We reviewed results from 4b through 4h on the Iterated Functions sheet. Every example had a single attracting fixed point (and possibly an additional repelling one) until 4f, which had two repellers. But what did these repellers repel to? It wasn’t infinity, it was a cycle of values between 0 and -1. This is our first example of a limit cycle, a surprising result in algebraic iterations where a pattern of steps is pushed into a repetitive cycle.
Then we pushed the function a little further, from x^2 – 1 to x^2 – 2. Still with two repellers, this time the system of iterations dissolved into complete chaos, careening randomly within the interval -2 to 2, never settling into one spot. Changing the value of the seed by only one ten-thousandth produced a completely different pattern of iterations, demonstrating the sensitivity to initial conditions that are characteristic of chaos.
We will see plenty more examples of weird behavior in our iterations in the days to come. For tomorrow, be sure to read the article posted yesterday.