Fractals & Chaos Recap for 12/3

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,…)

One important takeaway from this is that 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 (and the  Chain Rule from calculus) served as our proof of Question 2 posed early in our exploration of this pattern.

The implication of this is clear: after a = 3.0, the pattern of the orbit of the logistic function bifurcates to a two cycle. By a = 3.5 (more specifically at a = 3.449490…), it has bifurcated again to a four-cycle. Soon after (at a = 3.544090…), it bifurcates yet again to an eight-cycle, then to a 16-cycle (at a = 3.564407…), a 32-cycle (at a = 3.568759…), and so on. These bifurcations happen over increasingly shorter intervals for values of a. The fascinating thing is that the ratio between these intervals approaches a constant value, approximately 4.669202…, known as Feigenbaum’s Constant (this link goes to a Numberphile video about this idea that we watched a portion of in class today; the picture of orbit destinations that we have been drawing is known as the Feigenbaum Plot, which we will look at in detail tomorrow).

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