This month, the American Physics Society magazine, Physics Today, published an article about the 50th anniversary of the Lorenz model. At the link, you can read the entire article. In it, experts describe the history of chaos, Lorenz’s discovery of it, and some of the state of the field today, but with a great deal less technical jargon.
50 years ago, Edward Lorenz first captured the mathematical phenomena we now know as chaos, known popularly as the “butterfly effect“. Below is a picture from the Lorenz model exhibiting chaos. The idea of chaos boils down to highly structured behavior that cannot be predicted. No matter how precisely we measure, after some time we cannot know the state of the system. We can say that the system will stay in a certain region of weather; in the picture below, there are definitely places the trajectory does not visit. We observe this with weather models– the forecast is good for a couple of days, so-so for a couple of days after that, and completely inaccurate for any time farther in the future. Analogously, we can say that it will not be -100 C tomorrow. Appropriately, Lorenz’s discovery of chaos came about as he tried to develop a model for the weather. Chaos is all around us and can be observed in a number of systems.
At this link, you can play with a fun Lorenz model java applet. The trick with the applet is choosing the right parameters. Try setting the “spread” to 0.1, the “variation” to 20, the “number of series” to 2, and the “refresh period” to 100. Then push the button “reset the parameters” and “restart”. This will start 2 trajectories in the Lorenz model that differ by only 0.1. You will quickly see the two paths diverge and become completely unrelated. If you reduce the “spread” to 0.01, the same thing will happen, though it will take longer. As long as the spread is more than 0, the two paths will eventually diverge.
This is why we cannot predict the state of a chaotic system, because our ability to measure the state of the system is inevitably flawed. If we could measure the state of the weather to 99.99999% accuracy, that 0.00001% inaccuracy would eventually lead to divergence. And you can imagine that getting 99.99999% accuracy is much harder and more expensive than 99.9% accuracy.