Exploring Chaos & Randomness in Deterministic Dynamics

[frostysh]

For myself is horrible to imagine that indeterministic randomness can appear in deterministic, absolutely known and visible dynamics. I have briefly look on the mathematics behind it and understand almost nothing... Well I have a very basic knowledge in mathematics. There is so-called "Rannou's mapping" (Rannou F. — 1974, Astron. Astrophys., v. 31, p. 289) that says the chaos is not a result of computations precision — can somebody please say something about that? Is it true? Or is it questioning... Mathematics more rigorous than physics in that case, in physics is super hard to make experiment with such precision.

So please, if somebody can say something on a that "low level" to help me in understanding of this topic, thank you!

 

[Doc Al]

Have you looked at the Wiki page for Chaos Theory? They give a few simple examples of how some systems are incredibly sensitive to initial conditions, giving the appearance of randomness. Wiki: Chaos theory

 

[frostysh]

Wiki page for Chaos Theory

Yes I looked briefly, this explanation is totally not understandable for me. There is no determinism or chaos in just sensitivity, on abstract level I can reverse this process in time and obtain initial state. Or I am wrong?

What sensitivity they talking about if we are precisely know, with infinite precision, the initial conditions and this initial conditions is going to totally knowable, determined equation, HOW it can produce the randomness in the output with time? I was shook when I have first know about what is deterministic chaos means. And what "sensitivity"? We have some random fluctuation in start condition?

There is should be the only one dynamics:$$\text{INITIAL CONDITIONS} \Longleftrightarrow \text{EQUATION} \Longleftrightarrow \text{THE RESULT}$$That initial conditions is perfectly defined, the connections must have no any randomness, and the equations have no any randomness, so where is the chaos? Of course, of course, as to be said on the other scientific forum (the russian language one), I have zero knowledge and it is not a simple trivia, but still, as an attempt...

 

[Vanadium 50]

I have a marble on top of a perfectly conical hill. When it falls down the hill, how do I predict the direction it "chooses"?

 

[Doc Al]

That initial conditions is perfectly defined, the connections must have no any randomness, and the equations have no any randomness, so where is the chaos?

The chaos comes from extreme sensitivity to initial conditions, which came as a surprise to the first ones who discovered this. @Vanadium 50 gives a good example, as does the Wiki page (double-rod pendulum). Everything is completely determined, but ever so slight variations in initial conditions lead to wildly different consequences.

 

[phyzguy]

In an extreme case, read about "riddled basins of attraction." These are cases where you can end up with one of two possible outcomes (say A and B), depending on where you start. But, arbitrarily close to a point that ends up at A, there are points that end up at B, and arbitrarily close to points that end up at B, there are points that end up at A. So the points that end up at A or B are in some sense infinitely close together. So you would need infinite precision to predict whether an initial point will end up at A or B.

 

[anorlunda]

Yes I looked briefly, this explanation is totally not understandable for me. There is no determinism or chaos in just sensitivity, on abstract level I can reverse this process in time and obtain initial state. Or I am wrong?

I think you missed the key word "apparently" random. Deterministic yet apparently random.

 

[Filip Larsen]

Another (visual) example that may be helpful for understanding sensitivity on initial conditions is the process of mixing of colors (see also [1]).

Put a red and blue slab of clay on top of each other, fold them half-way back on top of each other and flatten them so they again take up same volume as before (like when making bread). As you repeat this folding and stretching of the two colored materials they will get more and more finely mixed. This process will be deterministic if the folding and stretching is done in a known way each time and if we assume there is no diffusion or similar processes going on for the particles of the material (i.e. mathematically the folding and stretching deterministically move each particle around directly and there is no particle-to-particle interaction).

Now, if you before the first fold put in two small black black dots very, very close to each other you can map how far apart these dots will be after first, second, and Nth fold. What you will then typically see is that the dots will seem to stay together (be correlated) for many folds and then suddenly over a few more fold start to visually separate from each other and then for the following fold end up being in "random" positions relative to each other (their positions are no longer correlated with each other). This is the effect of having sensitivity on initial conditions.

Thus, sensitivity on the initial conditions means that no matter how close you initially place the dots, as long as the distance is non-zero it will eventually grow (exponentially) until the distance becomes larger than some constant factor of the system size.

Please note here, that having sensitivity to the initial conditions is a necessary but not sufficient condition for having a chaotic system. That is, you can have sensitivity on initial conditions without the system exhibiting deterministic chaos. One other important condition for deterministic chaos is that there must be some kind of folding and stretching of the phase space, like the process of mixing of colored clay.

