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The Chaos Theory And Applications.

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Thinker13
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Posted 06/11/09 - 8:18 AM:
Subject: The Chaos Theory And Applications.
Hi,this thread an extension of our previous conversations.

http://www.thecouchforum.com/comments.php?id=1260&page=1

Thinker13 wrote:

Our discussion has still not entertained 'Chaos Theory'.It is interesting to note that semantics causes so many differences zen

libertygrl wrote:

it makes for interesting conversation, i find smiling face and hopefully will facilitate future communication the more we take time to explain our perspectives.

what are your thoughts on chaos theory?


Thinker13 wrote:
I would try posting them soon


It is not very soon,still,we could engage in a conversation.:



What is chaos theory?

Formally, chaos theory is defined as the study of complex nonlinear dynamic systems. Complex implies just that, nonlinear implies recursion and higher mathematical algorithms, and dynamic implies nonconstant and nonperiodic. Thus chaos theory is, very generally, the study of forever changing complex systems based on mathematical concepts of recursion, whether in the form of a recursive process or a set of differential equations modeling a physical system.

Chaos has already had a lasting effect on science, yet there is much still left to be discovered. Many scientists believe that twentieth century science will be known for only three theories: relativity, quantum mechanics, and chaos. Aspects of chaos show up everywhere around the world, from the currents of the ocean and the flow of blood through fractal blood vessels to the branches of trees and the effects of turbulence. Chaos has inescapably become part of modern science. As chaos changed from a little-known theory to a full science of its own, it has received widespread publicity. Chaos theory has changed the direction of science: in the eyes of the general public, physics is no longer simply the study of subatomic particles in a billion-dollar particle accelerator, but the study of chaotic systems and how they work.[/quote]

zenSo,Chaos Theory has become a full fledged science,which began as a theory initially.
zenIs it not interesting to note that in spite of being considered as one of the three most important theories of 20th century science(with Quantum mechanics and Relativity),it is still not being taught in universities,except in a few rare ones.




The Myth About The Chaos Theory:


The most commonly held misconception about chaos theory is that chaos theory is about disorder. Nothing could be further from the truth! Chaos theory is not about disorder! It does not disprove determinism or dictate that ordered systems are impossible; it does not invalidate experimental evidence or claim that modelling complex systems is useless. The "chaos" in chaos theory is order--not simply order, but the very ESSENCE of order.
It is true that chaos theory dictates that minor changes can cause huge fluctuations. But one of the central concepts of chaos theory is that while it is impossible to exactly predict the state of a system, it is generally quite possible, even easy, to model the overall behavior of a system. Thus, chaos theory lays emphasis not on the disorder of the system--the inherent unpredictability of a system--but on the order inherent in the system--the universal behavior of similar systems.



zenSo,it is important to make a note that this theory does not establish that disorder prevails,it does not explicate the impossibility of order but rather impact of initial parameters on the output of sensitive systems,while,the behavior of system can be said to have an essential,regular model no matter what are degrees of randomness.



An Example-Lorenz Attractors:

Thus, it is incorrect to say that chaos theory is about disorder. To take an example, consider Lorenz's Attractor. The Lorenz Attractor is based on three differential equations, three constants, and three initial conditions. The attractor represents the behavior of gas at any given time, and its condition at any given time depends upon its condition at a previous time. If the initial conditions are changed by even a tiny amount, say as tiny as the inverse of Avogadro's number (a heinously small number with an order of 1E-24), checking the attractor at a later time will yield numbers totally different. This is because small differences will propagate themselves recursively until numbers are entirely dissimilar to the original system with the original initial conditions.
However, the plot of the attractor will look very much the same.
Both systems will have totally different values at any given time, and yet the plot of the attractor--the overall behavior of the system--will be the same


http://library.thinkquest.org/3120/


Interesting History(Lorenz)

One day in 1961, he wanted to see a particular sequence again. To save time, he started in the middle of the sequence, instead of the beginning. He entered the number off his printout and left to let it run. When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.




