Is Chaos Truly Nonexistent in a Digital World?

[JD1]

Does " Chaos " really exist ?

Does " Chaos " really exist, i think not, there is an understanding of every situation, no matter how small, and a Digital view of this, is by way no means a true view, this is the only reason " Chaos " exists, because "Digital Fails to understand and gives false values as answers " :-)

regards

JD

 

[genericusrnme]

I'm not quite sure what you mean...
Chaos does exist because for systems like the double pendulum, for example, you need to know exactly the initial conditions to be able to predict the time evolution. In reality however, there is no way we can measure the initial conditions to that degree of precision.

 

[NascentOxygen]

Does " Chaos " really exist, i think not, there is an understanding of every situation, no matter how small, and a Digital view of this, is by way no means a true view, this is the only reason " Chaos " exists, because "Digital Fails to understand and gives false values as answers " :-)

regards

JD

Perhaps you should first define what you mean by chaos. Might be difficult, admittedly, to define something you don't believe exists, but give it a try...

 

[cavinmozarmi]

Do you mean something like "chaos theory" or entropy?

 

[Danger]

Anyone who had ever visited my home knows that not only does chaos exist, it's a way of life.

 

[mfb]

Chaos does not mean "unpredictable" (or even "not deterministic") - it is a usual consequence, but not a basic principle. Chaos means that arbitrary small deviations from initial conditions lead to large deviations after some time. This exists, you can show it with mathematics.

You can measure the initial conditions of some system with a precision of 5 decimal digits? Fine. You can predict the evolution for the next minute or whatever (depends on the system), afterwards the tiny uncertainty in the initial condition ruins your calculations. You improve the setup and now you can measure 10 decimal digits? Great, now you can predict the evolution for the next two minutes.
And so on - as long as there is any uncertainty (and there is always an uncertainty - at least when you reach the quantum level), the timescale of your predictions is limited.

 

[Rap]

Does " Chaos " really exist, i think not, there is an understanding of every situation, no matter how small, and a Digital view of this, is by way no means a true view, this is the only reason " Chaos " exists, because "Digital Fails to understand and gives false values as answers "

You are right, in a sense. First of all, "there is an understanding of every situation, no matter how small" is not true, in the sense that if we know initial conditions, we can predict the outcome at any time. That's only true in classical physics. But, if we restrict ourselves to classical physics, then yes, given initial conditions, the final outcome at any time can (in principle!) be calculated. In practice, when you have a "chaotic" system, you cannot. "Digital fails" only when you don't have enough digits. Use more digits, and you can calculate further out in time. The thing about chaos is that the number of digits grows exponentially as you move out linearly in time. That means that at some finite time, the number of digits you need will exceed the number of atoms in the universe. Even if we lived in an infinite universe, the time it takes to calculate your results will exceed the age of the universe. Does chaos exist? In a practical sense, the answer is always yes.

 

[mikeph]

It exists in every sense, even in classical mechanics.

 

[D H]

But, if we restrict ourselves to classical physics, then yes, given initial conditions, the final outcome at any time can (in principle!) be calculated. In practice, when you have a "chaotic" system, you cannot.

That is not what chaos is. Chaos does not mean unpredictable. It doesn't even mean random. In fact, truly random systems are not chaotic. A chaotic system is deterministic. Chaos simply means that two sets of initial conditions that are initially close to one another will eventually diverge markedly after some amount of time has passed (there's a bit more than that, but sensitivity to initial conditions is key). The Lyapunov time is typically used as a measure of time to become chaotic.

 

[JD1]

this is what i mean, even the smallest particles lookin as tho there is no order to their chaotic behaviour, there is, there has to be, the movement of one will cause a relative movement of another and so on and so forth, and if all the physical laws are taken into consideration, the movements of these particles can be foreseen, river water flowing and splashing off rocks looks chaotic, but again, take all the physical laws involved and this can be predicted, digital will never be able to achieve this, as you kindly put forward Rap, not even by the adding of digits, it just can't be done, so why use something thats, from what i can see, is a forgone conclusion of failure, and it just seems to me that with digital analysis, the word chaos is introduced when digital techniques fail to be able to give a true representation of the event, we live in an analogue world, and the movements of the smallest atoms, have order, and in order to see and understand this, better equipment is required, and in my eyes, any analogue values that are measured, should be kept as an analogue value, and not changed, this change causes the end representation to differ from the original, which in turn creates a different result,

regards

JD

 

[JD1]

That is not what chaos is. Chaos does not mean unpredictable. It doesn't even mean random. In fact, truly random systems are not chaotic. A chaotic system is deterministic. Chaos simply means that two sets of initial conditions that are initially close to one another will eventually diverge markedly after some amount of time has passed (there's a bit more than that, but sensitivity to initial conditions is key). The Lyapunov time is typically used as a measure of time to become chaotic.

ill get back to you on this Lyapunov time subject D H, I've just got to get me head round it, never heard of this, :-)

regards

JD

 

[ZapperZ]

You need to be careful of the terminology that you are using. This is a physics forum, and most of there terms have clear, definite definitions.

It certainly appears that you are not aware that the word "chaos" has a proper definition in physics.

Zz.

