Chaos Day

The Logistic Map

So, what does CHAOS really look like?!

What do you think, how about this?
Or this?
Is this chaotic behavior?

Okay, well, in order to get an idea of how bad things can get in this business, we need to get a feel for how nice they can be. So we need to talk about things called fixed points. Simply put, these are just points where f(x) = x. (You may want to use Newton's method to find this/these roots...). Note that the orbit of a fixed point under a given map, f, consists of only one point, the fixed point itself.

For example, x = 0 and x = 1 are the only fixed points of the logistic map, f(x) = ax(1-x).
Download the code for a Maple program which uses Newton's method to find the roots of an arbitrary function.

Attracting and repelling fixed points

A fixed point x^* of a one-dimensional map (in this case, any real-valued function which is also a map on some closed interval, and not just the logistic function) is called an attracting fixed point if the orbit of any point(s) near x^* converges (as a sequence) to x^*. In other words, if z is close to x*, then the orbit of z under the given function f converges to x*.

In the same spirit we can define a repelling fixed point as a fixed point x# with the property that points close to x# (but not equal to it) move away from x# as f is iterated. In other words, the orbit of x# under f stays away from x#. (In particular, the orbit cannot converge to x#).

Here's an example: Let a = 0.8, z = x_o = 0.75. Then x* = 0 is an attracting fixed point of the logistic map. Click here to see the first few points of the orbit of z = 0.75 under the map.

Here's another example: Let a = 1.4, z = x_o = 0.75. Then x* = 0 is a repelling fixed point of the logistic map. Click here to see the first few points of the orbit of z = 0.75 under this map.

Download the Maple code for iterating the logistic map.

Using some basic tools from Calculus one can prove that

If, at a fixed point x, we have

1. |f'(x)| < 1, then the fixed point is an attractor, whereas,
2. If |f'(x)| > 1, then the fixed point is a repeller.

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