Saturday, July 12, 2008

Characteristics of Chaotic Systems

Disclaimer – this is not meant to be a definitive article on chaos. This is merely to set in order the definition of chaos I've gathered from reading James Gleick's book.

The Lorenz attractor. Three simple equations produce a path is space that never repeats or intersects itself. First discovered by Edward Lorenz, a meteorologist who first studied such systems, this figure has become the universal symbol for chaos.


Traditional Thinking
The traditional thinking in science favors linear systems. The science we study in schools and colleges is essentially linear science. Linear means that the equations we use to describe phenomenon don't have powers of variables. For example, if some quantity depends on the temperature T of the system, we like to express this law in the form of some function of T but not T squared or T cubed.

This is because linear equations are easy to understand, easy to solve and easy to teach. They give our graduate students problems that can be clearly stated and solved and hence ideal to earn their PhD degrees in.

Thus, non linearity is usually the last chapter in any science textbook and even then, the attempted solutions to the problems are to reduce the non-linear equation to a linear one using some clever trick.

This hegemony of linearity in science has led us to believe that nature itself if linear and behaves in certain ways. There are two significant characteristics of linear systems that people start taking for granted.

1. Small changes in initial conditions mean small changes in the final result. If you were traveling 40 kms at the speed of 10 km per hour, you will reach your destination in 4 hours. If you travel at 11 km per hour, even without doing the math, we say that you'll reach in a little less than 4 hours. If you travel at 9 km per hour you will reach in a little more than 4 hours. No one in his right mind will claim that you might take a 100 hours to reach, if you vary your speed by 1 or 2 km per hour.

2. Systems tend to be monotonic or periodic. Linear systems are either monotonically increasing or periodic. If you heat a cup of coffee and leave it be, it slowly cools down to the room temperature and that is all there is to it. If you swing a pendulum, it oscillates at a steady, periodic rate. Scientists are habitual of looking for monotonic or periodic laws.

3. Simple governing equations lead to simple behavior. If you system is complex, the underlying math should also be so.

The Reality
The reality is that nature is non linear. In fact, even the term non linear is funny because the bulk of natural laws are non linear. We study a small set of natural laws, the linear ones, make them canon and then use the negation to define the most wide ranging phenomenon. (This reminds me of a discussion at Alfaaz. Fiction is only a small part of written texts. However we define this bulk of written stuff as non fiction.)

Thus, some of our 'intuition' about nature doesn't really hold.

1. Small changes in initial conditions can have very large effects on the final outcome. Everyone's heard of the butterfly effect. That's what I mean.

2. Nature can be non-periodic and non-repetitive. Take weather for example. Meteorologists keep on looking for repetitions in the weather. However, the weather each day is something we have never seen before. There are infinite combinations possible.

3. Simple equations can lead to complex behaviors. Chaos theorists have shown that even simple non-linear equations can show behavior that is infinitely varied and hence infinitely complex.

The Consequences
Thus, chaos theorists have shown that while non linear behavior may be complex, the underlying mathematics may be simple and by consequence, simple mathematics may underly complex behaviors such as weather and turbulence. The key idea is to get rid of this impartial affection for linearity and start looking at nature in another way. It is a paradigm shift, in Kuhn's words.

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