Question about chaos theory and determinacy?

[Frog march]

take a section of a fractal like this.
If I remember correctly it is produced by some sort of reiterative process, involving complex systems equations, with the coordinates being the input to the equation and the output being some result of feeding the result back into the equation.

My question is, can we know definitely where the boundaries are between the different types of output(represented by colour differences), even though it is a definite mathematical process, given that we can't enter infinitely detailed variables, in a computer?

This question came about by me thinking about the uncertainty principal, and how it might apply to the relationship between the world we live in and the world of mathematical.

[Neverfly]

Nope.

In order to definitely know the output, the input must be definite.

This is the main issue with chaos, sensitivity to initial conditions.

[Frog march]

Nope.

In order to definitely know the output, the input must be definite.

This is the main issue with chaos, sensitivity to initial conditions.

but you can put definite numbers into a computer program, as co-ordinates, you just can't put in an infinitely long decimal point number, say like pi.

[tdvance]

If you start with exact numbers and use exact methods (and either arbitrary precision or never exceed the precision of the machine), the final result will be exact.

If at any place there is an approximation, the final result might be way off even if the approximation was very close--chaotic systems work like that.

That's a problem with predicting, say, where the asteroid Apophos or whatever it is, will be when it passes Earth--we have fairly accurate information on where it is now and where all the planets are that affect it, but the error way out there in the decimal expansion blows up into "missing by miles" after several years.

And not only that, even if we did have the measurements really precise, we'd need to know measurements for interstellar medium, all the stars and their planets in our galaxy, and in Andromeda Galaxy as well! All that exerts a tiny amount of gravity, and can change things way, way out in the decimal expansion, but the magic of chaos is that this error gets blown up enough to ultimately matter.

I remember reading somewhere that if you want to shoot a cueball at a rack of 15 balls on, even a perfect pool table, and sink every ball on the break, you need to take into consideration positions of planets around stars in the Andromeda Galaxy in planning your shot.

[Frog march]

I should have been more clear, but I wanted to see if you'd spot what I was really asking.

I am primarily talking about computer models, where you can be as precises as you want, but you just can't have an infinitely long number like pi, as an input.

What I am suggesting is: if we can't definitely find the boundary exactly, can we even really say it has a definite boundary, even if it is a PURELY mathematical model.

If quantum particles can behave chaotically, I wonder if this has any implications for quantum wave functions....not that I know much about them....

[Neverfly]

I should have been more clear, but I wanted to see if you'd spot what I was really asking.

I am primarily talking about computer models, where you can be as precises as you want, but you just can't have an infinitely long number like pi, as an input.

What I am suggesting is: if we can't definitely find the boundary exactly, can we even really say it has a definite boundary, even if it is a PURELY mathematical model.

If quantum particles can behave chaotically, I wonder if this has any implications for quantum wave functions....not that I know much about them....

The ability to find the boundary has little relevance to whether or not one exists.

[Frog march]

The ability to find the boundary has little relevance to whether or not one exists.

Oh there HAS to be a boundary, but where is it?

[Neverfly]

Oh there HAS to be a boundary, but where is it?

Where is the Higgs Boson?

[Frog march]

Where is the Higgs Boson?

where you left it...neverfly, you're so disorganised.

[Frog march]

sorry NF, maybe I've got a dud idea here...

oh never mind

[Neverfly]

where you left it...neverfly, you're so disorganised.

I like to mes with those CERN folks minds... They think they're so smart... I'll show them!

sorry NF, maybe I've got a dud idea here...

oh never mind

I dunno what is going on in your head lately.
You've been bouncing around like a chaotic black hole ping pong ball.

Maybe you just need a sushi date?

[Frog march]

It's just that I've been stuck at the event horizon for what seems forever. it's such a two dimentional life. Sure, the Universe thinks that I have a fully dimensional life but if you ask me the Universe is over the limit most of the time; it's been hanging out too much with all that dark matter, taking light speed, and it's waist has been expanding too, really let its self go.

[Ken G]

If quantum particles can behave chaotically, I wonder if this has any implications for quantum wave functions....not that I know much about them....