The Butterfly Effect-sensitive dependence on initial variables


The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does. (Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)


This phenomenon, common to chaos theory, is also known as sensitive dependence on initial conditions. Just a small change in the initial conditions can drastically change the long-term behavior of a system. Such a small amount of difference in a measurement might be considered experimental noise, background noise, or an inaccuracy of the equipment. Such things are impossible to avoid in even the most isolated lab. With a starting number of 2, the final result can be entirely different from the same system with a starting value of 2.000001. It is simply impossible to achieve this level of accuracy - just try and measure something to the nearest millionth of an inch! From this idea, Lorenz stated that it is impossible to predict the weather accurately. However, this discovery led Lorenz on to other aspects of what eventually cam to be known as chaos theory..


The equations for this system also seemed to give rise to entirely random behavior. However, when he graphed it, a surprising thing happened. The output always stayed on a curve, a double spiral. There were only two kinds of order previously known: a steady state, in which the variables never change, and periodic behavior, in which the system goes into a loop, repeating itself indefinitely. Lorenz's equations are definitely ordered - they always followed a spiral. They never settled down to a single point, but since they never repeated the same thing, they weren't periodic either. He called the image he got when he graphed the equations the Lorenz attractor.



http://library.thinkquest.org/3120/


A few very interesting cases:


Chaotic Washing Machine:


A chaotic washing machine? This is just what Goldstar Co. created back in 1993. It was the world's first consumer product to exploit "chaos theory", which holds that there are identifiable and predictable movements in nonlinear systems. This washing machine is supposed to produce cleaner and less tangled clothes. The key to the chaotic motion is a small pulsator (which stirs the water) that rises and falls randomly as the main pulsator rotates. When released to the world market, it was expected to push Goldstar's share of the annual 1.5-million-unit washing machine market to 40% in 1993, compared to 39% for Samsung and 21% for Daewoo (Goldstar's major competitors). However, marketing is fierce in South Korea and Daewoo argues that Goldstar "was not the first" to commercialize chaos theory. Daewoo also built a "bubble machine" in 1990 which also used chaos theory that was the result of "fuzzy logic circuits." Fuzzy circuits make choices between zero and one, and between true and false. These factors control the amount of bubbles, the turbulence of the machine, and even the wobble of the machine. It is clear that chaos theory has not gone unnoticed in today's consumer world market



Chaos And The Stock Market:


According to respected authorities, stock markets are non-linear, dynamic systems. Chaos theory is the mathematics of studying such non-linear, dynamic systems. Chaos analysis has determined that market prices are highly random, but with a trend. The amount of the trend varies from market to market and from time frame to time frame. A concept involved in chaotic systems is fractals. Fractals are objects which are "self-similar" in the sense that the individual parts are related to the whole. A popular example of this is a tree. While the branches get smaller and smaller, each is similar in structure to the larger branches and the tree as a whole. Similarly, in market price action, as you look at monthly, weekly, daily, and intra day bar charts, the structure has a similar appearance. Just as with natural objects, as you move in closer and closer, you see more and more detail. Another characteristic of chaotic markets is called "sensitive dependence on initial conditions." This is what makes dynamic market systems so difficult to predict. Because we cannot accurately describe the current situation band because errors in the description are hard to find due to the system's overall complexity, accurate predictions become impossible. Even if we could predict tomorrow's stock market change exactly (which we can't), we would still have zero accuracy trying to predict only twenty days ahead.



A number of thoughtful traders and experts have suggested that those trading with intra day data such as five-minute bar charts are trading random noise and thus wasting their time. Over time, they are doomed to failure by the costs of trading. At the same time these experts say that longer-term price action is not random. Traders can succeed trading from daily or weekly charts if they follow trends. The question naturally arises how can short-term data be random and longer-term data not be in the same market? If short-term (random) data accumulates to form long-term data, wouldn't that also have to be random? As it turns out, such a paradox can exist. A system can be random in the short-term and deterministic in the long term.




A very interesting case:


Is The Coast Line Of Great Britain Infinite?http://library.thinkquest.org/3120/


A long, long time ago, fractal god Benoit Mandelbrot posed a simple question: How long is the coastline of Britain? His mathematical colleagues were miffed, to say the least, at such an annoying waste of their time on such insignifigant problems. They told him to look it up.

