Lorenz Attractor
(XZ Plane Projection)

3-Dimensional View

Construct a simple system: take a box, a simple solid rectangular solid. Within this box, place a homogenous, preferably elemental, gaseous substance. Heat the box, sit back, and observe.

What happens to the gas? It is, of course, common knowledge that warm gases rise, while cooler gases sink; and initially, the portions of the gas closest to the walls of the box (e.g. closest to the external heat source) will become heated and rise. At certain temperatures, the gas will begin to form cylindrical rolls spaced like jellyrolls lying lengthwise in the box. On one side of the box, the gas rises, and on the other, it sinks; the rising gases converge on one side and carry warmer gases up with them; as the gas cools, it falls on the other side of the box.

With a regularly applied temperature, a smooth box interior, and a system completely closed with respect to the gas itself, it might be expected that the circular motion of the moving gas should be regular and predictable.

Nature, however, is neither regular nor predictable. It turns out that the motion of the gaseous cylinders is chaotic. The rolls do not simply roll around and around in one direction like a steam-roller; they roll for a while in one direction, and then stop and reverse directions. Then, seemingly at random, the gas reverses direction again; these fluctuations continue at unpredictable times, at unpredictable speeds.

The Lorenzian Waterwheel

Most casual armchair scientists have no access to uniformly smooth boxes and elemental gases, much less instruments to measure the rotational speed of a moving cylinder of gas.

A metaphor for the gaseous system is found in the Lorenzian waterwheel. This is a thought experiment. Imagine a waterwheel, with an arbitrary number of buckets, usually more than seven, spaced equally around its rim. The buckets are mounted on swivels, much like Ferris-wheel seats, so that the buckets will always open upwards. At the bottom of each bucket is a small hole. The entire waterwheel system is then mounted under a waterspout.

The scenario is set: now we commence the action.

Begin the flow of water from the waterspout. At low speeds, the water will trickle into the top bucket, and immediately trickle out through the hole in the bottom. Nothing happens. Real boring. Increase the flow just a bit, however, and the waterwheel will begin to revolve as the buckets fill up faster than they can empty. The heavier buckets containing more water let water out as they descend, and when the water is gone, the now-light buckets ascend on the other side, ultimately to be refilled. The system is in a steady state; the wheel will, like a waterwheel mounted on a stream and hooked to a grindstone, continue to spin at a fairly constant rate.

But even this simple system, sans boxes or heated gases, exhibits chaotic motion. Increase the flow of water, and strange things will happen. The waterwheel will revolve in one direction as before, and then suddenly jerk about and revolve in the other direction. The conditions of the buckets filling and emptying will no longer be so synchronous as to facilitate just simple rotaton; chaos has taken over.

The explanation for the irregular movement of the gas lies at the molecular level. While the box sides may seem smooth and thus the flow of the gas should always be regular, at molecular levels the sides of the box are quite irregular due to the motion of atoms and molecules. After all, in any solid not at absolute zero, total entropy is positive and there must be some irregularity in the molecular structure of the sides of the box.

Molecular interactions are tiny, however. How would such fluctations as small as a slightly misplaced molecule affect the flow of the gas in such a profound way as to cause seemingly random motion? The theory behind how small deviations can lead to large deviations lies at the heart of chaos theory. The explanation is simple and, in retrospect, obvious explanation commonly known as sensitive dependence on initial conditions.

Edward Lorenz and Long-Term Meteorology

In the early sixties, a certain meteorologist named Edward Lorenz experimented with computer simulations of weather on a relatively primitive "Royal McBee" computer. His program used twelve recursive equations to simulate rudimentary aspects of weather; he entered several variables into his program each time he ran it, and watched to see what types of weather patterns such initial conditions would generate. He could print out graphs of fluctuating temperatures or other conditions, and his program captured the fancy of his fellow meteorologists.

But mousie, thou art no thy lane
In proving that foresight may be in vain;
The best-laid plans of mice and men
Gang aft a'gley.
--Robert Burns, "To a Mouse"
One day, Lorenz tried to recreate an interesting weather pattern, one he had seen previously, by re-entering the values the computer had previously calculated and reported. However, when he ran the program again, his results were different from the initial run. Lorenz suspected a bug; blown diode? burned-out vacuum tube? power surge? cosmic rays? After checking the two plots, however, he realized his "error"; on his previous computer printout, the one he had used to enter the initial conditions into the computer for the second trial run, the figures were printed with three significant digits. In the program, all values were calculated to six significant digits. Lorenz had assumed that the difference, only one part in a thousand, would be inconsequential; however, due to the recursive nature of the equations, little errors would first cause tiny errors, which would then affect the resulting next calculation a bit more, which would affect the output of the next run even more. The final result of a long string of recursive calculations would lead to a weather pattern totally different from the expected values.

