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Chaos Theory and Fractals

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Chaos Theory and Fractals

by Aishwarya » Fri Nov 15, 2002 7:06 pm

Why is it that seemingly similar atmospheric conditions result one day in nothing worse than squalls and showers and on another day in a full-blooded hurricane? What determines the pattern of turbulence in a stream of flowing water and what causes terrible fluctuations in an economy? Chaos theory is the branch of mathematics that attempts to deal with this apparent randomness in nature.<br><br>
The equations that underlie chaos theory are complex and have some bizarre properties. If you take two powerful computers made by different manufacturers, feed them both with identical figures say to 10 decimal places and then ask them to solve a \'chaotic\' equation, they will produce different results - not slightly different, but entirely different. The minute discrepancies in the way they handle calculations build up to produce massive differences in the results.<br><br>
Chaos theory might seem to sound the death knell for any attempt to make sense of the \'random\' and the \'unpredictable\'. But there is \'order\' in chaos. Chaotic equations produce mysterious results, known as \'FRACTALS\'. Fractals in fact are the shape of chaos, and very strange shapes some turn out to be. Most remarkably, they look very much the same at any level of magnification. Examine a fractal shape and it turns out to be made up of numerous smaller versions and so on, ad infinitum.
Fractals are more familiar than people suspect. Computers can generate convincing fractal landscapes by endlessly repetitive manipulation of a single initial set of figures. There is pattern and order, as well as fascination and beauty, in chaos.
Aishwarya
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Chaos Theory and Fractals

by Guest » Sun Nov 17, 2002 6:34 pm

Hey A, it\'s a wonderful topic.
Guest
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Chaos Theory and Fractals

by Gulmohar Hyderabadi » Mon Nov 18, 2002 6:07 am

In Mike Leigh\'s film: Nuts in May, a pair of campers named Keith and Candice Marie discover how a kind of modern-day vitalism colors our preconceptions about complex growth patterns. They take a trip to a local quarry to search for fossils in the ancient Purbeck limestone of Dorset. At the prompting of a quarryman, the unsusupectin couple find a delicate, plant like pattern traced out in the stone. \'Look at that - seaweed!\' exclaims Keith. \'Yar, well, \'tis not seaweed, see\' drawls the quarryman, adding that it is manganese oxide - a mineral - they are looking at. Keith is not convinced, \'It looks like a living organism to me.\' \'Yar,\' the quarryman rejoins, \'most people think that.\' Why did Keith and Candice jump to the wrong conclusion? (Nature 353, 409). There is pattern and order in chaos - sounds oxymoronic, though true. There is always an element of subjectivity in perceiving patterns. On the whole let\'s be concerned with patterns and forms in space, ones that we can see and perhaps touch. But of course there are all sorts of patterns - in a time sequence of events, in human behavior and interactions, in stories and myths. The word is a very plastic one. There are certain spatial images that we categorize as patterns - the repeating designs of wallpaper, carpets, chess board etc... This prompts the idea that a pattern might be regarded (not only) as a regularly repeating array of identical units, but also include arrays of units that are similar but not necessarily identical, and which repeat but not necessarily regularly or with a well-defined symmetry eg. ripples of sand at the seaward edge of a beach or in a desert. No two ripples are identical, and they are not positioned at exactly repeating intervals (i.e., periodically in space)-but nonetheless, it isn\'t too hard to persuade ourselves that this might be reasonably called a pattern, as we can recognize within it elementary units (the ripples) that recur again throughout space. The ripples are usually all of more or less the same width; but even this needn\'t be essential to qualify as a pattern. A mountain range has features of all sizes, from little crevasses to huge sweeping valley, but there is still something about the way it looks to us, from out of an aeroplane window, that allows us to see a pattern there. Cohen, J and Stewart, J (1994), The Collapse of Chaos, Penguin, London.
Gulmohar Hyderabadi
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Chaos Theory and Fractals

by EroXenoS » Mon Nov 18, 2002 12:47 pm

I think the best way for a complete layman to know something about Chaos Theory at a beginner\'s level is to read Michael Crichton\'s \"Jurassic Park\" and \"The Lost World\"...a brilliant way to explain things to a person from a non-science stream, and build up interest....i remember reading JP when i was in the 9th standard, and running out to buy a book on Chaos Theory.
EroXenoS
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Chaos Theory and Fractals

by Aishwarya » Mon Nov 18, 2002 2:43 pm

Anyone who has seen The Deathstar in \"The Return Of Jedi\" has seen a fractal shape.This is one the uses of Fractals and the Chaotic equations-generating convincing fractal landscapes by repetitive manipulation of a single set of initial figures, and ofcourse providing some clue to an understanding of natures apparent randomness.
Aishwarya
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Chaos Theory and Fractals

by laskar » Tue Nov 19, 2002 10:02 am

To the uniniated, welcome to the club!
Fractals?? why all these? look to the nature, you have fractals at work. No two snow flakes are identical, likewise
zebra stripes too, if you consider the simpleness of the curved lines on their bodies and to extend it a bit, lines on the fingers too have a pattern, but none are alike. Whatz the connection of Fractals to the Theory of Chaos in these cases???Got any idea?
laskar
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Chaos Theory and Fractals

