Listen to Ed Lorenz telling us about his concept of chaos in three steps. If a single flap of a butterfly's wings can be instrumental in causing a tornado, so also can all of the previous and subsequent flaps of its wings,

as can the flaps of the wings of millions of other butterflies, not to mention the activities of innumerable much more powerful creatures including our own species. We have already seen that the public understands this first point very well. If a flap of a butterfly's wings can be instrumental in causing a tornado,

it can equally well be instrumental in preventing it. Yes! But the third point is the most important, and this one provides work for scientists. More generally, I am proposing that, over the years, miniscule disturbances neither increase nor decrease the frequency of occurrences of various weather events such as tornadoes.

The most they may do is to modify the sequence in which these events occur. Let's return to Lorenz's model. Remember, each point in space represents a state of the atmosphere.

In some areas the weather is fine, and in others a hurricane blows. Suppose that the area corresponding to a hurricane is in this little ball.

Let's take an initial climate condition and let's observe the trajectory that results. Occasionally a hurricane occurs, that is, when the trajectory enters the ball. Let's measure the proportion of the time between 0 and T when a hurricane rages.

It seems difficult to guess at which specific moments the trajectory enters the ball, but the average time spent inside the ball converges to a limit as the interval of time measured becomes larger and larger.

Here the average is 5.1%.

Now we take a different initial condition and do the same calculation. Well, again the trajectory spends a certain amount of its time inside the ball and the average tends to a limit. Oh, how about that? The average is the same as before, 5.1%.

Let's try it with yet another trajectory. It works!

We obtain the same limit, and yet the trajectories are very different. They are sensitive to initial conditions, to the flaps of the wings of the butterfly. We add another ball, pink maybe, and a third one, green, perhaps corresponding to heat waves or to snow.

As a trajectory unfolds, it crosses the balls in a certain order. To see what happens, let's watch three trajectories, each starting from a different initial condition.

Look, on every trajectory, situations, hurricane, snow, heat wave, alternate incomprehensibly,

and in a different way for each trajectory, impossible to understand. But the proportions of yellow, green, and pink converge rapidly into the same limits. Here are the averages, 5.1%, 14.03%, and 7.4%.

It is as if we were randomly picking balls, yellow, pink, or green, from a bag filled to those proportions. Listen to Lorenz again. Minuscule disturbances neither increase nor decrease the frequency of events such as tornadoes.

The most they may do is to modify the sequence in which these events occur. This is a scientific statement. The purpose of the forecast has changed. We now try to predict averages, statistics, and probabilities.

The idea is that these statistics are, perhaps, insensitive to initial conditions. They are unaffected by the flapping of the wings of Brazilian butterflies. This hypothesis, as yet unproven, of the possible coexistence of a meteorological chaos

where the future movements are sensitive to initial conditions, with a statistical stability insensitive to initial conditions, took a long time to be formulated mathematically. Lorenz was probably a little embarrassed by the simplistic, to say the least, theoretical side of his 3-parameter atmosphere,

with the help of two physicists, Howard and Marcus. He developed a real physical system, that is, it is real, even if it is still simplistic and far removed from true weather phenomena.

Here is a mill. It consists of a wheel with buckets suspended evenly around it. Each bucket has a hole in the bottom, and the water will run out of the bucket. It runs more quickly when the water level is higher.

We open a valve at the top, and watch how the mill moves. It sometimes turns to the right, and sometimes to the left.

We seem to be totally incapable of predicting which way it will be turning in just a few seconds. The movement seems totally erratic, unpredictable, chaotic.

In fact, there is a relationship between the mill and the Lorenz attractor. Let's choose three numbers to describe the mill.

For instance, the angular velocity and the two coordinates of its center of gravity. The evolution of these numbers traces out a yellow curve in 3-space. Is it not incredible?

Our mill moves like a butterfly!

Let's change the initial position imperceptibly. Again, all the buckets are initially empty. The left wheel is 2 degrees off of vertical, while the right wheel is only 1.9996 degrees off of vertical.

We would need a good microscope to see the difference. Yes, there is clearly a sensitive dependence on initial conditions.

After a while, the two mills have completely different behaviors.

Let's see if Lorenz's statistical statement is true for the mill.

Our two mills take off from almost the same position. Let's measure their speed. For example, 25 times per second for 5000 seconds. So, 125,000 observations in total. For each observation, we note the rotational speed of the wheel.

One simple idea is to do what statisticians do. We use a bar chart to illustrate the distribution. We divide the possible speeds into 35 equal intervals. We count the number of times that the measured speed falls in each interval.

Here is the result. Some intervals appear to be visited more than others. Our two mills behave in very different ways. They have a sensitive dependence on initial conditions. But we see that the two sets of data, although different, are statistically identical.

The bar charts tend to become identical. It works! Lorenz seems to be right. When the statistics of a trajectory are insensitive to initial conditions, we say that the dynamics have a Sinai-Ruel-Boen, or SRB, measure.

The goal of a forecaster is then to determine these statistics. For example, let's observe the evolution of the temperature over time, the temperature being one of the coordinates in Lorenz's equation.

We draw another bar chart, as we did with the mill. For each temperature range, we note the proportion of observations that fall within this range. Even if we consider trajectories that are quite different from one another,

after a while, the charts tend to become identical. The Lorenz attractor has an SRB measure. It is as if the temperature was changing at random, but with very specific probabilities.

But there is a lot left to do mathematically and physically. We must find these probabilities and these distributions in order to be able to say something useful.

Let us hope that we have done justice to Lorenz, who did not merely say that the future depends heavily on the present.

Many people before him said that. He also, and more importantly, contributed the following idea, by refocusing on statistical issues science can still make predictions.