We reviewed results from 4b through 4f on the Iterated Functions sheet, making a brief detour to discuss the Golden Ratio which became surprisingly relevant for 4d. Every example had a single attracting fixed point 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, but for now please **finish question 4, parts h, i, and j**. We’ll be discussing these results on Monday when we return.

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