equations here: https://en.wikipedia.org/wiki/Kolmogorov_equations .

But I am trying to figure out the original philosophical sense.  Poincare had a great vision of deterministic world which is described formally as an interaction of many (infinite or finite?) particles via known laws (it was before the birth of quantum mechanics). Still a deterministic trajectory (even for much lower dimensional system) can become so complex that after a certain time it is better to describe its behaviour in probabilistic terms. In this case we use the term "chaotic", when motion of probability is used for description of a deterministic chaotic process. In this case we may think that the God knows where this process goes exactly, but no human with all its empiric is able to do better than having statistical description. Another contributor to the stochasticity is the phenomenon of bifurcations.

In the case of purely stochastic process we have to admit that somebody (God?) should be tossing a coin, and a priori we do not know where this path will go. The laws of quantum mechanics allow for this fundamental indeterminacy.

So the math can describe both. But what is the reality? In economic models it is very popular to have models with "nature tossing a coin", and you have good or bad luck. What will be your strategy? But the real process in this case is often (partly) known to somebody, not necessarily only to the God. In financial markets they might be market manipulation by big players while the rest believe in fundamental stochasticity.

In the case of mobile networks we face a fundamental question about motivation and predetermination of human actions. Economic theory teaches us to optimize solutions for everybody. So if we know all motivations of all people, we would know how the network of interactions will develop. But clearly no human is a supercomputer to have all this knowledge and to model its deterministic development.

Let me give a few more practical elaborations, also for those who have not seen the fundamental difference yet.

1.       Weather forecast. Here we have a complex dynamic system with chaotic behavior, so that Lyapunov exponent after a certain time (perhaps few days) is high enough so that only statistical description is possible. Still the weather for today and tomorrow is much more deterministic than a week from now.

2.       Shell game; https://en.wikipedia.org/wiki/Shell_game . Complex manipulation of three cups (clearly deterministic) makes an illusion that it is stochastic. People who play are often cheated by more experienced players.

3.       Forecast of a football match. There are so many trajectories, but the  God tosses no coin! (apart from strange decisions of referee to give penalty or not). Statistical approach may be correct, but there might be asymmetric information (like health status of a key player, and there were scandals with manipulation of horse runs). Then, like in the previous case, manipulation is not completely excluded. But on an average statistical approach works, and that is why betting takes place on high scale.

4. Growth of network of communications. Every person has some friends and gets new contacts, mostly in a predetermined manner, but sometimes randomly. A network for a certain individual is probably 90% deterministic (he rationally decides with whom to contact) and just a bit stochastic. If we use the deductive approach of Sherlock Holmes, we can even explain (almost) every contact. Then we have a combination of such networks, and overall super-network grows. If such networks would be purely stochastic, it would be impossible for certain organizations to track, for example, the networks of terrorists. But any scientist typically uses statistical description of such network because there is no public information on that (and if it would be, still complexity is too high).

In all examples we see quite complex dynamics, but stochasticity is not introduced by God, tossing coin in every moment. Stochastic model can be justified only if no human can have more deterministic vision (weather forecast in 1-2 weeks). Surely not for the case 2.

Yuri, yes, but might I suggest, the only difference in all your examples is the complexity of the math involved. Sometimes we can hope to derive a stochastic model of the process, other times we won't even attempt to, so we use statistics or experimental results. There's no fundamental difference, other than that.

Weather forecasting is one good example, because our models have indeed improved dramatically over time. It requires a supercomputer to run some of the most elaborate models, and they are not accurate beyond a certain time, but the trend has been clear. What seemed to be impossible to predict is becoming less impossible. What appeared to be chaotic is turning out to be ever more stochastic.

Forecasting even a football game could conceivably be a stochastic process, because in fact, depending how the ball approaches, the player will only be able to create a finite series of trajectories, within certain angles.

I'm not sure about whether a God would or would not use a coin toss, in these different examples you give. I would assume yes, if you must think in those terms, God could always toss the coin. Why wouldn't he? Every step of the way, a decision fork appears, a coin is tossed. I'm finding it hard to use any concept of what God might decide to do, or not to do, to define the difference between stochastic and chaotic, honestly.

And ultimately, Heisenberg's Uncertainty Principle claims that there's nothing truly deterministic, at all. So in fact, the universe is made up of processes which will vary in their stochastic complexity, and many will appear to be more deterministic than others.

But to me, this is a continuum. More or less chaotic, meaning harder or easier to model mathematically.

Albert, I think that we reached an agreement about the complexity of this problem. Still I do not know how many people understand that: it seems that there are not so many replies. I will try to be simpler in this answer. I also want to discuss some public issues of statistical science.

