While Lorenz was studying the mathematical structure of the data of chaotic behavior, he used time-series analysis to ferret out hidden details and employed the technique to help produce an image bearing three dimensions of information. The image (an attractor) soon became inextricably tied to the field of chaos (or nonlinear dynamics).

Here is a Lorenz-type image showing the beauty in this strange attractor. Within this image, one can detect an underlying fractal pattern. The apparently elliptical path is that followed by a particle drawing the trajectory in 3D. When this figure was drawn, the particle was seen looping around one lobe for a while and then jumping over to the other lobe as it continued looping. The path never retraced itself. (Though it appears to, that is simply a limitation of the drawing resolution of a computer screen.) This back-and-forth motion continued until the entire plot was generated. The image looks flat, but the tracing particle moved in three dimensions, alternately projecting out of the screen and then back in as it orbited the attractor.

Not all chaotic attractors exhibit a fractal (or self-similar) nature as clearly as the one seen here. Those attractors that do can be "zoomed-in" to reveal hidden structure, just like Mandelbrot fractals.

There is an obvious similarity between the Lorenz attractor and a butterfly. But the real reason chaos is forever associated with the butterfly effect is a paper Edward Lorenz presented at a session of the AAAS in December 1972. The title of the paper: "Predictability- Does the Flap of a Butterfly's Wings in Brazil Set Off a Tornado in Texas?"

Lorenz admitted that while one butterfly's flapping wings could trigger off a tornado in the Lone Star state, another butterfly's flapping wings could prevent it. (What this means is that when you hear about a tornado in Texas, it's a pretty safe bet that there must be an odd number of butterflies in Brazil—at least for the moment!)

So what is an attractor, and why is it so pivotal for studies of chaos?

Speaking of pivotal, imagine a pendulum moving back and forth in a slow, repetitive manner. In other words, the pendulum is doing what pendulums are supposed to do. Picture the pendulum moving from its extreme leftmost position to its extreme rightmost position. Draw a half circle with the outside (convex) curved surface facing down, like a half grapefruit sitting on a countertop, juicy side up. The semicircle represents the (almost) 180° motion. Now, picture the pendulum moving back from its extreme rightmost position to its extreme leftmost position and draw the other half of the circle with the convex surface facing up. You have drawn both a complete circle and a periodic attractor for the pendulum. Let the pendulum continue moving with no external source of energy until it slowly unwinds to a stop at mid-position. As it was slowing down, you could have drawn a slowly contracting spiral depicting its decreasing motion with a point at its center showing the pendulum's final position. That point is a fixed attractor. Finally, imagine that the pendulum has sufficient energy being input to occasionally travel more than 270° on a swing. The motion would soon become chaotic. Plotting that motion generates a chaotic or strange attractor. While, in theory, the location of any point on a chaotic attractor is determinable, in practice it usually is not, due to computational difficulties and other things. This deterministic chaos is seen in the Lorenz diagram show above.

More formally, a plotted attractor depicts an equilibrium state of a dynamic system. The system's state converges to the attractor. That is where the system spends its time. The orbit of the attractor is plotted in phase space where the coordinates of the space are determined by parameters of the system and usually are not simple x, y, and z axes.

I wanted to explore the appearance of the butterfly effect in simulated weather forecasting. Two approaches were adopted involving different types of mathematical analyses of the Lorenz equations behaving chaotically. In one approach, the stored output was mapped to a simple "weather" chart so that comparisons could be made with and without flapping butterflies. In the other approach, the same data was evaluated using contour plotting that highlights the smallest changes between alternate solutions. Both methods were carried out using Mathematica to solve the Lorenz triplet numerically (with NDSolve). The program was set up using 24 digits of working precision with very high accuracy and precision goals for the output results, as well. Without these provisions, disparities would pop up much too quickly due to round-off errors, etc. But, even that problem is representative of the insidious nature of the butterfly effect.

The three Lorenz equations

The first-order, coupled, nonlinear differential equations that helped start the study of chaotic phenomena are seen here. (The definitions for x(t), y(t), and z(t) are given. These were obtained from Mathematica Navigator by H. Ruskeepää.) I did not use those variables in the defined way, however. Instead, each solution for each of the three equations at each time step (t) was compared to a list of weather condition parameters I set up to pigeonhole the variables into temperature, humidity, and wind conditions. So, x(t) values were arbitrarily assigned to temperature, y(t) values were assigned to humidity, and z(t) values to wind conditions. Each table (described below) consisted of 150,000 24-digit numbers: 50,000 triplets of x(t), y(t), and z(t). The "t" time variable was incremented in steps of 0.001. The initial conditions for the weather parameters were set up as shown at right. In the study of these solutions that follows, bear in mind that the values being examined are points located on attractors that are similar in appearance to the Lorenz-type image seen above.

