January 02, 2009

Chaos Theory

Today I found a well-written paper titled Introduction to Chaos Theory, which explained why people use the analogy of a butterfly's wing flapping to explain the unpredictability of weather. When Edward Lorenz went back to rerun a test, the difference in the starting point for his second sequence was less than one thousandths off the original starting point for the first sequence, but the outcome was vastly different. He wondered why.

When he came back an hour later, the sequence had evolved differently. Instead of the same pattern as before, it diverged from the pattern, ending up wildly different from the original. (See figure 1.) Eventually he figured out what happened. The computer stored the numbers to six decimal places in its memory. To save paper, he only had it print out three decimal places. In the original sequence, the number was .506127, and he had only typed the first three digits, .506.

By all conventional ideas of the time, it should have worked. He should have gotten a sequence very close to the original sequence. A scientist considers himself lucky if he can get measurements with accuracy to three decimal places. Surely the fourth and fifth, impossible to measure using reasonable methods, can't have a huge effect on the outcome of the experiment. Lorenz proved this idea wrong.

This effect came to be known as the butterfly effect. The amount of difference in the starting points of the two curves is so small that it is comparable to a butterfly flapping its wings.

Tiny changes in the initial state can create huge differences in the outcomes for many natural systems.

So much of our world can be described using the chaos theory and it wasn't even discovered until the 1960s and not really appreciated until later.

Posted by Mary at January 2, 2009 12:26 AM | | Technorati links |