The Chaos Theory

The Chaos Theory - One of my favourite theories that can be explained with the simplest of examples. A theory which even the dumbest man on earth can understand.

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)

A student forgets to set his alarm clock for 8 am on the previous night. The next morning, two of his friends wait for him at the bus-stop. They turn up at the lecture thirty minutes late. The professor detains them after class in order to punish them. As a result, the professor gets late while going home. His wife isn’t happy that he’s late as they were supossed to go out. They end up having an argument. The next day, he’s in a very bad mood and decided to take it out on the students. He gives them the toughest of questions in a test and fails each and every one of them.
None of this would have hapenned if the student had set his alarm clock for 8 am.

One random act causes another…. causes another….. causes another….. and then the pattern emerges.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer, a Royal McBee LGP-30, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.
To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

For a dynamical system to be classified as chaotic, 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.
Just a small change in the initial conditions can drastically change the long-term behavior of a system. 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 behaviour.
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 Butterfly’s 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.

Initial conditions…. Strange attractors….. Deviations from the norm…. UTTER CHAOS!!!!