Let us consider the third term – statistic. It is description of reality using the tools of histogram, probabilities, density, correlations. This approach can be applied not only to stochastic but to purely deterministic cases (and not only for chaos). It is often used when we know nothing about functioning of something, but only the final outcomes.

Let us start from a purely deterministic case. Imagine traffic light that is programmed to change green to red every 30 seconds. Let this frequency be not exactly known to us (so that the period is 1 minute) but with some error, let say 0.1%. Then in 1000 minutes after start we will not know exactly if there is red or green light in this moment. We move to statistical description, 50% red and 50% green. This example is similar to shell game when one side keeps track of determinism and other starts to think in stochastic terms and loses. Asymmetric information is one of the keys.

Now let us consider the distribution of height for a group of people (or animals). At the ignorant level we can tell that the average height is 1.70 meters and the standard deviation is 0.2 m. But if we know genetic and nutrition information for each individual, this becomes much less random. We can explain most of this variance by proper regression.

I want to say that there are relatively few really stochastic processes, while in most of social applications we simply do not disclose a part of information to the public telling that all is random. (I am not talking about the cases like weather forecast, when nobody knows the truth better.)

For example, if you sent a job application, there is 10% statistical chance to get an interview and 1% a job. But it depends also on your education, race, sex, age,... Finally it may depend if you have private relations in this organization or not. In the end some people will never get job due to some hidden variable, while others get it with a higher probability. And statistical approach may not give an answer to this puzzle.

Statistics is used a lot in medicine, but again there is asymmetric information. There is statistics about expected time to live given one's age and country. If one fills answers about the way of life (smoking, drinking, overweight) it allows for more precise predictions. But only the God knows the true value exactly (and probably he is not tossing a coin).

There was a movie Coma (1978): https://en.wikipedia.org/wiki/Coma_(1978_film) . There is a statistical % of deaths after operation, and the cases of non-random death (criminal gang put some patients in coma and then used their organs for transplantation) can be within measurement error and not detected by statistics.

Voilà a most fascinating question, indeed. Normally scientists tend to parametrize the dynamics of thw world, whether deterministic or stochastic. I think that is a great mistake. Complexity cannot be parametrised, event though that is the standard procedure among  complexologists. They thus kill non-linearity. Analystical methods abound among scientists that do research on complex systems.

The lingua franca of science nowadays is not matematics any longer, but statistics. We have come to understand and explain the world in terms of distribution of probabilities. 

Normal distributions, as we already know, are not the only ones. Nonetheless, Gaussian distributions clearly dominate science at large. Call them, f.i. Bayesian networks, or Bell curve, and the like. At the end of the day it all comes to the same point.

The philosophical interpretation is very important here, not a minr issue, indeed.

Carlos, I agree that there is higher demand for statisticians today than for other mathematicians (I think that statistics is a branch of mathematics). However, I disagree that this is correct. If statisticians know only statistics (but not dynamical systems, for example), they will depict a wrong model, which will be used later in applied social sciences. If they would know basic physics and biology, they can look for other scientific regularities and keep statistics only for the part that cannot be explained by other tools.

Clearly, Gaussian is not a unique distribution. It works pretty well in physics when the number of interacting elements is very large. In social systems this is not always the case. Moreover, there is high asymmetry and heterogeneity. But since no other tool is well developed, we continue to use Gaussian assumptions in all econometric work.

You idea about being wrong to parametrize complexity may be correct philosophically, but it is still much better to have some parametric model, both statistical and deterministic, than to have no model at all. We always should be aware to what extent our model describes reality. Even a mixed statistic-deteministic model with parameters is only an approximation, but it is better than just statistical model.

I see the growing problem with the lack of multi-disciplinarity in education and scientific schools (and it is self-reinforcing because existing scientific school grows faster than a new one, even if it sets less socially important questions); see https://www.researchgate.net/publication/272357636_Using_the_Concept_of_Field_in_Space_to_Explain_Socio-Economic_Structures_and_their_Dynamics

About multi-disciplinarity you can see my other question:

https://www.researchgate.net/post/Do_you_think_that_education_in_physics_is_useful_for_social_scientists

Conference Paper Using the Concept of Field in Space to Explain Socio-Economi...

By and large, specialisation prevails in education. Cross-disciplionay approaches are still alternative, not dominant.

Your question focuses on a philosophical standpoint dear Yuri. Therefore, we should both agree that philosophically speaking parametrising is a big mistake when dealing with complex systems. That's the core of the point.

Historically speaking, statistics preceded quantum theory in grasping our world in terms of (distribution) probabilities. Over against the past, I believe this is a great achievement, for it allows us a better understanding of nature and reality.