Of course, Madelbrot had a reason for his peculiar question. Quite an interesting reason. Look up the coastline of Britain yourself, in some encyclopedia. Whatever figure you get, it is wrong. Quite simply, the coastline of Briutain is infinite.

You protest that this is impossible. Well, consider this. Consider looking at Britain on a very large-scale map. Draw the simplest two-dimensional shape possible, a triangle, which circumscribes Britain as closely as possible. The perimeter of this shape approximates the perimeter of Britain.

However, this area is of course highly inaccurate. Increasing the amount of vertices of the shape going around the coastline, and the area will become closer. The more vertices there are, the closer the circumscribing line will be able to conform to the dips and the protrusions of Britain's rugged coast.

There is one problem, however. Each time the number of vertices increases, the perimeter increases. It must increase, because of the triangle inequality. Moreover, the number of vertices never reaches a maximum. There is no point at which one can say that a shape defines the coastline of Britain. After all, exactly circumscribing the coast of Britain would entail encircling every rock, every tide pool, every pebble which happens to lie on the edge of Britain.

Thus, the coastline of Britian is infinite.



Why do long range predictions fail so frequently and we laugh on them?


LONG RANGE FORECASTING


Weather prediction is part of every new service. Three, maybe four days ahead. And sometimes they're right. But what about a week ahead, or a month? Naturally, Chaos scientists had a go at this problem. Their conclusions are not going to be much more use if you want to choose a fine day for your birthday party in three months!

There are many variables associated with the weather: temperature, air pressure, wind speed, wind direction, humidity and many more. The equations which control the weather involve all of these variables.

You can accurately put all these variables in an equation and calculate, with some degree of certainty the value of all the variables one second hence. These answers can be fed back in, and the values for the next second can be calculated one second hence. These answers can be fed back in, and the values for the next second can be calculated. Leave the poor computer go for long enough to do the iterations and you will know the weather one month later. Or will you?

Edward Lorenz tried this. Lorenz decided to run the program for longer. To do so he entered the values for halfway through the run and set the machine off again. But, the results soon deviated from the previous run. Lorenz found the reason was that he had put the values in accurate to three decimal places. The computer had calculated to six places. So a difference of one in a thousand was enough to change the output significantly. We can't measure the variables accurately enough to avoid the effects of chaos.

For ten years, Lorenz's paper on this result was ignored, despite Lorenz being aware that this was a crucial discovery. When he plotted the three key variables in three dimensional space, he gained a plot that came to be known as the Lorenz Attractor.




Do This Experiment:

GAS CHAMBER


Construct a simple system: take a box, a simple solid rectangular solid. Within this box, place a gaseous substance. Heat the box, sit back, and watch. What happens to the gas? Everyone knows that warm gases rise while cooler gases sink; and initially, the portions of the gas closest to the walls of the box will become heated and rise. At certain temperatures, the gas will begin to form cylindrical rolls spaced like jelly rolls lying lengthwise in the box. On one side of the box, the gas rises, and on the other, it sinks; the rising gases move to one side and carry warmer gases up with them; as the gas cools, it falls on the other side of the box. With a regularly applied temperature, a smooth box interior, and a system completely closed with regards to the gas itself, it might be expected that the circular motion of the moving gas should be regular and predictable. Nature, however, is neither regular nor predictable. It turns out that the motion of the gas is chaotic. The rolls do not simply roll around and around in one direction like a steam-roller; they roll for a while in one direction, and then stop and reverse directions. Then, seemingly at random, the gas reverses direction again; these changes continue at unpredictable times, at unpredictable speeds.


For Moviegoers:

Following movies are knowingly showing this effect,though there are many other movies,where you can speculate based on thesezen:

zenJurassic Park
zenBack To The Future
zenThe Butterfly Effect
zenChaos Theory



A few interesting chaos FAQs:

http://www.vismath.org/faq/chaosfaq.html

Proposed questions for our discussion:

zenWhat do you find interesting about this theory?
zenDo you think that your life may have been radically different had there been some differences in your upbringing,marriage and socializing?
zenAre you willing to explore it more? Do you think that this theory helps you understand this universe better?