The term "sensitive dependence on initial conditions" was coined to describe the phenomenon that small changes in a recursive system can drastically change the results of running that system. A term Lorenz coined to describe sensitive dependence on initial conditions is the "butterfly effect." This is another thought experiment which is hardly testable: imagine that there exist two earths, so that an incorporeal observer could compare events on one earth to another. Now imagine that both earths are identical except for one fact; in one, a butterfly flaps its wings somewhere in South America, and in the other, this butterfly remains still. One might think that such a small discrepancy between the two earths would be inconsequential; after all, nobody was there, nobody could even notice the butterfly's wings flapping, and air currents would be affected only minorly by such a miniscule event.

After a period of time which is impossible to calculate, however, the weather patterns of the two earths would be totally different. Why? Because of the difference caused by the flap of one butterfly's wings! The miniscule event affected air currents around that butterfly in a very miniscule fashion, true; but those tiny air currents affected in turn slightly larger air currents, which affected still larger air currents, and the small difference in air flow between the two earths exponentially increases to become a large difference. The wind patterns on the two earths, which started out otherwise identical and had every reason to remain identical in a nice deterministic manner, would now be different in every way. Eventually, as time multiplies the differences between the two earths, completely different weather patterns would emerge, all because of the fact that on one of the earths a butterfly in South America decided to flap its wings, while on another of the earth, the same butterfly did not.

The differences might not cause any major catastrophic events immediately; the thought experiment does not suggest that murdering a butterfly could cause a hurricane in a few minutes or a tornado in a few hours. However, the air currents and wind patterns would be different.

It is thus completely impossible, even in theory, to perform long-term weather prediction in any accurate manner. Unless a computer could be constructed which could monitor each individual atom on earth, even the smallest undetected anomaly could affect the weather in profound ways. Fascinated by this idea, Edward Lorenz began drifting away from meteorology and began exploring the realms of mathematics, looking for more unpredictable, nonlinear systems.

Lorenz's Attractor

At one point, Edward Lorenz was looking for a way to model the action of the chaotic behavior of the gaseous system first mentioned above. Lorenz took a few "Navier-Stokes" equations, from the physics field of fluid dynamics. He simplified them and got as a result the following three-dimensional system:

dx/dt = delta * (y - x)

dy/dt = r * x - y - x * z

dz/dt = x * y - b * z
Here delta represents the "Prandtl number." This number, which one absolutely does not have to know the meaning of, is the ratio of the fluid viscosity of a substance to its thermal conductivity (named after Ludwig Prandtl, a German physicist). The value Lorenz used was 10.

The variable r represents the difference in temperature between the top and bottom of the gaseous system. The value usually used in sample Lorenz attractors such as the one displayed here is 28.

The variable b is the width to height ratio of the box which is being used to hold the gas in the gaseous system. Lorenz happened to choose 8/3, which is now the most common number used to draw the attractor.

The resultant x of the equation represents the rate of rotation of the cylinder, y represents the difference in temperature at opposite sides of the cylinder, and the variable z represents the deviation of the system from a linear, vertical graphed line representing temperature.

[Owl Mask] Plotting the three differential equations requires the usage of a computer. Plotted on a three-dimensional plane, a shape unlike any other forms. Instead of a simple geometric structure or even a complex curve, the structure now known as the Lorenz Attractor weaves in and out of itself. Projected on the X-Z plane, the attractor looks like a butterfly; on the Y-Z plane, it resembles an owl mask. The X-Y projection is useful mainly for glimpsing the three-dimensionality of the attractor; it looks something like two paper plates, on parallel but different planes, connected by a strand of string. As the Lorenz Attractor is plotted, a strand will be drawn from one point, and will start weaving the outline of the right butterfly wing. Then it swirls over to the left wing and draws its center. The attractor will continue weaving back and forth between the two wings, its motion seemingly random, its very action mirroring the chaos which drives the process.

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