Pages: [1] 2 :: one page |
|
Author |
Thread Statistics | Show CCP posts - 0 post(s) |
Samirol
Blood Sweat and Tears
|
Posted - 2007.02.12 18:00:00 -
[1]
Edited by: Samirol on 12/02/2007 18:02:48 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. In biology chaos theory can explain how small random events may affect large ecosystems in an unpredictable way. Among the characteristics of chaotic systems, described below, is the sensitivity to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, exhibiting an exponential error dispersion, even though the system is deterministic in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, population growth and the vast variety of dissipative structures.
Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. (See the article on mythological chaos for a discussion of the origin of the word in mythology, and other uses.) A related field of physics called quantum chaos theory studies non-deterministic systems that follow the laws of quantum mechanics.
As well as being orderly in the sense of being deterministic, chaotic systems usually have well defined statistics. For example, the Lorenz system pictured is chaotic, but has a clearly defined structure. Weather is chaotic, but its statisticsùclimateùare not.
For a dynamical system to be classified as chaotic, most scientists will agree that it must have the following properties:
it must be sensitive to initial conditions, it must be topologically mixing, and its periodic orbits must be dense. Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.
Sensitivity to initial conditions is popularly known as the "butterfly effect", so called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. entitled Predictability: Does the Flap of a ButterflyÆs Wings in Brazil set off a Tornado in Texas? The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.
Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.
Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimensional torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is...
|
Samirol
Blood Sweat and Tears
|
Posted - 2007.02.12 18:07:00 -
[2]
Edited by: Samirol on 12/02/2007 18:03:57 a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.
Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, "mixing" is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.
Some dynamical systems are chaotic everywhere (see e.g. Anosov diffeomorphisms) but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.
An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor.
For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin.
While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the R÷ssler Map, which experiences period-two doubling route to chaos, like the logistic map.
Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the HTnon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure.
The PoincarT-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.
The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion.
Now...discuss.
|
AlexCA
Amarr De Valken BV Sylph Alliance
|
Posted - 2007.02.12 18:11:00 -
[3]
Edited by: AlexCA on 12/02/2007 18:07:40
Originally by: Samirol Edited by: Samirol on 12/02/2007 18:02:48 Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, population growth and the vast variety of dissipative structures.
And that ladies and gentlemen is why they're always wrong about the weather.
|
Frezik
Celtic Anarchy Anarchy Empire
|
Posted - 2007.02.12 18:35:00 -
[4]
Originally by: AlexCA Edited by: AlexCA on 12/02/2007 18:07:40
Originally by: Samirol Edited by: Samirol on 12/02/2007 18:02:48 Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, population growth and the vast variety of dissipative structures.
And that ladies and gentlemen is why they're always wrong about the weather.
Not quite. It means you can't make detailed, long-term predictions, because there's always a butterfly flapping its wings somewhere to screw you up. The butterfly won't have a great deal of effect over the next 3 days, will have a small but noticable effect over the following 3-5 days, and will completely throw you off past that. OTOH, you can make general, long-term predictions, like "it will be warmer than usual this summer".
I suspect that when looking at weather forcasts, people tend to pick out the times when they're wrong and focus on those. Human intuition tends to be very bad at statistics and probability, even after several years of college-level training in the field.
|
Hakera
Freelance Unincorporated Ushra'Khan
|
Posted - 2007.02.12 19:10:00 -
[5]
to me chaotic systems like the atmosphere are only such until we arrive at the point where we can model it more precisely to rule out the dynamics. The long term and short term seasonal models are becoming more reliable as each revision comes as we better understand albedo and radiative forcing for eg. Whilst it will never remove inherant complex and chaotic nature of such systems, better understanding does allow us to one day answer such questions like 'is man affecting the climate'.
