My Options Journal

15 October: Group Presentation and Pigeon Hole Principle

CHAOS THEORY
Zhang Wei (406), Elizabeth Liew (415), Lau Hui Ning (415)

"by small/ Accomplishing great things, by things deem’d weak/ Subverting worldly strong."
- Paradise Lost, Milton

HOW IT CAME ABOUT
- Meteorologist Edward Lorenz
- Weather prediction using computer programme
- Wanted to see a particular sequence but started from the middle to save time
- Entered .506 (3 dp) into the equation
- An hour later, the sequence that emerged was different from the one he expected
- Computer stored no.s to 6 dp (.506127)

- The slight difference in precision and accuracy in the initial conditions had a huge impact on the long-term outcome
- Produced vastly different results
- BUTTERFLY EFFECT
- Amount of difference in the starting point of the 2 curves is comparable to the flapping of a butterfly’s wings

CHAOS THEORY – What it is
- A small occurrence can produce unpredictable and drastic final results by triggering a series of increasingly significant events
- 2 vs 2.0000000001 -> final results entirely different
- Simulation model
- Owing to this extreme sensitivity, behaviour of chaotic systems appears to be random

The Lorenz Attractor
- Lorenz went further by modeling the location of a particle subject to atmospheric forces
- Obtained a series of equations
- When solved numerically, the movement of the particle seemed to be wild and random
- But he found out later that while the momentary behaviour is chaotic, a general pattern surfaced -->Lorenz Attractor
- Though the particle appears to move randomly, it actually obeys a deeper order

Summary: Chaotic systems
- Deterministic
- Very sensitive to initial conditions --> unpredictability
- Appear to be disordered and random, but beneath them all there is a sense of order and pattern

Applications: WEATHER FORECASTING
- Possible to predict the weather a week or a month ahead?
- Weather is the total behaviour of all the molecules in the earth’s atmosphere.
- Many variables: Air pressure, wind speed & direction, humidity, temperature etc
- We can enter the variables into an equation and calculate their values one second later. Then using this answer, calculate the weather conditions in the next second. (and so on…)

BUT…

- Remember Lorenz’s experiment
- Difference of 0.001 was enough to arise significant changes in output
- The location of one particle cannot be accurately pin-pointed, let alone a combination of particles and their subsequent evolution.
- Just like how Lorenz entered a 3 dp number, we are unable to measure every single variable accurately enough to avoid the effect of chaos.
- Hence, long range weather forecasting is practically impossible.

Chaos in the Solar System
- Scientists had once thought that planets are fixed in their orbits in a completely ordered, predictable, unchanging clockwork.
- But their equations never accurately predicted the movement in planets
- Chaos in solar system --> abrupt change in patterns of the orbital motion of planets
- Theoretical problem: Each body in the system is subjected to the gravitational force of (n) bodies --> indefinite (n-body problem)
- Hence, long-term evolution of the system is impossible to predict.

Attractors – Order within disorder

Chaos and Fractals
- Iterations/repetition of chaos results in a kind of fractal order.
- Many aspects of nature display fractal characteristics.

Other examples of Chaos
- Stock Markets
-Migratory patterns of birds
- Evolution of biodiversity
- Spread of vegetation across a continent
- The Human Mind
- Chaos-based graphics created and featured in movies

Reference
http://library.thinkquest.org/3120/
http://www.imho.com/grae/chaos/chaos.html
http://www.abarim-publications.com/ChaosTheoryIntroduction.html
http://library.thinkquest.org/26688/
http://www.zeuscat.com/andrew/chaos/sierpinski.html#FRACTAL
http://webecoist.com/2008/09/07/17-amazing-examples-of-fractals-in-nature/
Wikipedia.org
http://www.fortunecity.com/emachines/e11/86/solarsys.html

Conclusion for CHAOS THEORY Presentation
Nature is highly complex.
The only prediction we can make is that Nature is amazingly unpredictable.
Chaos Theory has managed to somewhat capture the beauty of the unpredictable and display it in the most awesome patterns.

There is beauty not only in complexity and simplicity; there's beauty in order and disorder too.

Chaotic systems may appear to be disordered, but the deeper sense of order and pattern beneath them is what makes them beautiful.

Our lives would probably be much more boring if there wasn't chaos; it is the unpredictable nature of things that makes the world interesting. At the same time, be thankful that we're still "safe" and "in control", for however disorder and chaotic things may seem, there is still an inherent sense of order, and we are able to capture and represent it.

I think this applies not only in the scientific/mathematical sense, the theory can also be brought into life -- when things appear to be chaotic, fret not, (they're meant to be like that and) we can always identify or seek some kind of order in it. Once one is willing to take charge and take a closer look at things, one will soon realise that many things aren't as bad as they might seem.

Also, chaos theory states that a seemingly insignificant small initial change can result in drastic differences in the outcomes. This is nevertheless true in daily life as well. Many disasters happen not because of a great preceding event, the ultimate causual factor can be traced down to a minor overlooking on our part, a moment of laziness, or a small letdown on our integrity and/or principles. Conversely, success is probably the cumulative effect of our attention to minute details.

“In nature, nothing is perfect and everything is perfect. Trees can be contorted, bent in weird ways, and they're still beautiful.”
Alice Walker quotes (American writer, b.1944)
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Pigeon Hole Principle

Notes:
Beauty of Mathematics: Finding order in complexity (parallels chaos theory?)
Using simple intuitive methods to solve difficult problems
Finding the "unchanged" in "changed"
PHP: If kn+1 pigeons are put into n holes, then one hole must contain at least k+1 pigeons.