It's not a crazy idea, there have been serious efforts to connect chaotic behavior with the uncertainty in quantum mechanics. I'm not sure if much came of it, because chaos has an underlying determinism that is quite a bit different in quantum mechanics. Still, it is certainly true that classical chaotic systems that show "attractors" leave an imprint of that attractor on the wave functions of quantum mechanical systems controlled by the same basic situation (same Hamiltonian), so quantum mechanics "knows about" classical chaos in at least some ways.

[Frog march]

with this thread, I was trying to find a way, in which a system could be infinitely dynamic, somehow feeding back on itself in such a way that there was the posibility of saying that at some point in the system there were actually no laws of physics, therefore alowing God to be conscious, and thus us.

You cuold say that as you rise in information system levels from this universe to what ever could be said to be above it, and onwards, but even if you said that there were no limit to the complexity of the system, ie infinite levels of more and more complexity, they would still all be coverned by laws.

I have a bit of a problem of God having free will, and at the same time knowing everything, even his own future acts, but without him having free will, how could we?

You could look at this as not about religion, but finding the ultimate source of consciousness, thus not breaking the rules. :/

To really be conscious, surely the reason one makes a decision ultimately can't be deterministic, even if within this universe it is.


Oh well, another ramble in the woods........not going to bother spell checkin.. probably spelled equasion with an 's' again

[Neverfly]

For one, FrogMarch, you are trying to rationalize a deity scientifically. That will never work.
Plus, it may cross some boundaries on BAUT- if not in chaos. Though it might cause chaos.

For two, you seem to be trying to rationalize the consciousness of a deity all wrong too.


no offense.

[Frog march]

For one, FrogMarch, you are trying to rationalize a deity scientifically. That will never work.
Plus, it may cross some boundaries on BAUT- if not in chaos. Though it might cause chaos.

For two, you seem to be trying to rationalize the consciousness of a deity all wrong too.


no offense.

you probably hit the nail on the head .

If I could explain the answer, then how could it be free of laws, free of determinism...

[Neverfly]

you probably hit the nail on the head .

If I could explain the answer, then how could it be free of laws, free of determinism...

Now... What were the odds I would hit that nail on the head?

[mugaliens]

Now... What were the odds I would hit that nail on the head?

I dunno... About the same as you'd hit your thumb?

<ducks beneath the arc of the thrown hammer...>

[Neverfly]

I dunno... About the same as you'd hit your thumb?

<ducks beneath the arc of the thrown hammer...>

About the same as nailing you on the head.

[Cougar]

If I remember correctly it is produced by some sort of reiterative process, involving complex systems equations, with the coordinates being the input to the equation and the output being some result of feeding the result back into the equation.

The image you have shown is apparently a section of the Mandelbrot Set, which is a set of points in the complex plane. Whether a certain point is or is not in the set is determined by plugging that point's coordinates into a fairly simple equation and then iterating as you have described. But the number of times one iterates is somewhat arbitrary. With the Mandelbrot Set, the iteration leads one to a decision: If the iterating values explode to infinity (i.e., super-large), then the point you started with is NOT in the set. If the iterating values remain bounded (less than super-large), then the starting point IS in the set. In this case, one can never be absolutely certain if a point near the boundary is in the set, since the next iteration might send the iteration to infinity.

My question is, can we know definitely where the boundaries are between the different types of output(represented by colour differences), even though it is a definite mathematical process, given that we can't enter infinitely detailed variables, in a computer?

With Mandelbrot, I'd say no. But iterating other equations, such as the logistic difference equation, is a deterministic exercise - but wholly deterministic systems can lead to unpredictable results.

With some systems, one can make a small perturbation, resulting in a small change to the system's state, as one would expect. With other systems, a small perturbation can lead to a massive change, contrary to common sense expectations. These are (poorly) coined chaotic systems, which are obviously highly sensitive to initial conditions.

Quantum physics and chaos theory seem to play in different ballparks, but here is a recent U of U article about them appearing to play together....

[GOURDHEAD]

With some systems, one can make a small perturbation, resulting in a small change to the system's state, as one would expect. With other systems, a small perturbation can lead to a massive change, contrary to common sense expectations. These are (poorly) coined chaotic systems, which are obviously highly sensitive to initial conditions.

