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Unit 13

The Concepts of Chaos

13.4 Sensitive Dependence

ROUNDING ERROR

Edward Lorenz is a noted mathematician and meteorologist. Throughout the mid-to-late 20th century he was a meteorological researcher at MIT. In the 1950s and 1960s, the study of meteorology was as much art as it was science. Weather forecasters could find certain patterns in weather systems that were somewhat tame and predictable, but there was always an element of surprise. It was thought that this was simply because the dynamics of the atmosphere were so complex, involving so many variables, that it was impossible to state with any precision at any one time what exactly was going on. Without knowing the initial conditions of the system, it was very hard to make exact predictions about what it would do next.

Lorenz hoped to gain some insight into the complexity of the weather by working with an extremely simplified version of a weather model and running it on the newly available computers. With a computer, he believed that he could have exquisite control over the initial conditions, allowing the modeling equations to function more-or-less free of measurement error. By looking at such an ideal and simplified system, he hoped to get a better idea of the fundamental phenomena that underlie the weather.

After considering a complicated, 12-equation model of how air moves, Lorenz chose to focus on a system employing just three equations, a simple model of convection rolls.

equation= σ(y-x)
equation = rx – y – xz
equation = xy – bz

Lorenz's model represented an extreme simplification of a weather system. Using simplified equations for convection currents, his model simulated various winds interacting. In the early days of scientific computing, this was a tedious process. He would input his equations and a set of initial conditions and then have the computer calculate what would happen as time moved forward in discrete steps. To make sense of his model's output, he would choose a specific variable, such as the direction of the west wind, and plot its behavior graphically. He watched as the wind shifted directions, a phenomenon represented by a wavy-line computer printout. This line represented a record of how that direction of the wind changed according to his mock-up, as calculated by the computer.

As the story goes, one day, he was forced to stop his calculations mid-simulation. When he returned a bit later, he decided to start the simulation again, using the values that had been generated and recorded at its stopping point, rather than starting the simulation over with the initial values. He entered the values from before as the initial conditions and was amazed by what happened. The simulation progressed as predicted for a while, but then quickly and inexplicably diverged from what he had seen in previous simulations.

Lorenz initially suspected that there had been a computer malfunction. In Newtonian determinism, there should be no difference between an interrupted and a non-interrupted test. Upon further investigation and reflection, Lorenz realized that there had been no malfunction; the discrepancy was due to a tiny rounding difference between the computer and the printer that displayed the data.

Lorenz's computer's memory was programmed to register six decimal places. For example, at the end of a round of simulation, the computer would output a number such as 0.506127. This number would then automatically be used as the initial condition for the next round of simulation. Lorenz's printout, on the other hand, displayed only three decimal places (a paper-saving feature), and it was this printout that he used to input the starting values when he re-started the interrupted experiment. Had the computer not been interrupted, it would have continued using the 6-digit number; Lorenz had assumed that inputting a 3-digit approximation would not change the results very much.

The difference between 0.506127 and 0.506 is a little more than one part in ten thousand. This is a miniscule deviation, the kind of discrepancy that scientists regularly ignore because they assume that small errors in input have only small effects on output. Lorenz found, however, that this tiny discrepancy had profound implications for the long-range behavior of his "simple" system. Lorenz had thought that perhaps computers would be the supreme data processors, capable of generating complete, accurate weather predictions. Nonetheless, he also knew that a computer's output is only as reliable as its input. Experimental scientists have long known that the initial conditions of a system can never be quantified with 100% accuracy. What Lorenz found in his computer simulations was that a small difference in initial conditions could result in large discrepancies between expected outcomes in certain systems. This concept, which came to be known as sensitive dependence, is the key trait of systems that exhibit chaos.

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BUTTERFLIES

To understand sensitive dependence a little better, Lorenz decided to look at the phase space of his system. He saw something much more complicated than the simple phase portraits that we observed in the previous section.

phase portrait

This phase portrait is three-dimensional, one dimension for each of the variables in Lorenz's equations. It represents how Lorenz's simplified weather system evolves through time. It is an abstract path consisting of points whose coordinates are determined by Lorenz's equations. If we imagine a particle sliding along this abstract path, that particle's behavior will be indicative of the behavior of the system in general.

What is remarkable about this object is that if you were to start two different particles off in almost but not quite exactly the same location and then allow them to flow along the curve, they would remain close to each other for a while but would at some point start to diverge in their paths very rapidly. This is just like the example of falling leaves from the introduction to this unit. Although the two leaves start out in almost, but not quite exactly, the same position, we all know that by the time they reach the ground, they can be very far apart indeed. In the present example, note also that the particles, even though they follow different paths, still stay somewhere close to this butterfly pattern. That is another hallmark of chaos: indeterminacy mixed with some notion of determinacy — that is bounded in space. Just as the leaf is sure to hit the ground eventually, chaotic behaviors are confined in their outcomes.

Chaotic unpredictability and sensitive dependence can arise in some nonlinear systems, but not all. They represent just a small part of the broader, mostly untamed, field of nonlinear dynamics. While the initial discoveries of chaotic behavior came from the realm of continuous dynamics, such as the motions of planets, chaos also arises in discrete time situations. Lorenz, for example, made his discovery by examining discrete-time solutions to his differential equations. These are situations in which a process is repeated for several steps, each step using the product of the step before as its initial condition. The mechanics of chaos can be better understood by looking at these iterative functions, and so it is to the subject of iteration that we will now turn our attention.

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Next: 13.5 Iteration


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