Sorry for the length of this post,still it makes for an interesting study.smiling face




Thank You

Edited by Thinker13 on 06/11/09 - 9:25 AM
libertygrl
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Posted 06/18/09 - 3:59 PM:

hi thinker,

thank you for taking pains to include so much interesting information in this topic, i think it does make for interesting study. conceiving of the universe in terms of fractal patterns has for many years been very useful and resonant for me. it was, in fact, michael crichton's "jurassic park" which first piqued my interest in chaos theory.

Nature, however, is neither regular nor predictable. It turns out that the motion of the gas is chaotic. The rolls do not simply roll around and around in one direction like a steam-roller; they roll for a while in one direction, and then stop and reverse directions. Then, seemingly at random, the gas reverses direction again; these changes continue at unpredictable times, at unpredictable speeds.

i'm curious to know whether you still feel that humanity is capable of evolving to the point where it is possible to know exactly when and where a time traveler from the future may have appeared in our distant past, using the analysis of frequencies. it seems to me that chaos theory has demonstrated that such a task could easily be impossible, due to the sensitive dependence on initial conditions.

in this case, the initial conditions would require not only an absolutely accurate reading of current frequencies, but also an absolutely accurate formulation to evaluate iterations of frequencies moving backward in time, in order to identify disturbances in the distant past.

i think scientists are faced with similar challenges today in analyzing large bodies of data collected by means of scientific studies. these are useful in identifying certain trends, but it is rare that any absolute determinations may be derived.

what are your thoughts?

smiling facelib
Thinker13
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Posted 06/23/09 - 5:39 AM:

libertygrl wrote:
thank you for taking pains to include so much interesting information in this topic, i think it does make for interesting study.


You are welcome,lib.zen

libertygrl wrote:
i'm curious to know whether you still feel that humanity is capable of evolving to the point where it is possible to know exactly when and where a time traveler from the future may have appeared in our distant past, using the analysis of frequencies. it seems to me that chaos theory has demonstrated that such a task could easily be impossible, due to the sensitive dependence on initial conditions.


As far as I recall,the moot point was not "possibility of detecting,exactly when and where time traveler from the future may have appeared" but rather 'evidence for existence of time travel'. Chaos theory does not reject 'patterns' but suggests extremely different behaviors depending on slight changes in input of a sensitive system,so there always remains a 'system behavior'.

lib wrote:
in this case, the initial conditions would require not only an absolutely accurate reading of current frequencies, but also an absolutely accurate formulation to evaluate iterations of frequencies moving backward in time, in order to identify disturbances in the distant past.


You are correct lib,still,detecting sounds of ancient times(for example,chants of Bhagvad Gita etc) and radiations/signals from aliens is even more complex a job,yet they have been talked there for some time now. Many a things are in their infancy,methinks.



smiling face Thank You smiling face
Panguix
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Posted 06/27/09 - 12:23 AM:

To divert slightly,

I am told by those who understand the topic better than I do that the entropy of the universe is increasing as a whole, rather than decreasing. This, I am told, is why a coffee cup smashes into a myriad jagged pieces, and does not reform itself after, as if someone filmed the event and played it in reverse. My studies have torn me in several directions, and I spend my time thinking about humanity as a collage of atoms and universal forces, a collaboration of molecules, a network of people, personalities, or thoughts.
In a thread of chaos, I wonder how our cultural phenomenon and behaviors relate to the every expanding entropy.
It is easy to see how some of our behaviors increase the "chaos" of the universe.
smiling face
But how do we relate chaos to the general functions of our minds?
Thinker13
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Posted 06/27/09 - 1:59 AM:

Hello Panguix,welcome to 'The Couch'smiling face.What is etymology of your name?

Panguix wrote:
In a thread of chaos, I wonder how our cultural phenomenon and behaviors relate to the every expanding entropy.
It is easy to see how some of our behaviors increase the "chaos" of the universe.


Indeed.


Panguix wrote:

But how do we relate chaos to the general functions of our minds?


