The Chaos Theory

[Richard.]

This is something I've been interested in recently but have very limited knowledge in. My basic understanding of it it is that:

 

Unpredictable events that seem to have no pattern will eventually reveal themselves to have a pattern in time.

 

Would anyone be willing to elaborate on that for me?

 

Thanks for your help :bouquet:

[lawcat]

Unpredictable events that seem to have no pattern will eventually reveal themselves to be predictable.

 

I would change your statement to one above. Chaos is lack of information. Chaos is not to be confused with unstable. The more information we have, the less chaotic something is. Chaos Theory is concerned with predictability of outputs, final conditions. Those depend on inputs, and processes within the system. If you do not know all the inputs and process, the system appears chaotic.

 

Nature is predictable, only conscious organisms can be unpredictable.

[Richard.]

So it is having enough information that all outcomes are predictable?

What do you mean about organisms being unpredictable? Is it because life can always make the other choice no matter how unlikely?

[Boerseun]

I would change your statement to one above. Chaos is lack of information. Chaos is not to be confused with unstable. The more information we have, the less chaotic something is. Chaos Theory is concerned with predictability of outputs, final conditions. Those depend on inputs, and processes within the system. If you do not know all the inputs and process, the system appears chaotic.

 

Nature is predictable, only conscious organisms can be unpredictable.

Not quite.

 

Thanks to Heisenberg, some things are inherently unknowable. For instance, if you know a particle's velocity, you cannot know it's position in space - and vise versa.

 

So there actually is a limit to the information you can have of any system. Whilst lawcat is right in principle, that "the more information we have, the less chaotic something is", there is a limit to the information that can be gleaned from any particular system - and thus a limit to the accuracy of predictions that can be made as to the system's future states. The further you look into the future, the more pronounced tiny little unpredictable quantum effects become, and might eventually express themselves in the bigger picture where Heisenberg don't feature. And because quantum motions are not predictable at all, there is a certain amount of chaos built into every system - and no amount of information gathering will improve your predictions, because there is a physical limit as to the information you can get.

 

But that's in the world of the very small.

 

Chaos theory in general pertains to systems which are incredibly sensitive to initial conditions. The smallest change in one of the initial variable will have a huge impact further down the line. This is the so-called "butterfly effect", and because the eventual outcome is so sensitive to initial conditions, the outcome is very hard to predict with any sort of accuracy. Here's a good primer.

[Buffy]

Boerseun is right, and it's also important to note that "patterns" do not make "predictability."

 

You can certainly show that over time a whole lot of photons will arrange themselves into pretty patterns when light is projected through two slits, but you will never, ever, ever, be able to accurately predict which slit a *particular* photon will go through (unless you're clever and say "both").

 

The first principle is that you must not fool yourself - and you are the easiest person to fool,

Buffy

[Richard.]

So...

 

The chaos theory is basically that the more inputs you have on a system, the more predictable the outcomes are? But you can never have all of the inputs so it will always be slightly chaotic. Systems tie into the Butterfly Effect in that little things that are insignificant at certain times in the system may have a major effect on things that happen down the line.

 

Is that about right?

[lawcat]

So...

 

The chaos theory is basically that the more inputs you have on a system, the more predictable the outcomes are? But you can never have all of the inputs so it will always be slightly chaotic. Systems tie into the Butterfly Effect in that little things that are insignificant at certain times in the system may have a major effect on things that happen down the line.

 

Is that about right?

 

More or less, Yes. Chaos Theory postulates that even if you know all inputs, and know all processes, the final condition is still a matter of chance, probability. So, in probability,as more processes are involved, the chance of error in prediction compounds. The more subsystems, the more outputs are calculated, the larger the error. Now, for some calculations we can model systems as large systems with a single process, and the chance of error is heavily reduced. The example is Newtonian Gravity calculation. But if we were to predict movement of earth through calculation of each quantum state, then the error would be large. Then, the scientists introduce some constant, or coefficient, to reduce the error to the proper scale, to fit the observed data.

[CraigD]

Chaos theory isn’t a scientific theory in the ordinary sense of explaining some specific collection of observable phenomena, but more of an informing principle for making scientific theories and less rigorous practical predictive models. To try to emphasize this interpretation, I’ll call it “the chaos principle”.

 

TCP is well described by the butterfly effect thought experiment.

