The Lorenz attractor

The Lorenz attractor is a strange attractor living in 3D space that relates three parameters arising in fluid dynamics. It is one of the Chaos theory's most iconic images and illustrates the phenomenon now known as the Butterfly effect or (more technically) sensitive dependence on initial conditions.

Let's try to get a grip on what the Lorenz attractor really is. It's named after the meteorologist Edward Lorenz who was trying to understand why weather prediction is inherently so difficult to predict very far out. To get to the crux of the issue, he did what many scientists do - he drastically simplified. In many ways, the atmosphere behaves like a fluid so he studied the simplest fluid with interesting dynamics he could think of - a pot of slowly heating water. As the water is heateded slowly, convection rolls begin. The situation might look something like so.

According to Lorenz, there are three quantities that characterize the state of the fluid:

I think the first two are fairly easy to understand but the last is more subtle, as the temperature gradient is simply indicated by the color in the picture. It's easier to see this effect in a simple graph.

When \(z=0\) so that there is no distortion from linearity of the temperature gradient, the convection picture looks like so:

Lorenz showed that the quantities \(x\), \(y\), and \(z\) are related via a set of differential equations:

\[ \begin{align} x' &= \sigma(y-x) \\ y' &= x(\rho-z)-y \\ z' &= xy-\beta z \end{align} \]

Note that \(\sigma\), \(\rho\), and \(\beta\) are parameters; they depend on conditions like the fluid, the heat input, the size of the pot, etc, but they are assumed constant throughout one experiment. In the simulations on this page, \(\sigma=10\), \(\rho = 28\), and \(\beta=8/3\).

Lorenz modeled this system on his computer and studied the results. It's impossible to describe the fortuitous nature of his discovery better than he did in his book The Essence of Chaos:

At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last decimal place. . . . The numbers I had typed in were not the exact original numbers, but were the rounded off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution. In today’s terminology, there was chaos.

Today, we can simulate this discovery very easily in Javascript. The little demo below generates a solution to the Lorenz equations and animates the result. Just hit the start button to begin the simulation. The blue point should start moving leaving a trace in its path. After a few moments, hit the round button. The blue point will stop and be replaced by two new points - a green point and a red point. The blue point is really just the continuation of the original blue point; it's been restarted but maintains the exact conditions that the previous blue point had when it stopped. The red point is another restart of the original blue point but rounds the state of the original blue point to the thousandths place just as Lorenz did. You should notice that the two new points are right on top of on another to start with; you can hit the stop button to take a closer look, if you like. After a time, though, they diverge from one another - first just a little but, eventually, the two have apparently completely independent paths.

Now, you might object that this model is way too simple. Of course, that objection is completely correct; the earth's atmosphere is much larger and more complicated than a little pot of water. But this objection gets to the very nature of chaos. Chaos does not arise from sheer size or complexity; rather, it can be a fundamental aspect of very simple systems.