This is a good question,Panguix,still,it can be fully understood if you read the text written below. Try reading this and still,if you feel difficulty in grasping it,tell me,we shall discuss it. It seems to me that our minds(dependent on organaism,brain etc) aren't isolated systems but rather in contact with various open systems,therefore,in spite of the local decrease in entropy,the total entropy of universe remains positive.smiling face

Applications to living systems
The second law of thermodynamics has been proven mathematically for thermodynamic systems, where entropy is defined in terms of heat divided by the absolute temperature. The second law is often applied to other situations, such as the complexity of life, or orderliness. However it is incorrect to apply the closed-system expression of the second law of thermodynamics to any one sub-system connected by mass-energy flows to another ("open system"). In sciences such as biology and biochemistry the application of thermodynamics is well-established, e.g. biological thermodynamics. The general viewpoint on this subject is summarized well by biological thermodynamicist Donald Haynie; as he states: "Any theory claiming to describe how organisms originate and continue to exist by natural causes must be compatible with the first and second laws of thermodynamics."

This is very different, however, from the claim made by many creationists that evolution violates the second law of thermodynamics. Evidence indicates that biological systems and evolution of those systems conform to the second law, since although biological systems may become more ordered, the net change in entropy for the entire universe is still positive as a result of evolution. Additionally, the process of natural selection responsible for such local increase in order may be mathematically derived from the expression of the second law equation for non-equilibrium connected open systems,arguably making the Theory of Evolution itself an expression of the Second Law.

Furthermore, the second law is only true of closed systems. It is easy to decrease entropy, with an energy source. For example, a refrigerator separates warm and cold air, but only when it is plugged in. Since all biology requires an external energy source, the Sun, there's nothing unusual (thermodynamically) with it growing more complex with time.

Entropy And Life
Complex System
It is occasionally claimed that the second law is incompatible with autonomous self-organisation, or even the coming into existence of complex systems. This is a common creationist argument against evolution.The entry self-organisation explains how this claim is a misconception. In fact, as hot systems cool down in accordance with the second law, it is not unusual for them to undergo spontaneous symmetry breaking, i.e. for structure to spontaneously appear as the temperature drops below a critical threshold. Complex structures, such as BĂ©nard cells, also spontaneously appear where there is a steady flow of energy from a high temperature input source to a low temperature external sink.

Furthermore, a system that energy flows into and out of may decrease its local entropy provided the increase of the entropy to its surrounding that this process causes is greater than or equal to the local decrease in entropy. A good example of this is crystallization. As a liquid cools, crystals begin to form inside it. While these crystals are more ordered than the liquid they originated from, in order for them to form they must release a great deal of heat, known as the latent heat of fusion. This heat flows out of the system and increases the entropy of its surroundings to a greater extent than the decrease of energy that the liquid undergoes in the formation of crystals.

An interesting situation to consider is that of a supercooled liquid perfectly isolated thermodynamically, into which a grain of dust is dropped. Here even though the system cannot export energy to its surroundings, it will still crystallize. Now however the release of latent heat will contribute to raising its own temperature. If this release of heat causes the temperature to reach the melting point before it has fully crystallized, then it shall remain a mixture of liquid and solid; if not, then it will be a solid at a significantly higher temperature than it previously was as a liquid. In both cases entropy from its disordered structure is converted into entropy of disordered motion.


Thank You
libertygrl
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Posted 06/28/09 - 9:31 AM:

interesting excerpts thinker thumb up

hi panquix, welcome smiling face

my thoughts are that entropy is actually a form of self-organization, that chaos, while appearing to be disorderly, is actually the most evenly distributed form of order toward which things can tend (or "aspire", if you will).
Thinker13
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Posted 06/28/09 - 2:11 PM:

libertygrl wrote:
interesting excerpts thinker thumb up


Thank you,lib.smiling face

lib wrote:
hi panquix, welcome smiling face


Gathered that his name is 'panguix' lib.wink



Thank You
libertygrl
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Posted 06/28/09 - 3:55 PM:

Thinker13 wrote:
Gathered that his name is 'panguix' lib.wink

Thank You

ah, i see. thanks for catching that. i'm curious how did you gather that panguix is male? smiling face
Thinker13
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Posted 06/29/09 - 11:17 AM:

libertygrl wrote:
ah, i see. thanks for catching that.


Welcome.

libertygrl wrote:
i'm curious how did you gather that panguix is male? smiling face


Did not. I use 'his' when in doubt,unlike Monk2400(who uses 'hesh' which is,a good practice,methinks).


Thank You
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