 

At its heart, TCP states that small differences in the state of a purely deterministic system that changes over time according to a well, or even completely, defined set of rules, may result in large differences in future states of the system. It contradicts and replaces an older principle, which we can reasonable call “the law of averages”, that played a similar guiding role in science prior to the emergence of TCP as a dominant principle. In contract to TCP, TLOE states that small differences in the state of a purely deterministic system that changes over time according to a well, or even completely, defined set of rules, result in small difference in future states of the system.

 

For example (picking a famous one in the history of science), according to TLOE, a sufficiently “high-resolution” model of the Earth’s weather – one, for example, that tracks the temperature and momentum of a representational sample of air and surface molecules, incoming solar and outgoing geothermal energy, etc – should, if a reasonably precise (vs. perfectly accurate) measurement of the real phenomena it models – the Earth and the Sun’s geological, meteorological, an heliological phenomena - is made and used to set the model’s variables, the model will be able to make an equally precise, accurate prediction of the weather far into the future. TCP states that, unless the model is nearly perfect and the measurements used to set its variables nearly exact, it will fail to make accurate predictions far into the future.

 

I’ve made a pretty sweeping claim to a concise definition of Chaos theory, which demands some strong support. Rather than attempting to support it with a technical argument, I’ll draw from the history of statements made by a prominent scientist, John von Neumann:

All stable processes we shall predict. All unstable processes we shall control.

Although commonly attributed to von Neumann, it’s questionable that he actually said this, or stated it for public consumption. It’s more likely to have been a quote attributed to his – something they believed he would say, by people who knew him and his principles attributed to him, such as Freeman Dyson. Even this is hard to precisely attribute, but was repeated enough during von Neumann’s lifetime that its reasonable to assume he didn’t disagree with it.

 

Thus, it’s reasonable to assume that von Neumann, and the generation of mathematical model makers he inspired, believed ca. 1950 that it was just a matter of time and technological progress in computers and measuring devices before it would be possible to make detailed weather predictions for any place on Earth for specific dates years into the future – and more dramatically, through practical engineering means, using these models, prevent detrimental weather, such as hurricanes, tornados, and floods and droughts.

 

Instead, as technology advances to the point where von Neumann and his contemporaries believe numeric modeling would begin to produce accurate forecasts hundreds of days ahead, it became apparent that, where these models and the actual weather they modeled were concerned, TLOE did not apply. Rather than small inaccuracies in the model and the measurements used to set it to match a known state of the real weather “averaging out” as the model was run, they increasingly perturbed the model, until after a short amount of modeled time – a few to ten or so days – the model badly disagreed with the modeled, predicting, for example, a precise large amount of rain on a day and a location that actually had no rain.

 

The failure of these models to work as von Neumann and his contemporaries had expected, informed as they were by TLOE, was a major influence in the emergence of TCP.

 

Here’s a partial list of points helpful in avoiding some common misperceptions about

Chaos:

• A model need not be complicated to be chaotic. For example, a simple computer model of a double pendulum – a simple pendulum on the end of another simple pendulum – is chaotic. The simple, event-based simulation we played with in 11819 is chaotic.
  
• Models don’t have to contain random or unknown variable to be chaotic. Thus, it’s not necessary for some intractable uncertainty, such as is found on a subatomic level in the physical universe, to exist or be significant for a system for it to be chaotic.
  Ideal systems can be chaotic.
  
• Chaotic systems are not necessarily unpredictable. For example, the double pendulum and “Newtonian bowling” examples mentioned above can be used to exactly predict a future state if the exact values of their initial variables are known. Unless a model intentionally or unintentionally contains true randomness – a harder thing to do than one might think – it must be completely predictable.
  If we know the exact initial variable of such a model, it ceases to be unpredictable, but is still chaotic.
  
• Whether a system is chaotic or not depends on what about it’s being measured at what time.
  For example, a double pendulum with constant friction can’t have its position predicted for all future instants if a small uncertainty about its initial position and velocity exists, but the future state in which it is no longer moving can be predicted with certainty, even if we know very little about its initial state.
  

Here are a couple of sources on von Neumann and mathematical weather modeling: Quotations by Von_Neumann

Before 1955: Numerical Models and the Prehistory of GCMs

[Kriminal99]

Chaos theory and it's purpose is best summarized in my opinion by putting it in terms of probability. There is a measurement called variance that determines how far away from the average (and how often) measurements of a certain type get. High variance results from situations where a small change in input factors causes a large change in output. A good example is a toin coss. Say you toss a coin twice, but the second time you use a slightly different amount of force, position it slightly differently in your hand etc. As a result, the coin hits an object on it's way down resulting in a completely different chain of events. Or maybe the slight difference causes it to hit the ground on its side rather than flatly, which also causes a completely different chain of subsequent events. Now the end position in the result space (which is folded into just two outcomes heads or tails) of the coin toss is totally different due to a tiny change in input factors.