 

[boneh3ad]

JD1, I still think you don't understand that chaos has a very specific meaning in mathematics and physics. You aren't using it correctly.

As for the rest of your idea regarding computational inaccuracy being the real root of chaos, that isn't true either. Look into the history of the Lorenz attractor. That is exactly what everyone thought then and Lorenz proved it was not a computational artifact but actually a real mathematical phenomenon.

 

[meldraft]

Also keep in mind that chaos is bound by the order of magnitude of the system. For instance, Poincare showed that, while there are chaotic elements in the 3 body system, the perturbations are bound by the order of magnitude. This is why the Earth doesn't disengage orbit from the sun. So, in that sense, no, for me chaos in the sense of wild unpredictability that most people realize it, does not 'really' exist.

 

[Rap]

That is not what chaos is. Chaos does not mean unpredictable. It doesn't even mean random. In fact, truly random systems are not chaotic. A chaotic system is deterministic. Chaos simply means that two sets of initial conditions that are initially close to one another will eventually diverge markedly after some amount of time has passed (there's a bit more than that, but sensitivity to initial conditions is key). The Lyapunov time is typically used as a measure of time to become chaotic.

I agree with everything you say.

You have to read more carefully what I wrote: "But, if we restrict ourselves to classical physics, then yes, given initial conditions, the final outcome at any time can (in principle!) be calculated. In practice, when you have a "chaotic" system, you cannot."

Perhaps I should have been more clear and said "...the final outcome of a chaotic system at ...". In other words, yes, its deterministic.

When I say "in practice", I mean that, for a chaotic system, if you sit down with a calculator of any finite size, there will be a finite time beyond which that calculator will be unable to predict the outcome to a given accuracy. (for sufficiently small accuracy)

 

[meldraft]

I agree with everything you say.

You have to read more carefully what I wrote: "But, if we restrict ourselves to classical physics, then yes, given initial conditions, the final outcome at any time can (in principle!) be calculated. In practice, when you have a "chaotic" system, you cannot."

Perhaps I should have been more clear and said "...the final outcome of a chaotic system at ...". In other words, yes, its deterministic.

When I say "in practice", I mean that, for a chaotic system, if you sit down with a calculator of any finite size, there will be a finite time beyond which that calculator will be unable to predict the outcome to a given accuracy. (for sufficiently small accuracy)

As far as I know there are no chaotic phenomena in digital systems. That's why we invented the control bit Also, what you describe is not a time dependent phenomenon but rather an order of magnitude dependent one, which redirects you to my previous post. Uncertainty here is bound by the order of magnitude, which allows you to be certain that up to a certain digit the result is accurate.

 

[Khashishi]

I think there's some misunderstanding about what chaos is.
A calculator has only a limited number of digits of resolution, so it is susceptible to chaos.
Let's say you want to calculate the recursion relation:
x_{0}=\pi/4
x_{n}=4x_{n-1}(x_{n-1}-1)
And you want to know what x_1000 is. If you punch in the first 20 digits of pi/4 into your calculator and calculate the result, it will be totally wrong. Note that your input is only off by less than 1 part in 10^19, yet every digit in your result is probably wrong.
Chaos doesn't mean randomness in physics. It means than a small deviation in the input value will produce a deviation in the output value which grows exponentially in time (until it saturates). When you punch in a number into a calculator, you generally truncate some digits past a certain point. This is typically a very small deviation in the input value, and you expect that your output is very close to the "true" output for the function. In a chaotic system, you might get an output that is close to the true output... initially. But as you evolve the system, the output will deviate from the true output more and more.

On the other hand, if space-time is quantized in such a way that it is possible to specify _exactly_ what the input position is, then perhaps chaos becomes unimportant.

 

[Rap]

I think there's some misunderstanding about what chaos is.
A calculator has only a limited number of digits of resolution, so it is susceptible to chaos.

No, I think physical systems are classified as chaotic (or at least the mathematical equations modelling them). This does not mean that they are not deterministic (i.e. final conditions can be calculated to arbitrary accuracy with a sufficiently "large" digital computer and "large" amount of time. Practically speaking, the "largeness" grows exponentially, quickly exceeding all practical bounds. Digital computers are not chaotic, nor "susceptible to chaos". The mathematical equations describing their behavior do not qualify as chaotic.

 

[davenn]

Anyone who had ever visited my home knows that not only does chaos exist, it's a way of life.

haha, sounds like my workshop at home too :)

it would be illogical to consider that chaos didnt exist

Dave

 

[meldraft]

I think there's some misunderstanding about what chaos is.
A calculator has only a limited number of digits of resolution, so it is susceptible to chaos.

Indeed, but notice what I wrote: it's bound by the order of magnitude. This means that up to a certain digit you can trust your result I have NEVER seen a calculator give different values for the same calculation. It may give the wrong value, but it will always be the same, no matter how many times you try it.

 

[JD1]

well guys, i feel should apologise first of all for jumpin the gun and making a statement on something i had a wrong view of, but at the same time, if i hadnt of done id more than likely still have the same view, so at least an education has took place, we do after all learn by our mistakes, how ever they might happen, and also thank you all for commenting, its been very interesting, and even proper giggled in places, ( Danger - Davenn ) lol, :-)