by hazard » Tue Nov 19, 2002 5:44 pm

The very name \"chaos theory\" seems to contradict reason, in fact it seems somewhat of an oxymoron. The name \"chaos theory\" leads the reader to believe that mathematicians have discovered some new and definitive knowledge about utterly random and incomprehensible phenomena; however, this is not entirely the case. The acceptable definition of chaos theory states, chaos theory is the qualitative study of unstable aperiodic behavior in deterministic nonlinear dynamical systems. A dynamical system may be defined to be a simplified model for the time-varying behavior of an actual system, and aperiodic behavior is simply the behavior that occurs when no variable describing the state of the system undergoes a regular repetition of values. Aperiodic behavior never repeats and it continues to manifest the effects of any small perturbation; hence, any prediction of a future state in a given system that is aperiodic is impossible. Assessing the idea of aperiodic behavior to a relevant example, one may look at human history. History is indeed aperiodic since broad patterns in the rise and fall of civilizations may be sketched; however, no events ever repeat exactly. What is so incredible about chaos theory is that unstable aperiodic behavior can be found in mathematically simply systems. These very simple mathematical systems display behavior so complex and unpredictable that it is acceptable to merit their descriptions as random.
hazard
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Chaos Theory and Fractals

by hazard » Tue Nov 19, 2002 5:46 pm

An interesting question arises from many skeptics concerning why chaos has just recently been noticed. If chaotic systems are so mandatory to our every day life, how come mathematicians have not studied chaos theory earlier? The answer can be given in one word: computers. The calculations involved in studying chaos are repetitive, boring and number in the millions. No human is stupid enough to endure the boredom; however, a computer is always up to the challenge. Computers have always been known for their excellence at mindless repetition; hence, the computer is our telescope when studying chaos. For, without a doubt, one cannot really explore chaos without a computer.

Before advancing into the more precocious and advanced areas of chaos, it is necessary to touch on the basic principle that adequately describes chaos theory, the Butterfly Effect. The Butterfly Effect was vaguely understood centuries ago and is still satisfactorily portrayed in folklore:

\"For want of a nail, the shoe was lost;
For want of a shoe, the horse was lost;
For want of a horse, the rider was lost;
For want of a rider, a message was lost;
For want of a message the battle was lost;
For want of a battle, the kingdom was lost!\"

Small variations in initial conditions result in huge, dynamic transformations in concluding events. That is to say that there was no nail, and, therefore, the kingdom was lost. The graphs of what seem to be identical, dynamic systems appear to diverge as time goes on until all resemblance disappears.
hazard
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Chaos Theory and Fractals

by hazard » Tue Nov 19, 2002 5:52 pm

anybody watched Final destination\'???????it \'s superb movie
hazard
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Chaos Theory and Fractals

by Gulmohar Hyderabadi » Tue Nov 19, 2002 7:39 pm

But although this increase in computer power has provided scientists with perhaps the most important technological tool currently at their disposal, I think it also serves to underline the phenomenal achievements of early researcher into complex systems, who had to rely on their exquisite intuition alone to deduce the essential physics of pattern-forming processes.
There is another factor, little emphasized but equally important, in the recent development of ideas in pattern formation. This is the maturation in the past two decades of a field of theoretical physics that provides much of the framework for understanding the features that accompany spontaneous patterns, such as abrupt, global changes of state and scaling laws. The field is the study of phase transitions and critical phenomena, and it is the center of physics today. All the theories do not collectively constitute a \'theory of patterns\' (same could be said of the mixed bag of concepts that are popularly touted as \'chaos theory\'.

Patterns live on the edge, where small changes can have large effects. This is what is \'the edge of chaos\'. Pattern appears when competing forces banish uniformity but cannot quite induce chaos. It sounds like a dangerous place to be, but it is where we have always lived.
(I am still unfinished, but got to go :-))
Gulmohar Hyderabadi
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Chaos Theory and Fractals

by Aparna » Wed Nov 20, 2002 4:26 pm

Chaos theory and Fractals are a result of man\'s dirge to know the Nature and this effort merely panders to his desire to conquer nature.What he very obviously cant fathom is the randomness of nature,he fails to understand a simple fact that nature or for that matter any creation of God is hard to simulate or understand leave alone put in the form of some mathematical equations.

Now coming to the \'Chaotic equations\',plausible, but do they have any practical use except for ofcourse generating lanscapes... what i want to suggest to Aishwarya and rest of you is that these topics might seem very techno and Hi-fi,but come to think of it what is the use? i could be wrong and i am waiting here to be proven wrong!
Aparna
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Chaos Theory and Fractals

by vivek of themavericks » Thu Nov 21, 2002 4:50 pm

All things are numbers - pythogorus.If i remember correctly, this statement forms the foundation of finding any kind of \'order\' in a chaotic environment. The understanding goes a step further from assuming nature as a linear entity as Newton did. A chaotic nature and universe has intrinsic order, much like hyderabad traffic. I don\'t know more than this
http://www.wiredbeats.com - Download Attitude for Free!

How will it end?
vivek of themavericks
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