An unusual weather map

Using Mathematica to solve the Lorenz equations with the initial conditions listed above, I created my first table of values. Then, two of the three initial conditions—x(t) and z(t)—were each changed by one part in a (U.S.) billion (0.000000001) and the equations were solved once more. A second table was created.

The results for the three equations as supplied by each table were tested by a program routine within Mathematica. Based on where the values lay, weather conditions were applied. For example, if the value of x(t) fell within a certain band of values, then the temperature might have been assigned as, say, "cool." Similarly, if y(t) met a particular criterion, its humidity value would be assigned as, perhaps, "wet." Finally, z(t) could get a wind rating of "calm," "breezy," or "windy."

The tables were compared and, as can be seen at right, at some point the tabulated "weather" began to vary between tables. As the calculations continued, deviations grew more obvious. The chart seen here is a composite of the two tables along with a location indicator. The three columns listed at the rightmost position of the chart represent the modified equation values calculated with the altered initial conditions. (This is almost as if a really, really big Brazilian butterfly had fluttered its wings causing changes in the amplitude of convective currents and the normal temperature deviations.) The three columns to their left represent the conditions in a world free of Brazilian butterflies. The extreme leftmost values listed under the column heading"LOC" (for location) are the values corresponding to the calculated equation values at a particular "t" value. For example, the three weather values given at "241" represent the calculations at a "t" value about one-quarter of the way between 0 and 1 (t=0.241). The values at, say, "45696" represent the calculated triplet of values at t = 45.696, and so on. These locations give you an idea of how far the table calculations proceeded up to the points shown.

Incidentally, I tried altering only the temperature difference between rising and falling air currents, y(t), at levels of 1 part in a quadrillion. While it required more calculations before disparities were observed, they soon made an appearance. And, while even one part in a (U.S.) quadrillion (0.000000000000001) doesn't quite approach the infinitesimal effect of a fluttering butterfly, the principle is clear.

Another—and far more sensitive—way of looking at the calculated tables is to employ a contour plotting routine (ListContourPlot) within Mathematica. Whereas I arbitrarily assigned "weather conditions" to varying x(t), y(t), and z(t) solutions using very broad bands of criteria (e.g., temperature values of, say, 0-30 not inclusive at the upper end are "very cold," 30-60 are "cool," 60-90 are "warm," and 90+ are "hot") contours can detect minute value differentials. Also, the contouring is reminiscent of weather-forecasting maps. To accomplish this method of presentation, the two tables generated above were subtracted from each other creating a third table. If both original tables were equal at each and every one of the 150,000 points, the new table would consist of zeros only. All 50 contour images seen below would then show flat, green fields. However, any deviations from zero that might exist would be assigned a color, in addition to being ringed with black contour lines.

Unlike the smooth variations with streamline contour motion seen on real weather maps, the Lorenz equations produce discrete and substantial differences between solutions of variations in "t," at least in the way I carried out the procedure. My animation shows discrete maps with each image representing the contour mapping of the differences between the two tables in unit increments of "t." That is, frame #1 corresponds to "t" varying from 0 to 1 inclusive. Frame #2 represents "t" varying from 1 to 2. And so on. There are 50 frames corresponding to "t" varying from 0 up to 50.

Also, no attempt was made to normalize the contours. Accordingly, the colored mapping in frame #7 does not necessarily compare to the color mapping in frame #8 except that the reds, oranges, etc., are always at the same relative locations within the data slice of 3000 values for that field. It is just the span of values that differs, more or less, in each frame. Therefore, each frame should be judged independently. (Normalizing tends to wash out the visual differences in most of the earlier frames, and they aren't as interesting or informative to view.)

Clearly, minute variations that occur in both computer and nature can accumulate to produce huge differences over time. These differences are the bugbears of reliable, long-term forecasts and are manifestations of sensitivity to initial conditions. Today, with the intention of depriving an oncoming hurricane of heat, groups of atmospheric and ocean scientists are attempting to modify small regions of the ocean's surface by means of polymers. Of course, any energy deprivation would be small. But if the technique is applied at the right spot and at the right time, it might succeed in starving a hurricane in mid-course. This could reduce the storm's intensity, thereby saving lives and property. For monarchs or meteorologists, chaos can work in mysterious ways!