[1] https://en.wikipedia.org/wiki/Horseshoe_map

 

[frostysh]

I have a marble on top of a perfectly conical hill. When it falls down the hill, how do I predict the direction it "chooses"?

Good model, I like it. In physics it's fine, and indeed it will be impossible to predict exactly how will be fall, but if we can add some abstract level we can easily determine. Let's say we have infinitely powerful zoom glass.,.

And indetermined becoming determined.

The chaos comes from extreme sensitivity to initial conditions, which came as a surprise to the first ones who discovered this. @Vanadium 50 gives a good example, as does the Wiki page (double-rod pendulum). Everything is completely determined, but ever so slight variations in initial conditions lead to wildly different consequences.

I have initial condition that I can set as $2,999... = 3,000...$, then I have changed it slightly to $5,000...$, the change is equal to $2,000...$, all this conditions and the difference have infinity precision, it's actually fractions with infinite mantise. I have obtain widely different, but still truly deterministic result, there is no chaos. Well, at least my poor knowledge say it to me...

In an extreme case, read about "riddled basins of attraction." These are cases where you can end up with one of two possible outcomes (say A and B), depending on where you start. But, arbitrarily close to a point that ends up at A, there are points that end up at B, and arbitrarily close to points that end up at B, there are points that end up at A. So the points that end up at A or B are in some sense infinitely close together. So you would need infinite precision to predict whether an initial point will end up at A or B.

And I have read it! And I hope I uderstood good the basic idea... And I can say that I can always choose infinitely small $b_{\varepsilon}$ which will be a border for the point $p$, in other words we can direct the radius $\varepsilon$ to the zero, and zero is by definition $0 < \varepsilon$, so the point $p$ is preciously defined.

I think you missed the key word "apparently" random. Deterministic yet apparently random.

The point is there is a book which I understand almost nothing, the monography of some brain-like mathematicians, but it actually says that word "definitely" or "mathematically rigorously" should be used instead of apparently. There is two way to show it (and I don't even know if it mathematical prove or not), first way is crazy thing that called Solomonoff-Kolmogorov randomness, the second insane thing is called Rannou's mapping. I will put some screen shot of the book and the paper of Rannou F. 1974 year, which is somekind of a topological crazines which is shows that. The article is in public access on the suite of the journal so I have attached it to the post.

The screen shot with a topological "proof" —

Lichtenberg and Lieberman, "Regular and Chaotic Dymanics", 1992 year, pages 306-310.

I will try to translate, bu I have a poor knowledge of English:

"Rennou (she) has defined "the randomness" of periodical trajectories in the next way. It's have only $M!$ one-to-one mappings $M$ points on themselves. Writing the same probability $\dfrac{1}{M!}$ to any of that mappings, we obtain a "random" mapping. This mapping is have a next statistical properties:

1. The probability of the trajectory with the length of $n$, that exit from the given point $(a, b)$, is equal $\dfrac{1}{M}$ and not depends on $n$.
2. The average length of the trajectory $\dfrac{M + 1}{2}$.
3. The average number of the all trajectories is approximately equal $\ln{M} + \gamma$, where $\gamma = 0,577\ldots$ — the constant of Euler.

The numberly modelling of the mapping... has proved this properties of "randomly" mapping. This results is serve as the confirmation of that, the chaotic motion which is observed in hamilton's systems is a consequent of of their dynamics but not the result finitelity of calculus..."

How to understand it?

Another (visual) example that may be helpful for understanding sensitivity on initial conditions is the process of mixing of colors (see also [1]).

Put a red and blue slab of clay on top of each other, fold them...

A very nice analogy! But the point of abstract level is that we precisely knows the position of two points, with an infinity precision, and the folding proces is deterministic, now matter how complicated, so we suppose to obtain a totally predictable and one-to-one in terms of initial points result.

 

[Doc Al]

I have initial condition that I can set as 2,999...=3,000...2,999...=3,000...2,999... = 3,000..., then I have changed it slightly to 5,000...5,000...5,000..., the change is equal to 2,000...2,000...2,000..., all this conditions and the difference have infinity precision, it's actually fractions with infinite mantise. I have obtain widely different, but still truly deterministic result, there is no chaos. Well, at least my poor knowledge say it to me...

But what if you had a system where the difference in outcome between initial conditions 3.000 and 3.0001 was surprisingly large? Also: Don't keep saying "truly deterministic" means "no chaos". That misses the point! (No one is denying determinism here.)