|
Tarquin Tarquinius
Gallente Escorts of Eve
|
Posted - 2007.02.12 19:10:00 -
[6]
my brain hurts
-----
Traditional morality is just a clever way for the weak masses to shackle the strong individual. -- Callicles |
Khemical
Gallente Supernova Security Systems
|
Posted - 2007.02.12 19:20:00 -
[7]
Discuss what, Professor? You pretty much laid the intellecutal (cut and paste) smack down. There's not thing left to discuss. If you want to spark debate, perhaps you might want to do a point/counterpoint setup and then we can really get this party started. Like chaos theory vs fate?
|
ReaperOfSly
Gallente Lyrus Associates Interstellar Starbase Syndicate
|
Posted - 2007.02.12 19:27:00 -
[8]
You haven't really left much to discuss tbh.
If you'd asked some sort of question about it, then all us maths geeks could spam identical answers at you --------------------------------------------------------------------
|
Der Alt
|
Posted - 2007.02.12 21:49:00 -
[9]
Originally by: Frezik
Originally by: AlexCA Edited by: AlexCA on 12/02/2007 18:07:40
Originally by: Samirol Edited by: Samirol on 12/02/2007 18:02:48 Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, population growth and the vast variety of dissipative structures.
And that ladies and gentlemen is why they're always wrong about the weather.
Not quite. It means you can't make detailed, long-term predictions, because there's always a butterfly flapping its wings somewhere to screw you up. The butterfly won't have a great deal of effect over the next 3 days, will have a small but noticable effect over the following 3-5 days, and will completely throw you off past that. OTOH, you can make general, long-term predictions, like "it will be warmer than usual this summer".
I suspect that when looking at weather forcasts, people tend to pick out the times when they're wrong and focus on those. Human intuition tends to be very bad at statistics and probability, even after several years of college-level training in the field.
Its official, we must exterminate all the butterflies of the world to restore cosmic order. We must strike first, before they all flap in tandum and destroy us all with their WMD (Wings of Mass Destruction.).
|
Alvar Ursidae
Amarr Decisive Outcomes
|
Posted - 2007.02.12 22:09:00 -
[10]
All I know, it was chaos killed the dinosaur's baby!
-=services ò eve-stuff =-
|
|
Ralus
eXceed Inc. INVICTUS.
|
Posted - 2007.02.12 23:52:00 -
[11]
ok since theres no actual question in the op heres one for you all, what is the fundamental cause of chaos? Personally I believe that chaos stems from the Heisenberg uncertainty principle, but where else does chaos stem from.
Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
hmmmm I dunno,
|
Derovius Vaden
|
Posted - 2007.02.13 02:39:00 -
[12]
Originally by: Ralus ok since theres no actual question in the op heres one for you all, what is the fundamental cause of chaos? Personally I believe that chaos stems from the Heisenberg uncertainty principle, but where else does chaos stem from.
Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
hmmmm I dunno,
Its impossible to completely remove the affects of chaos from ones calculations, the moment you measure something, be it length, time, velocity, you have invariably altered it. To gain 100% perfection in your observations, one must be outside the system. And seeing as the system can be defined as no larger than the universe, good luck.
|
Thales Archon
Gallente Aliastra
|
Posted - 2007.02.13 02:49:00 -
[13]
DS, are you sick? I'm worried at the fact that you have let a physics thread go uncorrected
|
Dark Shikari
Caldari Imperium Technologies Firmus Ixion
|
Posted - 2007.02.13 02:55:00 -
[14]
Originally by: Thales Archon DS, are you sick? I'm worried at the fact that you have let a physics thread go uncorrected
Corrected? What?
-[23] Member-
EVE-Trance Radio! (DSTrance channel ingame) |
Nekuva
The SMITE Brotherhood Curse Alliance
|
Posted - 2007.02.13 07:34:00 -
[15]
Originally by: Dark Shikari
Originally by: Thales Archon DS, are you sick? I'm worried at the fact that you have let a physics thread go uncorrected
Corrected? What?
Holy sht. I'm convinced that DS is either apart of the matrix or some sort of genie.
