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Unit 13

The Concepts of Chaos

13.5 Iteration


In everyday life, we rarely perceive any boundary between one moment and the next, but, instead, perceive time as flowing continuously. Some processes, however, can be broken into discrete steps. Folding and kneading dough is a good example of this; each fold is more or less an instantaneous event and the time between folds serves as a boundary.


Folding dough can, therefore, be modeled, approximately, in "discrete time." You can think of discrete time as something like a sequence of snapshots, whereas continuous time is more like a movie. Discrete time breaks a process up into the inputs and outputs at individual, separated moments in time (or space). A discrete dynamical system generally takes at each moment the output of a given step to be used as the input for the next step in the process. This process is called iteration; complete a step by performing an action that generates a new value, then use that new value as the starting point as you repeat the same action. Repeat this process for as long as you like.

Imagine a flake of pepper on the surface of the dough. As we knead and fold the dough the pepper flake gets moved about, its location changing from discrete moment to the next. A computational analogy would be as follows: We start with a number; "stretch" it; chop off a bit we don’t need; and end up with a new number. The stretching will be accomplished by multiplication, the chopping will be a modular arithmetic action. Our process will be to multiply the starting number by ten, then take the result, modulo 1. (Recall from the unit on primes and modular arithmetic that "modulo 1" is the mathematical way of saying "remove the integer part.") This eliminates any whole numbers that might be in the result, leaving only a decimal number to begin the next iteration.

Let’s start our process with a decimal input, 0.506127.

First we stretch it by multiplying by ten:

10 × 0.506127 = 5.061270

Next we take the result mod 1:

5.061270 mod 1 ≡ 0.061270

We now use this result as the starting point for the next iteration:

0.061270 × 10 = 0.612700

0.612700 mod 1 ≡ 0.612700 (no change, because there was no whole number component of the number)

So far so good, but what does this have to do with chaos? If we take two numbers that are almost but not quite exactly the same, say 0.12345 and 0.12349, and perform this iterative stretching and chopping process, we will see the essence of chaos unfold before our eyes. The following table records the evolution of the iterative process, and the image that follows demonstrates the divergence of the two values on a series of number lines.


number lines

Notice that the two numbers start out virtually indistinguishable, with the difference between them being only 4 parts in 100,000, hardly something to note. As the iterative process begins to unfold, the numbers stay relatively close to one another. After the first iteration, they differ by only 4 parts in 10,000. They continue to remain relatively close to one another all the way up to the end of the 3rd iteration. After four rounds of stretching and chopping, the numbers no longer resemble one another at all; their initial difference has been amplified by a factor of 10,000 and one is now nearly twice the value of the other. This is the essence of sensitive dependence.

Notice that in this system, there was a particular point, namely the 4th iteration, at which time the divergence of the values escalated quickly. We can call this breakpoint the threshold of chaos. In the study of nonlinear dynamics, other, more complicated systems, can have similar thresholds of chaos. These thresholds are determined by the system and the exact values chosen as initial conditions. An important question to explore is "when does chaos set in?" Stated in other terms, the question is "how do I know when a system is predictable and when it is not?" To see how one might answer these questions, we are going to look at a famous model that involves the rise and fall of the populations of various wild animal species.

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Next: 13.6 The Logistic Map

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