 

Some people think of coin toss as "random" which to them is a magical word that means there is no rhyme or reason to how it turns out. However the above example shows how a perfectly deterministic chain of events can cause such a random outcome. By dispelling the naive notion that most of what we see around us is due to some magical unexplainable randomness, we also dispel many notions that depended on such naive interpretations. Determinism and the lack of free will as a consequence is a good example.

 

Chaos theory dispels the barrier of randomness along the path of observing the world as entirely mechanical.

 

 

Not quite.

 

Thanks to Heisenberg, some things are inherently unknowable. For instance, if you know a particle's velocity, you cannot know it's position in space - and vise versa.

 

So there actually is a limit to the information you can have of any system. Whilst lawcat is right in principle, that "the more information we have, the less chaotic something is", there is a limit to the information that can be gleaned from any particular system - and thus a limit to the accuracy of predictions that can be made as to the system's future states. The further you look into the future, the more pronounced tiny little unpredictable quantum effects become, and might eventually express themselves in the bigger picture where Heisenberg don't feature. And because quantum motions are not predictable at all, there is a certain amount of chaos built into every system - and no amount of information gathering will improve your predictions, because there is a physical limit as to the information you can get.

 

There is a difference between something being predictable or observable and it being determined. There is no way to rule out that unobservable actions are determined by some process we cannot be aware of. Some people try to argue that if you can never observe the way in which something is determined, than there is no reason to think that it is determined. I disagree. In my opinion there is no alternative concept to propose. Everything that we know and the way that we know it is dependent on determinism.

 

When people try to speak of things being undetermined, they often refer to statistical models of quantum mechanics. These are usually people that never really understood "randomness" in the macro world to begin with before studying quantum mechanics. We can create randomness without quantum phenomenon, understand what it means, and construct probability theory based on it. Statistical observations of something that is undetermined would be nonsense.

 

Thinking about something being undetermined is nonsense. You can't use a bunch of metaphors made up of things that are fully determined, using a mind that depends 100% on the principles of determinism to function correctly to try and represent something undetermined. We don't know what that means, we can't know what that means, if it even means anything as opposed to being a nonsensical linguistic construction.

 

So the point is, while chaos theory might not be useful in the realm of quantum physics due to limits on what we can observe, it still may apply.

[uso]

But why the chaos 'theory' is 'deterministic' and 'chaos' at the same time? Does chaos theory explain that things in the future cannot be exactly predicted?

[CraigD]

But why the chaos 'theory' is 'deterministic' and 'chaos' at the same time? Does chaos theory explain that things in the future cannot be exactly predicted?

Chaotic systems are considered deterministic because The future state of one can be exactly predicted if its initial/present state is known exactly.

 

If the initial state of the system is not known exactly, but rather with some small uncertainty, uncertainty about its future state becomes much greater with time. This "sensitive dependence on initial conditions" is characteristic of a chaotic system.

[uso]

Chaotic systems are considered deterministic because The future state of one can be exactly predicted if its initial/present state is known exactly.

 

If the initial state of the system is not known exactly, but rather with some small uncertainty, uncertainty about its future state becomes much greater with time. This "sensitive dependence on initial conditions" is characteristic of a chaotic system.

 

Thanks for the reply. As i grasp it, the first or initial conditions should be known in order to make exact prediction. Does it mean that there can be first cause or causes that arose without presence of any conditioning cause? can you please give me an example of chaotic system?

[Qfwfq]

I think it might be necessary to clarify what Craig said.

 

There can be simple systems for which (as long as they are not disrupted from outside) the uncertainty keeps increasing gradually. Consider for example a projectile with some initial position and velocity, as long as no further forces are applied to it. There isn't some finite time scale at which even the tinyest initial uncertainty causes total unpredictability.

 

Does it mean that there can be first cause or causes that arose without presence of any conditioning cause? can you please give me an example of chaotic system?

It isn't a matter of causes arising out of nothing. It is all a matter of uncertainty getting radically amplified. This does not depend on how relations between causes and effects work in the system, only on the complexity of the dynamic evolution they result in overall.

[uso]

I think it might be necessary to clarify what Craig said.

 

There can be simple systems for which (as long as they are not disrupted from outside) the uncertainty keeps increasing gradually. Consider for example a projectile with some initial position and velocity, as long as no further forces are applied to it. There isn't some finite time scale at which even the tinyest initial uncertainty causes total unpredictability.