-_-_-_-_-_-_-_-_-_-_-_-_-_-
Give me ISK.... |
KingsGambit
Caldari Knights
|
Posted - 2007.02.13 13:52:00 -
[16]
Originally by: Alvar Ursidae All I know, it was chaos killed the dinosaur's baby!
I thought it was Chuck Norris. -------------
My T2 Shop |
Crumplecorn
Gallente Eve Cluster Explorations
|
Posted - 2007.02.13 14:02:00 -
[17]
Originally by: Ralus Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
You've already made the choice, you're just here to understand why. ----------
IBTL \o/ Fix the ******* map! Privateers FTW |
Caol
The Nest Interstellar Alcohol Conglomerate
|
Posted - 2007.02.13 14:55:00 -
[18]
Originally by: Ralus ok since theres no actual question in the op heres one for you all, what is the fundamental cause of chaos? Personally I believe that chaos stems from the Heisenberg uncertainty principle, but where else does chaos stem from.
Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
hmmmm I dunno,
If you could model everything, then I would say yes you could predict the future and understand chaos. As for the initial conditions to that model? well, if you could model everything in the first place you probably have a piece of kit that could try every "reasonable" intial condition there is so giving you a set of "reasonable" futures.
Unfortunately, what is perceived to be and what occurs are usually different. In other words, what is measured experimentally and what the human mind proposes to be taking place (the model so to say) usually diverge to some statistical degree. If you had a model that precisely agreed with the real world in every respect (how you would every verify this is another debate) then (see first paragraph) you don't have a problem (with telling me the future) ... as no such "model" exists at present you'll always be out by whatever error and your prediction of the future will be wrong unless one thing happens: your incredibily lucky and your incorrect model gives you a correct prediction.
Can or will you ever be able to perfectly measure a physical system? Quantum theory says no. Would be interesting to know if there is anyway around the uncertainty principle however. Physicists?
Btw, Coffee + science topics = rambling...
|
ReaperOfSly
Gallente Lyrus Associates Interstellar Starbase Syndicate
|
Posted - 2007.02.13 17:10:00 -
[19]
Originally by: Ralus ok since theres no actual question in the op heres one for you all, what is the fundamental cause of chaos? Personally I believe that chaos stems from the Heisenberg uncertainty principle, but where else does chaos stem from.
Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
hmmmm I dunno,
Well chaos is more or less based on the uncertainty principle. In a dynamical system, you can't measure the position and velocity of the particles. That means it's impossible to accurately measure the initial conditions of the system. A chaotic system is one which is sensitive to those initial conditions, so it's impossible to accurately predict the state it will be in much later on.
Of course, it is possible to predict to some degree of accuracy, if you don't try to predict too far ahead.
For example, the weather is a chaotic dynamical system. The met-office will take readings from all its weather stations and plug them into their simulation. They will then adjust the readings slightly, and plug that into the simulation aswell, and so on and so forth. This will return a large number of possible future states. They look at these states and see if there are any prominent features that occur on most of them, such as high/low pressure points, distinctive wind patterns, etc. The ones which "are most likely to occur" are then broadcast to the public by the weatherman
Of course, this is a probabalistic approach. Sometimes, the unlikely end states occur and the weatherman gets it wrong . --------------------------------------------------------------------
|
Strangely Brown
Cult of the Purple Wolf
|
Posted - 2007.02.13 17:16:00 -
[20]
I think Mr Tim Bisley and co explain it best.
Chaos Theory
|
|
Mtthias Clemi
Gallente Infinitus Odium Curse Alliance
|
Posted - 2007.02.13 18:05:00 -
[21]
Chaos is my only true master...
-------------------------------------------- Stay away from my signature all of ya!!! IM WARNING YOU!!