Most physical systems that you play with in an intro physics course are well-behaved. So small changes in starting point make only a small change in outcome. Like releasing a simple pendulum: Releasing it at θ = 10.00 versus 10.01 degrees won't make a big difference. But with some chaotic systems you would get surprisingly different responses with only tiny modification of initial conditions.

 

[frostysh]

But what if you had a system where the difference in outcome between initial conditions 3.000 and 3.0001 was surprisingly large? Also: Don't keep saying "truly deterministic" means "no chaos". That misses the point! (No one is denying determinism here.)

Most physical systems that you play with in an intro physics course are well-behaved. So small changes in starting point make only a small change in outcome. Like releasing a simple pendulum: Releasing it at θ = 10.00 versus 10.01 degrees won't make a big difference. But with some chaotic systems you would get surprisingly different responses with only tiny modification of initial conditions.

The number $2,000...$ is an abstract thing, with same point a can choose$$\dfrac{1}{10^{10^{10^{10^{10}}}}}$$the only thing, its must be finite, in case of infinitely small difference, $2,999...$ is equal to $3,000...$, this is from Fichtengolz's first volume of the book of mathemathics for physicists, so called "Dedekind cuts".

The point is in physics, as I know, we can suppose the no enclosed systems in Universe, this is somekind of postulate of chaos. No matter which system we chose, there is will be some uncertainity due to some external stuff... Indetreminism of "the state of the system" it's called, because we are totally, absolutely have zero knowledge about the external system which interact with this particular one. If we know something? The Universe is large, we can always choose another system and so on...

But the point is "deterministic chaos" is on abstract level similar (or even congruent) to indeterminism, to absolute chaos, to randomness, and there above even some mathematical proof of something like that, at least I have understood it like that.

 

[Vanadium 50]

@frostysh , are you asking a question or telling us an answer?

 

[frostysh]

I am not telling! I am trying to understand this horrible "deterministic chaos"! Well, with poor my own mathematical level, that because I need some help... And simple analogies.

 

[boneh3ad]

The bottom line is that a chaotic system is completely deterministic but is extremely sensitive to initial conditions (ICs). With perfect knowledge of the ICs, we would have perfect knowledge of any later state of the system. Since perfect knowledge of the ICs is impossible, two ICs that look effectively the same subject to measurement precision can still produce wildly different outcomes (i.e. appear random). These systems are governed by completely deterministic (and sometimes fairly simple) equations (e.g. the Duffing equation, the Van der Pol oscillator) yet exhibit such extreme sensitivity to initial conditions that their outputs appear random under certain conditions.

Strogatz is a very accessible text on the topic.

 

[frostysh]

The bottom line is that a chaotic system is completely deterministic but is extremely sensitive to initial conditions (ICs). With perfect knowledge of the ICs, we would have perfect knowledge of any later state of the system. Since perfect knowledge of the ICs is impossible, two ICs that look effectively the same subject to measurement precision can still produce wildly different outcomes (i.e. appear random). These systems are governed by completely deterministic (and sometimes fairly simple) equations (e.g. the Duffing equation, the Van der Pol oscillator) yet exhibit such extreme sensitivity to initial conditions that their outputs appear random under certain conditions.

Strogatz is a very accessible text on the topic.

I will look for the book, but your post, how it's say, don't know word on English, it is contradiction, the point is determinism disallow "randomness", probabilistic things in physics is used when something is unknown.

Totally different the case of the book that I have posted above — on abstract level has been said that the chaos is a property no the finite measurment case (sensitivity is excluded from the conclusion), but the dynamics it'self! There is NO any "sensitivity" to initial condition there if to say hard which is "causing chaos", chaos is the property of dynamics (without initial condition) itself, there is an abstraction from the "sensitivity"... There is topological "proof" of someking, and in addition some "proofs" from Solomonoff-Kolmogorov definition of randomness.
— But the problem is, the mathematical level of this proofs is so high, that I can barely understand even a single word there, but me very curious about so I need somebody who can on simple analogies explain Rennou's mapping...

Let point this again:

— If the chaos is the result of the "sensitivity" from the initial conditions, I am saying that this sensitivity is finite, and our knowledge is infinitely precise, "randomness" in the usual meaning of this word disappearing instead!
— But the some brainly mathematicians saying: no matter of my infinite precision — the chaos will appear! Topology and Information theory, looks like saying that...

OFFTOPIC: By the way, I know much more better film than Cubrick's "Bomb..." about the same topic, much more complicated and interesting — "Fail Safe" in 1964, and reaincarnation of this film in 2000.

 

[anorlunda]

I think the question has been adequately answered several times, but the OP is not accepting what he has been told. The thread has become repetitive, so I am going to close it.