Fractals and Chaos Recap for 12/4

We explored cycles a bit more in class today, noting that if {p,q} are the two values of a 2-cycle, then by definition (p) = q and (q) = p. Put another way, ((p)) = p and ((q)) = q. What’s important to note, though, is that the reverse is not true. If ((n)) = n, then n could be a member of a two cycle (n,m,n,m,n,m,…or it could just be a fixed point: (n,n,n,n,n,n,…)

There were two important takeaways from this algebraic definition of a 2-cycle. The first is that the local slope (Instantaneous Rate of Change) to all points of a 2-cycle, or indeed any cycle, is always the same. As a result, all points of a cycle will pass an IRC of -1 simultaneously, meaning all points of a cycle bifurcate simultaneously. This served as our proof of Question 2 posed early in our exploration of this pattern.

The second takeaway is graphing the function y = ((x)) can be a way of finding new cycles. Fixed points on the graph of y((x)) that are not common with the graph of y = (x) will be the parameters of our 2-cycle. By extension, graphing y = f (((x))), or any number of nested iterations, will give us a tool of finding new cycles. More crucially, this also shows us why cycles spontaneously appear from chaos. Explore the graph here. For a = 3.84, the “wiggliness” of the graph of y = f (((x))) is enough for the fingers of the graph to touch the line y = x. But for a < 3.84, it isn’t. The moment that a becomes large enough for those fingers to touch the line y = x is the moment that a 3-cycle is born!

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