PEW PEW PEW PEW!
|
Shigsy
Caldari Four Horsemen
|
Posted - 2007.02.13 18:28:00 -
[22]
Damn, and I thought this thread would be about Splinter Cell Your Signature exceeds the max filesize limit of 24,000 bytes. Mail us if you have questions -Eldo |
Nicocat
Caldari New Age Solutions New Age Solutions Amalgamated
|
Posted - 2007.02.13 18:46:00 -
[23]
Originally by: Shigsy Damn, and I thought this thread would be about Splinter Cell
Bloody hell, you beat me to it. ----------------------------
Originally by: Splagada SEED ME DADDY
Down with alts! One character per account per IP! |
Wild Rho
Amarr Sharks With Frickin' Laser Beams
|
Posted - 2007.02.13 18:54:00 -
[24]
Edited by: Wild Rho on 13/02/2007 18:52:02 You lot make me feel really dumb.
Edit: BUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUUHHHHHHHHHHHHHHHH
I have the body of a supermodel. I just can't remember where I left it.
|
Frezik
Celtic Anarchy Anarchy Empire
|
Posted - 2007.02.13 22:06:00 -
[25]
Originally by: Ralus ok since theres no actual question in the op heres one for you all, what is the fundamental cause of chaos? Personally I believe that chaos stems from the Heisenberg uncertainty principle, but where else does chaos stem from.
Can chaos ever be eliminated, would it at some point be possible to calcualte all variables in the universe, and with this calculation would it be possible to see the future?
hmmmm I dunno,
Around the early 1900's, a theory called "Scientific Determinism" was rather popular, with Einstein as one of its main supporters. This is the theory that if you knew all the variables, and know how all the variables fit into the equation, you could predict everything that would ever happen or had happened in the universe. Modeling that system might be computationally infeasible, but it would at least be theoretically possible.
Scientific Determinism was killed off by Heisenberg Uncertainty. Einstein hated it, even though Relativity was a key advancement towards Quantum Physics. This is the origin of his phrase "God does not play dice". Einstein supported a "hidden variables" interpretation of Quantum Physics, which stated that Uncertainty is wrong--we just don't know all the variables in the equations yet.
Current research indicates that the hidden variable interpretation is simply wrong and Uncertainty is upheld. God does play dice. However, all other interpretations thus far presented have been either unable to show reproducible results or have outright failed under experimental scrutiny.
Stepping away from the strange world of Quantum Physics, Chaos Theory demonstrates that small effects can build exponentially over time. Previously, it was largely assumed that such small effects would be canceled out by larger effects.
In principle, calculating the weather and calculating the orbit of planets should work the same. You take the current state of the system, build a mathematical model of functions, feed it into a computer, and get the future state of the system out. However, we can accurately predict the positions of planets for centuries in the future or past while we have a hard time figuring out the weather a week from now.
The reason is that there are just too many variables to account for in the weather model. The movements of individual air particles hit hard because there are so many of them. While there are some small particles floating in space, there aren't enough of them to have any measurable effect for several thousand or even millions of years.
|
Rusgar
The Syndicate Inc INVICTUS.
|
Posted - 2007.02.14 07:58:00 -
[26]
tldr
|
Scoundrelus's Alt
|
Posted - 2007.02.14 18:55:00 -
[27]
Why didn't you just link people to: Where you copied and pasted it from
|
Frezik
Celtic Anarchy Anarchy Empire
|
Posted - 2007.02.14 21:19:00 -
[28]
Originally by: Scoundrelus's Alt Why didn't you just link people to: Where you copied and pasted it from
But then, how would you be able to convince people that you're smart?
|
Derovius Vaden
|
Posted - 2007.02.14 23:23:00 -
[29]
Originally by: Frezik
Originally by: Scoundrelus's Alt Why didn't you just link people to: Where you copied and pasted it from
But then, how would you be able to convince people that you're smart?
Talk about your biblicially large and well-used e-*****?
|
Constantine Arcanum
IMPERIAL SENATE Pure.
|
Posted - 2007.02.14 23:34:00 -
[30]
I'm thick
|
|
|
|
|
Pages: [1] 2 :: one page |
First page | Previous page | Next page | Last page |