 

It isn't a matter of causes arising out of nothing. It is all a matter of uncertainty getting radically amplified. This does not depend on how relations between causes and effects work in the system, only on the complexity of the dynamic evolution they result in overall.

 

Thanks for clarification. I want to know one more thing. Is there any difference between 'chaos' and 'deterministic chaos'?

[uso]

it is said that if the initial conditions are exactly known, in any chaotic system, the prediction can be deterministic. But what about the other condition that come later and affect the chain? Does that mean if any prediction can be exact, all these factors initial and later must be known? From the today's condition, something can be predicted. But does not that prediction get changed by tomorrow's conditions?

[CraigD]

As i grasp it, the first or initial conditions should be known in order to make exact prediction. Does it mean that there can be first cause or causes that arose without presence of any conditioning cause?

The term “first cause” usually refers to a an old philosophical argument for the existence of a supernatural creator of nature. I’ve never heard the term “conditioning cause”.

 

Neither are commonly used in math or mathematical physics, unlike “initial” or “starting conditions” or “state”. I wouldn’t use them, to avoid confusing mathematics and philosophy.

 

can you please give me an example of chaotic system?

A compound pendulums with particular starting positions and velocities can be chaotic. The same pendulum with different ones can be non-chaotic.

 

The weather – whether it’s raining or not on a given day in a given place, for, example – is chaotic. Prior to about 1970, it – or, more precisely, computer models of it – was assumed to be non-chaotic. The discovery by Ed Lorenz that it wasn’t is considered to have been the beginning of chaos theory (though the mathematical foundations for chaos theory were laid around 1900), so it’s reasonable to say that attempting to predict the weather far in advance using computer models is the “mother of chaos theory”.

 

Many iterative mathematical functions are chaotic, arguably the most famous being

[math]x_{n+1} = x_0 + x_n^2 -y_n^2[/math]

[math]y_{n+1} = y_0 +2 x_n y_n[/math]

which determining if a point (x₀, y₀) is contained in the Mandelbrot set

 

Is there any difference between 'chaos' and 'deterministic chaos'?

As used in mathematics, as we’re using it in this thread, I’d say there’s no difference.

 

Chaos isn’t, like many mathematical terms, a specific, well-defined one, like, say “sign”, “parity”, “closed”, “greater than”, etc. It’s to some extent a principle advising caution about making some kinds of assumptions.

 

It’s hard to precisely categorize these kinds of assumptions, but I’ll try.

 

One is assumptions that systems that have only small differences in their state will continue to have only small differences – that is, non-chaotic. This is true of so many systems, prior to widespread awareness of the concept of chaos, we intuitively imagined that nearly all systems were like this.

 

Another, which follows from the first, are assumptions that chaotic systems aren’t very important, so, while we can create ideal models, like compound pendulums, that are chaotic, these oddities aren’t useful or important for describing things we really care about.

 

Since nobody’s much surprised that non-deterministic systems – ones that are subject to strong, external changes – have large differences over time, it doesn’t make much sense to talk about such systems being chaotic.

 

Looked at from another perspective, a good definition of a chaotic system is one critically sensitive to initial conditions. As the initial conditions of a system with lots of non-deterministic future inputs doesn’t depend much on initial conditions, it doesn’t make sense to call them chaotic.

 

In short, the term “non-deterministic chaos” doesn’t make sense, in the chaos theory sense of the term “chaos”.

[CraigD]

it is said that if the initial conditions are exactly known, in any chaotic system, the prediction can be deterministic. But what about the other condition that come later and affect the chain? Does that mean if any prediction can be exact, all these factors initial and later must be known?

Yes – though, as I explain in the next paragraph, there’re really no such thing as “factors later”.

 

From the today's condition, something can be predicted. But does not that prediction get changed by tomorrow's conditions?

In deterministic mathematical systems, the system must include everything that can affect its state. So the initial state of the sub-system that produces “tomorrow’s conditions”, along with the rules that allow the model to determine those conditions from that initial state, must be part of the whole system.

 

In other words, deterministic systems must be closed.

 

Consider the following very simple models, written as pseudo-code.

 

Set A = 0

Set X = 1

Begin loop

Set A = A + X

End loop

 

This is a closed system, and is deterministic. For any future time T, in units of number of completed loops, we can with complete confidence predict that the value of A is T.

 

Set A = 0

Begin loop

Get input X

Set A = A + X

End loop

 

This is an open system, and is non-deterministic. Without knowing what the value of each externally inputted X is, we can’t predict the value of A for any future T.