Tuesday, February 15, 2011

Chaos Theory

Chaos. The word itself evokes feelings of disorder, of things that are not orderly arranged, a jumbled up room full of stuff, stripes of paint seemingly without reason on a canvas, the results of the actions of satan, uninterpretable perceptions, everything that cannot be described with a simple description or looks untidy. The scientific meaning of chaos however is slightly different. It's not so much about being tidy, but about losing predictability and periodicity. The interesting thing is that from a scientific perspective mos
t, if not all, things around us have chaotic properties and are in one sense or another chaotically interfering with one another. Chaos theory researches the effect of sensitivity to initial conditions, which is when a very slight error in a volume, speed or other characteristic may lead to profound differences in the outcome of results over a longer period of time. Lorenz first discovered that certain systems are highly sensitive to initial conditions when he tried to predict the weather. He ran the simulation once and then printed results. At some point he wanted to verify his findings by running the algorithm again and to his astonishment, even after he verified that the numbers were the same, the outcomes were significantly different. The only difference was that the interpretation of the numbers by the computer were slightly truncated somewhere at the 1000th decimal number.

Normal periodic and linear systems do not typically amplify these errors, but just show a similar, linear difference in the outcome. Basically, your result is slightly off. What Lorenz found here is that after some point in time, the system started behaving completely differently from the initial run of the process. Sensitivity to initial conditions is what he discovered and he came up with a strong analogy for the phenomenon; the "Butterfly Effect". The analogy is that sensitivity to initial conditions could mean that a butterfly flapping its wings in Brazil could in theory cause a tornado in Texas to occur.

Other interesting discoveries were made by the russian Belousov, who mixed up a couple of chemicals together and discovered that it changed color to yellow, but then back again. Not only that, it was actually oscillating between clear and yellow. This phenomen had never been witnessed and at that time was seen as impossible. For that reason, his paper that he submitted to a journal was straight-out rejected. Even after a revisal nobody wanted to publish the results on the basis of lack of evidence. It was only years later after informal circulation in Moscow that eventually the results were picked up by Western scientists, who improved the experiments further and demonstrated that a petri-dish with a certain solution of chemicals may eventually demonstrate autonomous oscillation, autonomous meaning without induction of external disturbances. Thus, a system which switches between states in a temporal manner. The actual patterns that occur in such dishes *may* look like the following. The interesting bit is that this is dependent of..... the exact initial conditions!

As for the pattern itself... there's another great scientist called Benoit Mandelbrot, who's not a typical mathematician in the sense that he knew algebra very well :). He studied in Paris in the 2nd world war, so naturally the study was frequently interrupted. Also, he wasn't always that much interested in doing math tables and all that, but instead he had a great visual attention to detail. This made him look at coastlines and mountains and discover recurrences of smaller details in larger ones and come up with the idea of a very simple formula, describing a hugely complex shape overall. He called that a fractal:

The idea is that a very simple formula, z <=> z^2+c, gives rise to the picture above (calculated in the complex plane of course and where the result does not escape to infinity. The figure is self-similar in the sense that one can zoom in on the image and discover that the same shape is in many other different smaller locations at a fraction of the size, but in this case equal to the first one.

The interesting idea here emerges that very simple rules of interaction between elements can produce hugely complex systems at a larger scale. The complexity of the figure and the simplicity of the equation should give you some idea of that power. The relationship between the two has always been quite clear from an intuitive perspective, but reviewing these mathematical details suddenly changes that.

Chaos theory has put the world of Newtonian physics upside down. The idea of being in control of particular phenomena or occurrences just because we are able to predict it (to some extent).

The notions of chaos and order are not necessarily exclusive. In the majority of cases, when scientists mention chaos they do not mean "100% randomness" in their discourse, but they probably refer to: "some chaotic elements involved that deny a straightforward linear solution to the problem". This is because 100% randomness in systems yields no patterns whatsoever, just white noise. Therefore, there is a grey area between the notions of order and chaos and in many cases, when you feed energy into a system that behaves periodical, at some point you'll push it into chaos, where it'll behave unpredictably, but may eventually return to predictability and periodicity again, although that pattern of order may be different from the one you had before. Many systems, given a certain feed of energy, swing back between the two forever. This is what the Lorenz attractor at the top demonstrates, as well as demonstrating how the system is highly dependent on initial conditions (here, interpret this as infinitesimally small differences in the initial condition, the reciprocal of infinitely large).

What is different in mathematics when you compare Newtonian physics with Chaos Theory?
  • The expressions in chaos are very simple, but recursive.
  • Chaos math usually deals with interactions between systems or elements.
  • Newtonian physics require orderly systems to be able to predict what happens.
  • Chaos has its own cycles and may skip from apparent order to chaos and flip between the edge of chaos and back without warning.
  • When you put too much energy into chaotic systems, they become totally unstable and generate totally unpredictable results, leaning towards randomness the more energy you put in.

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