Could this apply to the cumulative effect of the various parameters and processes that drive changes to Earth's climate?

[Cougar]

Could this apply to the cumulative effect of the various parameters and processes that drive changes to Earth's climate?

Sure. That's the prototypical chaotic system, as rediscovered and re-"popularized" back in 1963 in an unintentional discovery by Ed Lorenz, who went on to publish...

"...his seminal work titled Deterministic Nonperiodic Flow, in which he described a relatively simple system of equations that resulted in a very complicated dynamical object now known as the Lorenz attractor."

[mugaliens]

Nope.

In order to definitely know the output, the input must be definite.

This is the main issue with chaos, sensitivity to initial conditions.

This isn't necessarily true, as chaos theory includes the concept of a locus. For example, a river might be the water locus of a valley. The input could be anywhere in the valley, including in the river, on the slopes, upstream, or downstream. In chaos theory, however, the locus for the entire valley is the mouth of the delta, and that's a definate output that's relatively independant of the input.

[Neverfly]

This isn't necessarily true, as chaos theory includes the concept of a locus. For example, a river might be the water locus of a valley. The input could be anywhere in the valley, including in the river, on the slopes, upstream, or downstream. In chaos theory, however, the locus for the entire valley is the mouth of the delta, and that's a definate output that's relatively independant of the input.

True, but pretty irrelevant to the OP...

[Disinfo Agent]

The image you have shown is apparently a section of the Mandelbrot Set, which is a set of points in the complex plane. Whether a certain point is or is not in the set is determined by plugging that point's coordinates into a fairly simple equation and then iterating as you have described. But the number of times one iterates is somewhat arbitrary. With the Mandelbrot Set, the iteration leads one to a decision: If the iterating values explode to infinity (i.e., super-large), then the point you started with is NOT in the set. If the iterating values remain bounded (less than super-large), then the starting point IS in the set. In this case, one can never be absolutely certain if a point near the boundary is in the set, since the next iteration might send the iteration to infinity.

The Mandelbrot set can be defined as a limit, and in that sense you can be absolutely certain whether a given point belongs to it or not. For example, the point c=0 belongs to the Mandelbrot set. An interesting remark in Wikipedia:

The intersection of M with the real axis is precisely the interval [-2, 0.25]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family [...] In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot set.

That's in theory. But for most points in the complex plane it's very difficult to tell in practice whether they belong to the set or not. The best you can do is get an approximate answer by having a computer do a couple of iterations, and classifing the points based on how close or far the associated sequence is to converging after those iterations (this is often coded with colours).

In short: pictures like the one Frog March posted are only approximations to the actual set.

[tdvance]

It's often easy to tell a point is outside the set--if the function ever goes past, I think it's 2, it can be proven it goes to infinity. So, you can draw a set that is at worst slightly larger than the Mandelbrot set.

That is one interesting thing about mathematics--even if you can't count to infinity, you can still prove a lot about what happens there.

[mugaliens]

True, but pretty irrelevant to the OP...

Huh?

(image removed)Take a section of a fractal like this.
If I remember correctly it is produced by some sort of reiterative process, involving complex systems equations, with the coordinates being the input to the equation and the output being some result of feeding the result back into the equation.

Yes. And the arms are one locus, or attractor, of many which guide the formation of the fractal.

This question came about by me thinking about the uncertainty principal, and how it might apply to the relationship between the world we live in and the world of mathematical.

The equation is said to be chaotic if a slight change of x and/or y produce a drastically different result.

Some people think it means the equations are not predictable. Not true! If you enter a, b, and c into a chaotic equation, you'll get the same answer every time, provided you use the same value for a, the same value for b, and the same value for c. If you change any of them slightly, however, all bets are off, as the value could be adjacent, or it could be on the other side of the page, with a different value(s).

One aspect of chaotic equations is that, generally, they're not back-solvable. That is, given a point (x,y) with a value (z), there are usually many values for a, b, and c that would satisfy the equation.

This quality is the basis of public key/private key encryption - but that's another thread entirely...